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    <title>DEV Community: Aman Kr Pandey</title>
    <description>The latest articles on DEV Community by Aman Kr Pandey (@_a_m_a_n_pandey).</description>
    <link>https://dev.to/_a_m_a_n_pandey</link>
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      <title>DEV Community: Aman Kr Pandey</title>
      <link>https://dev.to/_a_m_a_n_pandey</link>
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    <item>
      <title>429: TOO MANY REQUESTS</title>
      <dc:creator>Aman Kr Pandey</dc:creator>
      <pubDate>Thu, 05 Mar 2026 18:28:28 +0000</pubDate>
      <link>https://dev.to/_a_m_a_n_pandey/429-too-many-requests-53nh</link>
      <guid>https://dev.to/_a_m_a_n_pandey/429-too-many-requests-53nh</guid>
      <description>&lt;p&gt;As systems continue to scale to serve millions of concurrent users, the need for controlled and predictable API access has never been more critical. Rate limiting is a traffic management technique that restricts the number of requests a client can make to a server within a defined time period. It serves as a foundational safeguard for system reliability, fairness, and security in production environments.&lt;/p&gt;

&lt;h2&gt;
  
  
  The Case for Rate Limiting
&lt;/h2&gt;

&lt;p&gt;An API without rate limiting is vulnerable. The consequences of unbounded request traffic can range from low performing micro-services to complete service outages. The primary motivations for implementing rate limiting include:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Protection against denial-of-service (DoS) attacks&lt;/strong&gt; : Malicious actors can flood endpoints with requests, exhausting server resources and crashing the system.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Prevention of resource monopolization&lt;/strong&gt; : A single client/user can degrade the experience for all other users&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Cost management&lt;/strong&gt; : Uncontrolled traffic directly affects the cost of the infrastructure used to deploy the application.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Enforcement of API usage plans&lt;/strong&gt; : Rate limiting enables business models built around different service plans (e.g., free vs. premium plans) to increase frequency of requests in a given time period.&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Rate Limiting Algorithms
&lt;/h2&gt;

&lt;p&gt;Several well established algorithms exist for implementing rate limiting, each with distinct trade-offs in terms of accuracy, memory usage, and burst tolerance.&lt;/p&gt;

&lt;h3&gt;
  
  
  Token Bucket
&lt;/h3&gt;

&lt;p&gt;Now, imagine a bucket of fixed size in which water is flowing at constant rate and for each request you take out one glass of water out of it. If you have sufficient water in the bucket to fill the glass - you accept the request, else rate limit it. &lt;/p&gt;

&lt;p&gt;Now, in the bucket replace water with tokens, and each request consumes one token. If the bucket is empty, the request is rate limited and if bucket is full, tokens are dropped/overflown.&lt;/p&gt;

&lt;p&gt;To implement this algorithm we require two parameters:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;capacity&lt;/strong&gt;: size of bucket or the maximum number of tokens the bucket should hold at a time.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;refillRate&lt;/strong&gt;: the rate at which tokens are refilled in the bucket.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;It is usually better to have different bucket for different API endpoints. Key users of this algorithm include Amazon AWS (API Gateway), Stripe, OpenAI, Cloudflare, and Netflix (Zuul/Envoy).&lt;/p&gt;

&lt;h2&gt;
  
  
  Leaky Bucket
&lt;/h2&gt;

&lt;p&gt;Now, we have the same bucket again, but instead of taking out a glass of water every time we make a hole at bottom, from where the water leaks out at constant rate (i.e. the request to be processed). Question should be if request is leaking out, what is going into the bucket, tokens or request? In this case request. In this algorithm, whenever a new request arrives we add a glass of water to bucket, and then the water leaks out of the hole at constant rate. What if the bucket is full? The water overflows and the request is dropped.&lt;/p&gt;

&lt;p&gt;To implement this algorithm we require two parameters:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;capacity&lt;/strong&gt;: size of bucket or the maximum number of request the bucket should hold at a time.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;outFlowRate&lt;/strong&gt;: the rate at which requests are leaking/processed out of the bucket.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Shopify, Uber and NGINX are known to use leaking bucket algorithm in their APIs.&lt;/p&gt;

&lt;h2&gt;
  
  
  Fixed Window Counter
&lt;/h2&gt;

&lt;p&gt;The Fixed Window Counter algorithm divides time into fixed, different intervals called windows and tracks how many requests arrive within each one. Every window has a counter starting at zero. Each incoming request increments it, and if the counter exceeds the configured limit, the request is rejected until the next window begins and the counter resets.&lt;/p&gt;

&lt;p&gt;To implement this algorithm we require two parameters:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;counter&lt;/strong&gt;: configured limit of count of request&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;timeWindow&lt;/strong&gt;: the interval of each window&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;This algorithm is resource efficient and easy to understand. The only issue with this algorithm is that a bust of traffic at the edges of window, it allows more requests than allowed quota.&lt;/p&gt;

&lt;h2&gt;
  
  
  Sliding Window
&lt;/h2&gt;

&lt;p&gt;As discussed, fixed window algorithms has a major issue, i.e. more no of requests at the edges of window. We use sliding window algorithm to overcome this hurdle.&lt;/p&gt;

&lt;p&gt;There are actually two distinct algorithms under the "sliding window" umbrella:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;The Sliding Window Log stores a timestamp for every request and checks how many fall within the rolling window, it is accurate but memory in-efficient at scale.
​&lt;/li&gt;
&lt;li&gt;The Sliding Window Counter, covered in depth below, is the production-grade choice for most APIs.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The Sliding Window Counter is a hybrid approach, it combines the memory efficiency of fixed windows with the accuracy of true sliding windows. Instead of storing individual timestamps, it maintains just two counters: one for the current window and one for the previous window.&lt;/p&gt;

&lt;p&gt;When a request arrives, the algorithm estimates how many requests have been made in the "rolling" window using a weighted average:&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;estimatedCount=(prevCount×(1−windowProgress))+currCount
\text{estimatedCount} = \left(\text{prevCount} \times (1 - \text{windowProgress})\right) + \text{currCount}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;estimatedCount&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;(&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;prevCount&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;×&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;windowProgress&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord text"&gt;&lt;span class="mord"&gt;currCount&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;Where &lt;code&gt;windowProgress&lt;/code&gt; is how far you are into the current time window (a value between 0 and 1).&lt;/p&gt;

&lt;p&gt;While being a resource efficient and distributed system ready algorithm (with redis integration to store timestamps and counter), it has certain drawbacks also. It assumes uniform request distribution in the previous window, which may not always hold true. In edge cases, a few extra requests may slip through due to the estimation&lt;/p&gt;

&lt;h2&gt;
  
  
  Comparison: All Major Algorithms
&lt;/h2&gt;

&lt;div class="table-wrapper-paragraph"&gt;&lt;table&gt;
&lt;thead&gt;
&lt;tr&gt;
&lt;th&gt;Algorithm&lt;/th&gt;
&lt;th&gt;Memory&lt;/th&gt;
&lt;th&gt;Burst Handling&lt;/th&gt;
&lt;th&gt;Accuracy&lt;/th&gt;
&lt;th&gt;Complexity&lt;/th&gt;
&lt;/tr&gt;
&lt;/thead&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;Fixed Window&lt;/td&gt;
&lt;td&gt;O(1)&lt;/td&gt;
&lt;td&gt;Poor (boundary burst)&lt;/td&gt;
&lt;td&gt;Low&lt;/td&gt;
&lt;td&gt;Low&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Sliding Window Log&lt;/td&gt;
&lt;td&gt;O(n)&lt;/td&gt;
&lt;td&gt;Excellent&lt;/td&gt;
&lt;td&gt;Exact&lt;/td&gt;
&lt;td&gt;Medium&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Sliding Window Counter&lt;/td&gt;
&lt;td&gt;O(1)&lt;/td&gt;
&lt;td&gt;Good&lt;/td&gt;
&lt;td&gt;~Approximate&lt;/td&gt;
&lt;td&gt;Medium&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Token Bucket&lt;/td&gt;
&lt;td&gt;O(1)&lt;/td&gt;
&lt;td&gt;Good (controlled)&lt;/td&gt;
&lt;td&gt;Good&lt;/td&gt;
&lt;td&gt;Medium&lt;/td&gt;
&lt;/tr&gt;
&lt;tr&gt;
&lt;td&gt;Leaky Bucket&lt;/td&gt;
&lt;td&gt;O(1)&lt;/td&gt;
&lt;td&gt;Excellent (smoothed)&lt;/td&gt;
&lt;td&gt;Good&lt;/td&gt;
&lt;td&gt;Medium&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;&lt;/div&gt;

&lt;h2&gt;
  
  
  Conclusion
&lt;/h2&gt;

&lt;p&gt;There is no "best" rate limiting algorithm, we should always select the right algorithm based on requirement. For most distributed, high-traffic APIs, the Sliding Window Counter is the best suited: O(1) memory, redis-friendly, and good performance to boundary bursts. When accuracy is non-negotiable, upgrade to Sliding Window Log. When bursty-but-bounded traffic is the normal in your APIs, Token Bucket performs best. Understanding these trade-offs is what separates a system that survives traffic spikes from one that crashes under it. Check out the implementation of these algorithms &lt;a href="https://github.com/AmanPandey0320/gatekeeper/tree/master/src/main/java/com/kabutar/gatekeeper/ratelimiter/algorithm" rel="noopener noreferrer"&gt;here&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Thanks for reading, stay tuned!&lt;/p&gt;

</description>
      <category>systemdesign</category>
      <category>java</category>
      <category>solidprinciples</category>
      <category>backend</category>
    </item>
    <item>
      <title>A solid explanation to SOLID Principle</title>
      <dc:creator>Aman Kr Pandey</dc:creator>
      <pubDate>Wed, 31 Dec 2025 12:03:20 +0000</pubDate>
      <link>https://dev.to/_a_m_a_n_pandey/a-solid-explanation-to-solid-principle-42j8</link>
      <guid>https://dev.to/_a_m_a_n_pandey/a-solid-explanation-to-solid-principle-42j8</guid>
      <description>&lt;p&gt;A piece of software rarely fails because the "the algorithm was wrong". It mostly fails slowly because features become hard to add, bugs reappear and even a small change feels risky. SOLID principles give a five very useful practical ways to write code so that it remains understandable and and can adapt to new changes for a significantly longer period of time.&lt;/p&gt;

&lt;h2&gt;
  
  
  What is SOLID?
&lt;/h2&gt;

&lt;p&gt;It's definitely not one among the three states of matter. SOLID here stands as acronym of 5 design principle that help us write highly maintainable code. The principles are as follows:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;S&lt;/strong&gt;ingle responsibility Principle&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;O&lt;/strong&gt;pen-Close Principle&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;L&lt;/strong&gt;iskov's Substitution Principle&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;I&lt;/strong&gt;nterface Segregation Principle&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;D&lt;/strong&gt;ependency Injection Principle
Together, they encourage loose coupling, clear responsibilities, and stable abstractions, which are essential for long-lived code.&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Single Responsibility Principle
&lt;/h2&gt;

&lt;p&gt;Single Responsibility Principle (SRP) states that a class should have one and only one reason to change. Let's say for example we are implementing a class to manage library. An example of bad code that does not follow SRP would be:&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;This class has four reasons to change:&lt;br&gt;
1.Change in validation logic for user data&lt;br&gt;
2.Change in email templates or email api/client&lt;br&gt;
3.Change in log format&lt;br&gt;
4.Change in DB schema will change the query inside &lt;em&gt;saveToDatabase&lt;/em&gt; method&lt;/p&gt;

&lt;p&gt;Apart from these it UserService class will be hard to reuse as no one will want to trigger email, and save to database operation when validating user data. This kind of classes are also a testing nightmare as you will have to mock logger, email and db clients.  &lt;/p&gt;

&lt;p&gt;Better idea would be to have separate class for each one of them.&lt;br&gt;
Below is the example of good code that follows SRP, with each responsibility handled with separate classes.&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;In the above code each class has only one reason to change and each class is independent of each other, so anyone can use login to validate user or sending email separately, with this email and logging logic can be used in different parts of the code.&lt;/p&gt;

&lt;p&gt;So our first mantra to achieve good design is &lt;strong&gt;Ek class Ek Kaam&lt;/strong&gt;.&lt;/p&gt;

&lt;h2&gt;
  
  
  Open/Close Principle
&lt;/h2&gt;

&lt;p&gt;Open/Close Principle(OCP) states that classes should be open for extension but closed for modification. This principle encourages us to use abstraction instead of concrete class so when we need new behaviors we can add new class instead of modifying the existing behavior of the tested code. This reduces risk while adding new features as you are not modifying the already tested code.&lt;/p&gt;

&lt;p&gt;Let's understand OCP via shapes to find the area. The below code calculates the area of the shapes like rectangle and circle. &lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;Let's say if you need to add a new shape, for example a square, if will change the implementation of &lt;em&gt;calculateArea&lt;/em&gt; in &lt;code&gt;AreaCalculator&lt;/code&gt; class and this is a violation of OCP.&lt;/p&gt;

&lt;p&gt;Now a good code that does not violate OCP for the same feature would be as follows.&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;Here we have created an abstract class &lt;code&gt;Shape&lt;/code&gt; that has a abstract method &lt;em&gt;area&lt;/em&gt;. This method is overridden in different shape classes like &lt;code&gt;Rectangle&lt;/code&gt;. The &lt;code&gt;AreaCalculator&lt;/code&gt; class only knows that is is receiving a Shape and it has to call area method on it without worrying about the implementation of area of each shape. In this design it will be easier to introduce new shape like Square with changing implementation of any other class.&lt;/p&gt;

&lt;p&gt;So, in order to keep this principle in mind remember the mantra - &lt;strong&gt;Extend karo, edit nahi&lt;/strong&gt;.&lt;/p&gt;

&lt;h2&gt;
  
  
  Liskov's Substitution Principle
&lt;/h2&gt;

&lt;p&gt;Liskov's substitution principle (LSP) suggestes that a child class can substitute a parent class without changing the correctness or expected behavior of the code. The core idea is, if &lt;code&gt;S&lt;/code&gt; is a child class of &lt;code&gt;T&lt;/code&gt;, then any piece of code that works with &lt;code&gt;T&lt;/code&gt; as object should always work with &lt;code&gt;S&lt;/code&gt; as object with out breaking the code or changing the behavior of the code.&lt;/p&gt;

&lt;p&gt;Let's understand this with this simple example. Suppose you are making a bird game with different birds that can fly or walk and the following code controls these behaviors&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;&lt;code&gt;Bird&lt;/code&gt; contract promises &lt;code&gt;fly()&lt;/code&gt; works for any bird, but &lt;code&gt;Ostrich&lt;/code&gt; breaks this promise by throwing an exception. Clients cannot safely substitute Ostrich for Bird and &lt;code&gt;Ostrich&lt;/code&gt;'s &lt;code&gt;fly()&lt;/code&gt; changes the behavior of the code.&lt;/p&gt;

&lt;p&gt;Now, we know that all birds can walk but all birds can not fly. So why not have multiple abstraction for each behavior and then a &lt;code&gt;Bird&lt;/code&gt; class implementing only the common behavior, like something explained below.&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;


 

&lt;p&gt;In this code &lt;code&gt;makeWalk()&lt;/code&gt; method expects a super-type &lt;code&gt;Bird&lt;/code&gt; but sub-classes like &lt;code&gt;Ostrich&lt;/code&gt; and &lt;code&gt;Sparrow&lt;/code&gt; can substitute it without changing the behavior &lt;code&gt;walk()&lt;/code&gt; in respective classes.&lt;/p&gt;

&lt;p&gt;In order to remember this easily keep in mind &lt;strong&gt;Agar bache ka behavior parents ke jaise hai to wo party me ja sakta hai&lt;/strong&gt;.&lt;/p&gt;
&lt;h2&gt;
  
  
  Interface Segregation Principle
&lt;/h2&gt;

&lt;p&gt;The Interface Segregation Principle (ISP), part of the SOLID design principles, states that clients should not be forced to implement interfaces/method they do not use. Instead of large, general-purpose interfaces, we should create smaller, specific ones according to client. This approach reduces tight coupling and makes systems easier to maintain.&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;In the above code, a single &lt;code&gt;IWorker&lt;/code&gt; interface requires all clients to provide &lt;code&gt;eat()&lt;/code&gt;, even robots that only work. This violates ISP, creating dummy methods and tight coupling.&lt;/p&gt;

&lt;p&gt;A goof practice would be to have separate interfaces &lt;code&gt;Workable&lt;/code&gt; and &lt;code&gt;Eatable&lt;/code&gt;, and client can implement any combination of the interface as per their requirement. So we refactor the above code to follow ISP.&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;




&lt;p&gt;This reduces coupling and supports extensibility, like adding &lt;code&gt;Rechargeable&lt;/code&gt; for robots.&lt;/p&gt;

&lt;p&gt;To easily remember this keep in min &lt;strong&gt;Divide and Rule&lt;/strong&gt;: Divide the interface and rule the SOLID principle!&lt;/p&gt;

&lt;h2&gt;
  
  
  Dependency Injection Principle
&lt;/h2&gt;

&lt;p&gt;This is one of my favourite and most widely used principle, the Dependency Injection Principle (DIP). DIP states that high-level components should not depend on low-level components, both should depend on abstractions. Let's understand with an example.&lt;br&gt;
Let's you are writing a code for making Tea. You can have different types of Tea: Masala Tea, Ginger Tea, Gold tea etc. A simple code to do this would be as follows.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;MasalaTeaMaker&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="nf"&gt;makeTea&lt;/span&gt;&lt;span class="o"&gt;()&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="s"&gt;"Masala Tea ready!"&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;TeaShop&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;private&lt;/span&gt; &lt;span class="nc"&gt;MasalaTeaMaker&lt;/span&gt; &lt;span class="n"&gt;maker&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;MasalaTeaMaker&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;  &lt;span class="c1"&gt;// Hardcoded dependency!&lt;/span&gt;

    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="nf"&gt;orderTea&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="n"&gt;type&lt;/span&gt;&lt;span class="o"&gt;)&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"masala"&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;equals&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;type&lt;/span&gt;&lt;span class="o"&gt;))&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
            &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;maker&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;makeTea&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
        &lt;span class="o"&gt;}&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="s"&gt;"Only Masala available"&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;  &lt;span class="c1"&gt;// Rigid!&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;The problem here is tight coupling of &lt;code&gt;MasalaTeaMaker&lt;/code&gt; with &lt;code&gt;TeaShop&lt;/code&gt;, and adding new type of TeaMaker would be risky as it would need us to update &lt;code&gt;TeaShop&lt;/code&gt; class, for example if we add &lt;code&gt;GingerTeaMaker&lt;/code&gt; also in the code&lt;br&gt;
&lt;/p&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;MasalaTeaMaker&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="nf"&gt;makeTea&lt;/span&gt;&lt;span class="o"&gt;()&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="s"&gt;"Masala Tea ready!"&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;GingerTeaMaker&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="nf"&gt;makeTea&lt;/span&gt;&lt;span class="o"&gt;()&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="s"&gt;"Ginger Tea ready!"&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;TeaShop&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;private&lt;/span&gt; &lt;span class="nc"&gt;MasalaTeaMaker&lt;/span&gt; &lt;span class="n"&gt;masalaTeaMaker&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;MasalaTeaMaker&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
    &lt;span class="kd"&gt;private&lt;/span&gt; &lt;span class="nc"&gt;GingerTeaMaker&lt;/span&gt; &lt;span class="n"&gt;gingerTeaMaker&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;GingerTeaMaker&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;

    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="nf"&gt;orderTea&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="nc"&gt;String&lt;/span&gt; &lt;span class="n"&gt;type&lt;/span&gt;&lt;span class="o"&gt;)&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"masala"&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;equals&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;type&lt;/span&gt;&lt;span class="o"&gt;))&lt;/span&gt; &lt;span class="o"&gt;{&lt;/span&gt;
            &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;masalaTeaMaker&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;makeTea&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
        &lt;span class="o"&gt;}&lt;/span&gt;&lt;span class="k"&gt;else&lt;/span&gt; &lt;span class="k"&gt;if&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="s"&gt;"ginger"&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;equals&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;type&lt;/span&gt;&lt;span class="o"&gt;)){&lt;/span&gt;
          &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;gingerTeaMaker&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;makeTea&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
        &lt;span class="o"&gt;}&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="s"&gt;"Un-supported tea type"&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt; 
    &lt;span class="o"&gt;}&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;Now, &lt;code&gt;TeaShop&lt;/code&gt; is high level which should depend on TeaMaker via abstraction and not through concrete class. So we will no refactor this code and create an interface &lt;code&gt;TeaMaker&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;

&lt;/p&gt;
&lt;div class="ltag_gist-liquid-tag"&gt;
  
&lt;/div&gt;





&lt;p&gt;&lt;code&gt;TeaMaker&lt;/code&gt; interface lets &lt;code&gt;GingerTeaMaker&lt;/code&gt;, &lt;code&gt;GoldTeaMaker&lt;/code&gt; implement it. &lt;code&gt;TeaShop&lt;/code&gt; depends on abstraction via injection. In the above code injection is done through a constructor, but injections can also be done via setter methods if needed.&lt;/p&gt;

&lt;p&gt;As a memory map you can remember &lt;strong&gt;Abstraction pe depend karo class pe nahi&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;With this we have completed SOLID Principles, for any queries feel free to use comments.&lt;/p&gt;

&lt;p&gt;Thanks for reading, stay tuned!&lt;/p&gt;

</description>
      <category>solidprinciples</category>
      <category>designpatterns</category>
      <category>lld</category>
      <category>design</category>
    </item>
    <item>
      <title>Lazy Heap: Lazy Deletion</title>
      <dc:creator>Aman Kr Pandey</dc:creator>
      <pubDate>Mon, 29 Dec 2025 15:03:35 +0000</pubDate>
      <link>https://dev.to/_a_m_a_n_pandey/lazy-heap-lazy-deletion-47hi</link>
      <guid>https://dev.to/_a_m_a_n_pandey/lazy-heap-lazy-deletion-47hi</guid>
      <description>&lt;p&gt;A lazy heap is a heap-based priority queue that delays some operation on heaps like deletions, merges, or structural cleanup, to future operations in order to simplify implementation or avoid time limit exceeded. This “do it later” strategy comes at the cost of slightly worse worst-case bounds or extra memory for simpler code and often faster execution in practice.&lt;/p&gt;

&lt;h2&gt;
  
  
  What is Lazy heap?
&lt;/h2&gt;

&lt;p&gt;A lazy heap is a heap variant where certain operations are postponed until it is absolutely necessary. Instead of eagerly fixing the heap after every update, a lazy heap records pending changes and executes them up only when they affect operations like &lt;code&gt;top&lt;/code&gt; or &lt;code&gt;pop&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;Common types of laziness includes:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Lazy deletion&lt;/strong&gt;: Mark elements as deleted and physically remove them only when they reach the top of the heap.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Lazy merging&lt;/strong&gt;: Splice two heap root lists together in O(1) and consolidate later, as in lazy binomial or Fibonacci heaps.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Lazy consolidation&lt;/strong&gt;: Avoid tree-linking in multi-tree heaps until an extract-min forces cleanup&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Here we will cover most frequently used lazy technique, i.e. Lazy deletion with the help of java's &lt;code&gt;PriorityQueue&lt;/code&gt;.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="kd"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;LazyHeap&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;{&lt;/span&gt;
    &lt;span class="kd"&gt;private&lt;/span&gt; &lt;span class="kt"&gt;int&lt;/span&gt; &lt;span class="n"&gt;size&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="kd"&gt;private&lt;/span&gt; &lt;span class="nc"&gt;PriorityQueue&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="kd"&gt;private&lt;/span&gt; &lt;span class="nc"&gt;Map&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="nc"&gt;Integer&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;

    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nf"&gt;LazyHeap&lt;/span&gt;&lt;span class="o"&gt;(){&lt;/span&gt;
        &lt;span class="n"&gt;heap&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;PriorityQueue&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;();&lt;/span&gt;
        &lt;span class="n"&gt;removed&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;HashMap&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&amp;gt;();&lt;/span&gt;
        &lt;span class="n"&gt;size&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;

    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="nf"&gt;LazyHeap&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="nc"&gt;Comparator&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="n"&gt;c&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;
        &lt;span class="n"&gt;heap&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;PriorityQueue&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&amp;gt;(&lt;/span&gt;&lt;span class="n"&gt;c&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;
        &lt;span class="n"&gt;removed&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="k"&gt;new&lt;/span&gt; &lt;span class="nc"&gt;HashMap&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&amp;gt;();&lt;/span&gt;
        &lt;span class="n"&gt;size&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;

    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="kt"&gt;int&lt;/span&gt; &lt;span class="nf"&gt;size&lt;/span&gt;&lt;span class="o"&gt;(){&lt;/span&gt;&lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;size&lt;/span&gt;&lt;span class="o"&gt;;}&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="nf"&gt;poll&lt;/span&gt;&lt;span class="o"&gt;(){&lt;/span&gt;
        &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
        &lt;span class="n"&gt;remove&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="nf"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;(){&lt;/span&gt;
        &lt;span class="k"&gt;while&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;getOrDefault&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;(),&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;)&amp;gt;&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;
            &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;poll&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
            &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;put&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt;&lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;get&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="o"&gt;)-&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;
        &lt;span class="o"&gt;}&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="nf"&gt;add&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;
        &lt;span class="n"&gt;size&lt;/span&gt;&lt;span class="o"&gt;++;&lt;/span&gt;
        &lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;add&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
    &lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="nf"&gt;remove&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;
        &lt;span class="n"&gt;size&lt;/span&gt;&lt;span class="o"&gt;--;&lt;/span&gt;
        &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;put&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt;&lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;getOrDefault&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;)+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The class maintains three key fields:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;code&gt;PriorityQueue&amp;lt;T&amp;gt;&lt;/code&gt; heap: Standard min-heap storing all elements (live + marked-deleted).&lt;/li&gt;
&lt;li&gt;
&lt;code&gt;Map&amp;lt;T, Integer&amp;gt;&lt;/code&gt; removed: Tracks removal counts for each element using &lt;code&gt;getOrDefault(num, 0) + 1&lt;/code&gt;.&lt;/li&gt;
&lt;li&gt;
&lt;code&gt;int size&lt;/code&gt;: Logical count of live elements (increases on &lt;code&gt;add&lt;/code&gt;, decreases on &lt;code&gt;remove&lt;/code&gt;).&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Key Operations
&lt;/h2&gt;

&lt;p&gt;&lt;code&gt;add(T obj)&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="nf"&gt;add&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;
    &lt;span class="n"&gt;size&lt;/span&gt;&lt;span class="o"&gt;++;&lt;/span&gt;  &lt;span class="c1"&gt;// Logical size increases&lt;/span&gt;
    &lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;add&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;  &lt;span class="c1"&gt;// Push to heap&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Simply adds to both logical size and physical heap in &lt;em&gt;O(logn)&lt;/em&gt; time. No validation is done here, just simple add to heap.&lt;/p&gt;

&lt;p&gt;&lt;code&gt;remove(T obj)&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="nf"&gt;remove&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;
    &lt;span class="n"&gt;size&lt;/span&gt;&lt;span class="o"&gt;--;&lt;/span&gt;  &lt;span class="c1"&gt;// Logical size decreases immediately&lt;/span&gt;
    &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;put&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;getOrDefault&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;)+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;  &lt;span class="c1"&gt;// Mark as deleted&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The &lt;code&gt;remove()&lt;/code&gt; method updates counters in &lt;em&gt;O(1)&lt;/em&gt; expected time (HashMap) without touching the heap structure. Multiple calls to remove the same element are handled by the counter, where as a real deletion in heap structure would have costed us &lt;em&gt;O(logn)&lt;/em&gt; time.&lt;/p&gt;

&lt;p&gt;&lt;code&gt;peek()&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="nf"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;(){&lt;/span&gt;
    &lt;span class="k"&gt;while&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;getOrDefault&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;(),&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;)&amp;gt;&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="o"&gt;){&lt;/span&gt;  &lt;span class="c1"&gt;// While top is marked deleted&lt;/span&gt;
        &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;poll&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;  &lt;span class="c1"&gt;// Physically remove it&lt;/span&gt;
        &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;put&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="o"&gt;,&lt;/span&gt; &lt;span class="n"&gt;removed&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;get&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="o"&gt;)-&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;  &lt;span class="c1"&gt;// Decrease counter&lt;/span&gt;
    &lt;span class="o"&gt;}&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;heap&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="na"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;  &lt;span class="c1"&gt;// Now top is real top of the heap&lt;/span&gt;
&lt;span class="o"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The cleanup in heap structure happens here, where all delete elements are removed from heap. Repeatedly pops already marked as removed top elements until finding a real top element that is not marked as removed. The overall time low if deletions don't cluster at the top.&lt;/p&gt;

&lt;p&gt;&lt;code&gt;poll()&lt;/code&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight java"&gt;&lt;code&gt;&lt;span class="kd"&gt;public&lt;/span&gt; &lt;span class="no"&gt;T&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;peek&lt;/span&gt;&lt;span class="o"&gt;();&lt;/span&gt;  &lt;span class="c1"&gt;// Triggers cleanup, gets live minimum&lt;/span&gt;
&lt;span class="n"&gt;remove&lt;/span&gt;&lt;span class="o"&gt;(&lt;/span&gt;&lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;);&lt;/span&gt;  &lt;span class="c1"&gt;// Marks it as deleted (logical removal)&lt;/span&gt;
&lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;obj&lt;/span&gt;&lt;span class="o"&gt;;&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This method combines &lt;code&gt;peek()&lt;/code&gt; cleanup with logical removal of the top element.&lt;/p&gt;

&lt;h2&gt;
  
  
  Sample problems to practice
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;&lt;a href="https://leetcode.com/problems/sliding-window-median" rel="noopener noreferrer"&gt;Leetcode 480. Sliding Window Median&lt;/a&gt;&lt;/li&gt;
&lt;li&gt;&lt;a href="https://leetcode.com/problems/find-median-from-data-stream" rel="noopener noreferrer"&gt;Leetcode 295. Find Median from Data Stream&lt;/a&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Thanks for reading, stay tuned!&lt;/p&gt;

</description>
      <category>dsa</category>
      <category>algorithms</category>
      <category>programming</category>
      <category>java</category>
    </item>
    <item>
      <title>All about Gradient Descent</title>
      <dc:creator>Aman Kr Pandey</dc:creator>
      <pubDate>Sat, 27 Sep 2025 06:11:39 +0000</pubDate>
      <link>https://dev.to/_a_m_a_n_pandey/all-about-gradient-descent-15bo</link>
      <guid>https://dev.to/_a_m_a_n_pandey/all-about-gradient-descent-15bo</guid>
      <description>&lt;p&gt;Linear regression is an important algorithms in machine learning. It models the relationship between a dependent variable (target) and one or more independent variables (features) by fitting a linear equation (for example a line) to observed data. This blog will walk through the mathematics behind linear regression models.&lt;/p&gt;

&lt;h2&gt;
  
  
  What is Linear Regression?
&lt;/h2&gt;

&lt;p&gt;Linear regression finds a linear function that best predicts the target variable &lt;em&gt;y&lt;/em&gt; based on predictor variables &lt;em&gt;x&lt;/em&gt;. The model equation is:&lt;br&gt;


&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;y=m0+m1x1+m2x2+…+mnxn
y = m_0 + m_1 x_1 + m_2 x_2 + \ldots + m_n x_n
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
where 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;m0m_0 &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the intercept and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;m1,m2,…,mnm_1, m_2, \ldots, m_n &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 are the coefficients (weights) for features 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;x1,x2,…,xnx_1, x_2, \ldots, x_n &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. In linear regression we aim to find optimal value for 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;m0,m1,m2,…,mnm_0, m_1, m_2, \ldots, m_n &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 to converge the cost function.
&lt;h2&gt;
  
  
  But, What is Cost function?
&lt;/h2&gt;

&lt;p&gt;The cost function in linear regression is a mathematical measure of how well the model's predicted values match the actual target values in the training data. It quantifies the difference (error) between the predicted values 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(y^)(\hat{y}) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and the true values 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(y)(y) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, representing this error as a single number that the learning algorithm tries to minimize. Now, how does learning algorithm minimizes it? The answer is using &lt;strong&gt;Gradient descent&lt;/strong&gt;.&lt;/p&gt;
&lt;h2&gt;
  
  
  Gradient Descent
&lt;/h2&gt;

&lt;p&gt;Gradient descent is used to find the global minima of the cost function, lower the cost better the model fits into the data set. But, how it finds the global minima? Remember functions, differentiation, partial derivatives etc. ? We use these mathematical concepts to achieve global minima. Lets understand the mathematics behind it.&lt;/p&gt;

&lt;p&gt;Let's consider a simple cost function, a parabolic function 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;y = x2 −4x − 3y\ =\ x^{2\ }-4x\ -\ 3 &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt; &lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt; &lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;span class="mspace mtight"&gt;&lt;span class="mtight"&gt; &lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;4&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="mspace"&gt; &lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt; &lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;3&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F32eggboak8lw7ntvog4r.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F32eggboak8lw7ntvog4r.png" alt="Parabolic Cost function"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Now from the graph it is clearly visible that at 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;x=2,y=−7x = 2, y = -7 &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;−&lt;/span&gt;&lt;span class="mord"&gt;7&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, we have the global minima. But, it is not possible to trace graph and observe the global minima for all cost functions because cost functions can be as complex as Mean squared error function with y depending on not just x but multiple independent set of variable like 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;x0,x1,x2,…,xnx_0, x_1, x_2, \ldots, x_n &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. For example,&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;J(y^)=12N∑i=1N(y^i−yi)2
J(\hat{y}) = \frac{1}{2N} \sum_{i=1}^{N} (\hat{y}_i - y_i)^2
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;where, the predicted values is 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(yi^)(\hat{y_i}) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, the true values is 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;(yi)(y_i) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;J(yi)J(y_i) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is the cost function, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;NN &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is number of data points and &lt;em&gt;n&lt;/em&gt; is no of feature in each data point. You see, it is very difficult to plot this cost function and observe the local minima, now here comes mathematics to rescue us.&lt;/p&gt;

&lt;p&gt;In gradient descent, the core idea is to move in the direction of steepest slope based on calculus (gradient at a point or the slope). So, by walking down the hill step by step we will reach the global minima. Now, how do we find this mathematically. So lets take example of Mean Squared Error (MSE) function, our cost function for linear regression,&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;J(y^)=12N∑i=1N(y^i−yi)2
J(\hat{y}) = \frac{1}{2N} \sum_{i=1}^{N} (\hat{y}_i - y_i)^2
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;2&lt;/span&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord accent"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="accent-body"&gt;&lt;span class="mord"&gt;^&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;for regression line 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;y=mx+by = mx + b &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, our goal here is to find optimal values of &lt;em&gt;m&lt;/em&gt; (slope) and &lt;em&gt;b&lt;/em&gt; (intercept) to reduce the cost function. So lets take partial derivative of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;J(m,b)J(m,b) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. So, substituting 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;yi=mxi+by_i = mx_i + b &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and taking partial derivative with respect to &lt;em&gt;m&lt;/em&gt;, we get,&lt;/p&gt;


&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∂J(m,b)∂m=1N∑i=1N(xi((mxi+b)−yi))
\frac{\partial J(m,b)}{\partial m} = \frac{1}{N} \sum_{i=1}^N (x_i((mx_i+b) - y_i))
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;((&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;))&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
again, taking partial derivative of 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;J(m,b)J(m,b) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 with respect to &lt;em&gt;b&lt;/em&gt;, we get,&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∂J(m,b)∂b=1N∑i=1N((mxi+b)−yi)
\frac{\partial J(m,b)}{\partial b} = \frac{1}{N} \sum_{i=1}^N ((mx_i+b) - y_i)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;((&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
Let 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;mcurrm_{\text{curr}} &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;curr&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;bcurrb_{\text{curr}} &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;curr&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 be current values of slope and intercept and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;α\alpha &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 be the rate of learning, then out new slope and intercept will be,&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;mnew=mcurr−α.∂J(m,b)∂m
m_{\text{new}} = m_{\text{curr}} - \alpha.\frac{\partial J(m,b)}{\partial m}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;new&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;curr&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mord"&gt;.&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;bnew=bcurr−α.∂J(m,b)∂b
b_{\text{new}} = b_{\text{curr}} - \alpha.\frac{\partial J(m,b)}{\partial b}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;new&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;curr&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mord"&gt;.&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;We keep iterating over this process till our cost function converges, and once the cost function converges we get our optimal value for slope (m) and intercept (b). &lt;/p&gt;

&lt;p&gt;Now, this was just to fit a simple straight line containing one variable 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;xx &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, what is we have &lt;em&gt;n&lt;/em&gt; number of variables? well in that case we make use of linear algebra with multi-variate calculus.&lt;/p&gt;
&lt;h2&gt;
  
  
  The General Case
&lt;/h2&gt;

&lt;p&gt;In real life you will not get data whose outcome will depend on a single parameter. In real life, we will have &lt;em&gt;n&lt;/em&gt; number of independent parameters on which out outcome would depend. So, how to express these in terms of mathematical equation? Here comes linear algebra and vectors. So, lets say for our regression line 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;y=mx+by = mx+b &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
, we will try to express each terms in form of matrix for example,&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;X=[x1], M=[mb]
X = \begin{bmatrix} x \newline 1 \end{bmatrix}
, \space 
M = \begin{bmatrix} m &amp;amp; b \end{bmatrix}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;&lt;span class="delimsizing size3"&gt;[&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;&lt;span class="delimsizing size3"&gt;]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt; &lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;&lt;span class="delimsizing size1"&gt;[&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;&lt;span class="delimsizing size1"&gt;]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
and &lt;em&gt;Y&lt;/em&gt; being our outcome matrix, we can now express our line as a dot product of two matrix,&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Y=M⋅X⇠(Eq.1)
Y = M \centerdot X \quad \dashleftarrow (Eq. 1)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;Y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin amsrm"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel amsrm"&gt;⇠&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;Eq&lt;/span&gt;&lt;span class="mord"&gt;.1&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;
&lt;br&gt;
Expanding out idea further we can now express a multi-vatiate regression line 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;y=m0+m1x1+m2x2+…+mnxn
y = m_0 + m_1 x_1 + m_2 x_2 + \ldots + m_n x_n
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;y&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;…&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;+&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 as,&lt;br&gt;

&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;X=[1x1x2⋅⋅⋅xn-1xn], M=[m0m1⋅⋅mn-1mn]
X = \begin{bmatrix} 1 \newline x_1 \newline x_2 \newline \cdot \newline \cdot \newline \cdot \newline x_{\text{n-1}} \newline x_n \end{bmatrix}
, \space 
M = \begin{bmatrix} m_0 &amp;amp; m_1 &amp;amp; \cdot &amp;amp; \cdot &amp;amp; m_{\text{n-1}} &amp;amp; m_n \end{bmatrix}
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎣&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎡&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;2&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋅&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋅&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋅&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;n-1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;x&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;&lt;span class="delimsizing mult"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎦&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="delimsizinginner delim-size4"&gt;&lt;span&gt;⎤&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mpunct"&gt;,&lt;/span&gt;&lt;span class="mspace"&gt; &lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="minner"&gt;&lt;span class="mopen delimcenter"&gt;&lt;span class="delimsizing size1"&gt;[&lt;/span&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mtable"&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;0&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋅&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;⋅&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;n-1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="arraycolsep"&gt;&lt;/span&gt;&lt;span class="col-align-c"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;m&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;n&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose delimcenter"&gt;&lt;span class="delimsizing size1"&gt;]&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;and to express our line, we can take dot product of &lt;em&gt;M&lt;/em&gt; and &lt;em&gt;X&lt;/em&gt; same as Eq.1 mentioned above. Now lets try to express all our calculation in terms of matrix.&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∇MJ(M)=1N∑i=1NXiT(M⋅Xi−Yi)
\nabla_M J(M) = \frac{1}{N} \sum_{i=1}^{N} {X_i}^T(M \centerdot X_i - Y_i)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∇&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;M&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mop op-limits"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;span class="mrel mtight"&gt;=&lt;/span&gt;&lt;span class="mord mtight"&gt;1&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="mop op-symbol large-op"&gt;∑&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;N&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;T&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin amsrm"&gt;⋅&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;X&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;Y&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;i&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;where, 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∇MJ(M)\nabla_M J(M) &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∇&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;M&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 is &lt;a href="https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant" rel="noopener noreferrer"&gt;Jacobian&lt;/a&gt; expression for 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∂J(m,b)∂m\frac{\partial J(m,b)}{\partial m} &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;m&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;J&lt;/span&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;m&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
 and 
&lt;span class="katex-element"&gt;
  &lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;∂J(m,b)∂b\frac{\partial J(m,b)}{\partial b} &lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mopen nulldelimiter"&gt;&lt;/span&gt;&lt;span class="mfrac"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="frac-line"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord mtight"&gt;∂&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;J&lt;/span&gt;&lt;span class="mopen mtight"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;m&lt;/span&gt;&lt;span class="mpunct mtight"&gt;,&lt;/span&gt;&lt;span class="mord mathnormal mtight"&gt;b&lt;/span&gt;&lt;span class="mclose mtight"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mclose nulldelimiter"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/span&gt;
. Similarly our new &lt;em&gt;M&lt;/em&gt; would be&lt;br&gt;

&lt;/p&gt;
&lt;div class="katex-element"&gt;
  &lt;span class="katex-display"&gt;&lt;span class="katex"&gt;&lt;span class="katex-mathml"&gt;Mnew=Mold−α∇MJ(M)
M_{\text{new}} = M_{\text{old}} - \alpha\nabla_M J(M)
&lt;/span&gt;&lt;span class="katex-html"&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;new&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mrel"&gt;=&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mtight"&gt;&lt;span class="mord text mtight"&gt;&lt;span class="mord mtight"&gt;old&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;span class="mbin"&gt;−&lt;/span&gt;&lt;span class="mspace"&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="base"&gt;&lt;span class="strut"&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;α&lt;/span&gt;&lt;span class="mord"&gt;&lt;span class="mord"&gt;∇&lt;/span&gt;&lt;span class="msupsub"&gt;&lt;span class="vlist-t vlist-t2"&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;span class="pstrut"&gt;&lt;/span&gt;&lt;span class="sizing reset-size6 size3 mtight"&gt;&lt;span class="mord mathnormal mtight"&gt;M&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-s"&gt;​&lt;/span&gt;&lt;/span&gt;&lt;span class="vlist-r"&gt;&lt;span class="vlist"&gt;&lt;span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span class="mord mathnormal"&gt;J&lt;/span&gt;&lt;span class="mopen"&gt;(&lt;/span&gt;&lt;span class="mord mathnormal"&gt;M&lt;/span&gt;&lt;span class="mclose"&gt;)&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;
&lt;/div&gt;


&lt;p&gt;We keep iterating over this process till our cost function converges, and once the cost function converges we get our optimal value for &lt;em&gt;M&lt;/em&gt;.&lt;/p&gt;

&lt;p&gt;So, this is it, we have covered the mathematics behind gradient descent and how to apply it in optimizing the cost function of models. We will further discuss its implementation using python, till then take care!&lt;/p&gt;

</description>
      <category>ai</category>
      <category>machinelearning</category>
      <category>linearregression</category>
      <category>regression</category>
    </item>
    <item>
      <title>Scalable Load Balancing with Consistent Hashing</title>
      <dc:creator>Aman Kr Pandey</dc:creator>
      <pubDate>Sun, 27 Jul 2025 17:45:27 +0000</pubDate>
      <link>https://dev.to/_a_m_a_n_pandey/scalable-load-balancing-with-consistent-hashing-d52</link>
      <guid>https://dev.to/_a_m_a_n_pandey/scalable-load-balancing-with-consistent-hashing-d52</guid>
      <description>&lt;p&gt;In the world of micro-services, load balancing is a cornerstone to achieve scalability. There are various load balancing algorithms that distributes incoming traffic to multiple servers. This prevents a single server from becoming a bottleneck and ensuring quick response time and less outage. Among multiple load balancing algorithms consistent hashing is an powerful technique of load balancing&lt;/p&gt;

&lt;h2&gt;
  
  
  Traditional Load balancing algorithms
&lt;/h2&gt;

&lt;p&gt;Imagine a simple scenario with three servers (A, B, C) and a traditional load balancer using a round-robin or modulo-based approach. When a request comes, it is directed to a server based on a simple algorithm. This works fine until you need to add a new server (D), remove an existing one or a server crashes down due to some technical issue.&lt;/p&gt;

&lt;p&gt;With traditional methods, adding or removing a server often needs a complete remapping of requests. This can lead to:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Cache Invalidation&lt;/strong&gt;: If your servers cache data, a remapping means a large portion of your cache becomes invalid, leading to a surge of requests to the backend database as server did not find any cache for remapped request. So, Many requests simultaneously hitting the database can overwhelm it.&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Downtime/Low performance&lt;/strong&gt;: The remapping process can introduce latency or even brief periods of unavailability.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;These issues become amplified in microservices architectures and cloud environments where services are constantly being scaled up and down.&lt;/p&gt;

&lt;h2&gt;
  
  
  Hello! Consistent hashing
&lt;/h2&gt;

&lt;p&gt;Consistent Hashing is a special kind of hashing that minimizes the impact of adding or removing a server on the overall distribution of request over hash of servers. Instead of remapping every request only a small fraction of requests are affected.&lt;/p&gt;

&lt;h3&gt;
  
  
  How it works
&lt;/h3&gt;

&lt;p&gt;Think of a a ring, that represents your hash space for servers and requests. Both server and requests are mapped to the hash ring using a hash function.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Server Placement&lt;/strong&gt;: Our first task is to place the servers on the hash ring. Each server is hashed using a hash algorithm and placed on the specific position on the ring&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Placement of requests&lt;/strong&gt;: For incoming requests too we hash them and place them on the ring. Request and server may/may not point to same position on the ring.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Server assignment&lt;/strong&gt;: Once both server and request is mapped to the ring, each request gets the next available server on the ring, your direction or traversal may be clockwise or anticlockwise. The first server we encounter while traversing that server gets allocated to the request.&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fasemhkodmembgo0y5kjy.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fasemhkodmembgo0y5kjy.png" alt=" " width="599" height="583"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Lets take reference to above image, it has a ring with hash space of &lt;code&gt;n = 100&lt;/code&gt; and four servers (A,B,C,D), number within them represent the index/hash where are located on the ring and we are traversing in clockwise direction on the ring.&lt;/p&gt;

&lt;p&gt;So, if any incoming request with hash 10 will be allocated to just next server in clockwise direction, in this case its Server B.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Note:&lt;/strong&gt; Hash space should be large enough to avoid collision based on your hashing algorithm.&lt;/p&gt;

&lt;h3&gt;
  
  
  How this makes the difference
&lt;/h3&gt;

&lt;p&gt;Now, let us see what happens when we add or remove a server:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Adding a Server&lt;/strong&gt;: When a new server is added to the ring, it only takes over a portion of the requests from its neighbour. The vast majority of other requests remain mapped to their original servers. For example, if we add a Server E which has value 40 between Server B and Server C. Only requests of value greater that hash value 20 and less that equal to 40 will be remapped to Server E, rest all mapping will remain same.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Removing a Server&lt;/strong&gt;: When a server is removed, its requests are simply re-mapped to next clockwise neighbour. Again, only a localized change occurs. Again, if we remove Server E all the request getting assigned to E will automatically get re-mapped to Server C.&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;This localized remapping is the core benefit of Consistent Hashing.&lt;/p&gt;

&lt;h2&gt;
  
  
  Benefits for Load Balancing:
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Reduced re-mapping&lt;/strong&gt;: This reduces the number of requests that need to be re-mapped when servers are added or removed.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Improved Cache Utilization&lt;/strong&gt;: Preserves a significant amount of cached data, reducing the load on backend systems.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Enhanced Scalability&lt;/strong&gt;: Allows for seamless scaling up and down of server capacity without significant performance issues.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Reduced Thundering Herd&lt;/strong&gt;: Prevents a sudden surge of requests to the database.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Better Availability&lt;/strong&gt;: Contributes to more stable and available systems during scaling events.&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Drawbacks of Consistent Hashing
&lt;/h2&gt;

&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Uneven distribution&lt;/strong&gt;: With the distribution of requests on the ring may not be even, it may happen that a server in ring gets most of the requests and other does not get any traffic at all. To mitigate this we use virtual nodes. Instead of mapping physical server directly on the ring we map multiple virtual nodes on the ring that point to the same physical. This helps a single physical server to appear at multiple places on the ring, and thus avoid uneven distribution of requests. The quality of hash function used also has a significant affect on distribution.&lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Performance overhead&lt;/strong&gt;: Repeated hashing of requests and traversing the ring even with using best data structures and optimised algorithms may introduce slight performance overhead as compared to simple hashing techniques. &lt;/p&gt;&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;Cold start issue&lt;/strong&gt;: When a new server is added to the ring, it is likely to have an empty cache for that hash in the backend, so this may increase load on backend databases/cache for quite some time.&lt;/p&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Implementation of Consistent Hashing
&lt;/h2&gt;

&lt;p&gt;While the concept is straightforward, implementing a robust Consistent Hashing solution often involves libraries or frameworks that handle the complexities of managing the hash ring, virtual nodes, and server discovery, but for better understanding we will implement a simple consistent hash scheduler.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Create a class in java &lt;code&gt;ConsistantHashScheduler&lt;/code&gt;
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;public class ConsistantHashScheduler{
    private static final Logger logger = LogManager.getLogger(ConsistantHashScheduler.class);

    private ArrayList&amp;lt;Server&amp;gt; nodes;
    private ArrayList&amp;lt;Integer&amp;gt; keys;
    private ArrayList&amp;lt;Server&amp;gt; servers;
    private int maxPoolSize;

    public ConsistantHashScheduler(ArrayList&amp;lt;Server&amp;gt; servers, int maxPoolSize) {
            logger.info("Initializion a consistent hash scheduler with pool size: {}",maxPoolSize);
        this.maxPoolSize = maxPoolSize;
        this.servers = servers;
            this.keys = new ArrayList&amp;lt;&amp;gt;();
        this.nodes = new ArrayList&amp;lt;&amp;gt;();
    }
}
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;ul&gt;
&lt;li&gt;Add hash function of your choice
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;private int getHash(String ip) throws NoSuchAlgorithmException {
        MessageDigest digest = MessageDigest.getInstance("SHA-256");
            byte[] hashBytes = digest.digest(ip.getBytes(StandardCharsets.UTF_8));

            int hash = 0;
            for (int i = 0; i &amp;lt; 4; i++) {
                hash &amp;lt;&amp;lt;= 8;
                hash |= (hashBytes[i] &amp;amp; 0xFF);
            }
            return (hash &amp;amp; Integer.MAX_VALUE) % this.maxPoolSize;
    }
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This method hashes an IP using SHA-256, extracts a 32-bit integer, and scales it for pool size, ensuring even distribution.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Now, implement &lt;code&gt;addServer&lt;/code&gt; method.
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;public void addServer(Server server) throws Exception {
        if(keys.size() == this.maxPoolSize) {
            throw new Exception("Max server pool size overflow");
        }

        logger.info("adding new server: {}",server.toString());

        int key = this.getHash(server.getUrl());
        int index = Collections.binarySearch(this.keys, key);



        if(index &amp;gt; 0 &amp;amp;&amp;amp; this.keys.get(index) == key) {
            throw new Exception("Collosion occured for server "+ server.toString());
        }

        int insertionPoint = -(index+1);

        this.keys.add(insertionPoint,key);
        this.nodes.add(insertionPoint,server);

        return;
    }
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This &lt;code&gt;addServer&lt;/code&gt; method adds a new server to a Consistent Hashing ring. It calculates the hash of server, finds its insertion point in a sorted list of keys to maintain the ring order, and then inserts both the hash and the server. It handles pool size limits and hash collisions.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Add &lt;code&gt;removeServer&lt;/code&gt; method.
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;public void removeServer(Server server) throws Exception {

        if(this.keys.size() == 0) {
            throw new Exception("No nodes in the server pool");
        }

        int key = this.getHash(server.getUrl());

        int index = Collections.binarySearch(this.keys, key);

        if(index &amp;lt; 0) {
            throw new Exception("Unknown server, not present in nodes");
        }

        this.keys.remove(index);
        this.nodes.remove(index);

        return;

    }
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This &lt;code&gt;removeServer&lt;/code&gt; function deletes a server from the Consistent Hashing ring. It hashes the URL of server, finds its position via binary search, and removes its key and the server object from the sorted lists. It validates against an empty pool or unknown servers.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Let's schedule some requests.
&lt;/li&gt;
&lt;/ul&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;public Server schedule(HttpExchange exchange) throws NoSuchAlgorithmException {
        int key = this.getHash(exchange.getRemoteAddress().getHostName());
        int index = Collections.binarySearch(keys, key);

        int nodePoint;

        if(index &amp;gt;= 0) {
            nodePoint = index;
        }else {
            nodePoint = ((-(index+1))%this.keys.size());
        }
        return this.servers.get(nodePoint);

    }
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This &lt;code&gt;schedule&lt;/code&gt; method is core logic of a load balancer. It hashes the incoming request's client IP, then uses binary search on the sorted hash ring (keys) to find the appropriate server. If no exact match, it calculates the next clockwise server using modulo, returning the chosen Server object.&lt;/p&gt;

&lt;p&gt;The summary of the above implementation can be found &lt;a href="https://gist.github.com/AmanPandey0320/8d591decb73e8e7c628bda9f6ab0beb4" rel="noopener noreferrer"&gt;here&lt;/a&gt;.&lt;/p&gt;

&lt;h2&gt;
  
  
  At last
&lt;/h2&gt;

&lt;p&gt;Consistent Hashing offers an efficient approach to load balancing. By minimising the impact of frequent changes, it ensures greater stability and availability, allowing the systems to scale better. If you are building or optimizing a distributed system, understanding and implementing Consistent Hashing is an important tool in your skill set.&lt;/p&gt;

</description>
      <category>loadbalancer</category>
      <category>hashing</category>
      <category>systemdesign</category>
      <category>programming</category>
    </item>
    <item>
      <title>PKCE Demystified: How OAuth 2.0 Became More Secure for Public Clients</title>
      <dc:creator>Aman Kr Pandey</dc:creator>
      <pubDate>Mon, 23 Jun 2025 13:18:57 +0000</pubDate>
      <link>https://dev.to/_a_m_a_n_pandey/pkce-demystified-how-oauth-20-became-more-secure-for-public-clients-37nb</link>
      <guid>https://dev.to/_a_m_a_n_pandey/pkce-demystified-how-oauth-20-became-more-secure-for-public-clients-37nb</guid>
      <description>&lt;p&gt;OAuth2.0 has been a very popular protocol among developers for delegating authorization for modern apps. But when it comes to mobile apps or single page application (SPAs), the traditional Authorization code flow done with the help of &lt;code&gt;client_id&lt;/code&gt; and &lt;code&gt;client_secret&lt;/code&gt; has a serious security vulnerability: &lt;strong&gt;Code Interception&lt;/strong&gt;.&lt;/p&gt;

&lt;p&gt;Here enters: &lt;strong&gt;PKCE (Proof Key for Code Exchange)&lt;/strong&gt; — a security enhancement that protects public clients from code interception attacks. This post breaks down what PKCE is, how it works, and how to implement it.&lt;/p&gt;

&lt;h1&gt;
  
  
  Need of PKCE Flow
&lt;/h1&gt;

&lt;p&gt;In the traditional OAuth2.0 Authorization code flow, the client first sends authorize request to authorization server with &lt;code&gt;client_id&lt;/code&gt;, the server then responds back with a &lt;code&gt;authorization_code&lt;/code&gt; and then the client sends a request with the &lt;code&gt;authorization_code&lt;/code&gt; and &lt;code&gt;client_secret&lt;/code&gt;. Now, the problem here is that public client like mobile apps and SPAs can not store secrets securely and this makes them vulnerable to interception attacks. So, your &lt;code&gt;authorization_code&lt;/code&gt; and &lt;code&gt;client_secret&lt;/code&gt; once they get intercepted by an attacker can use them to initiate token exchange and access resources. PKCE flow mitigates this by adding a dynamic challenge and verification in each authorization request. This ensures that only the app has the ability to get authorized not any attacker even if they have intercepted the &lt;code&gt;authorization_code&lt;/code&gt;.&lt;/p&gt;

&lt;h1&gt;
  
  
  How PKCE Flow works
&lt;/h1&gt;

&lt;p&gt;Let's walk through the PKCE flow and how it works:&lt;/p&gt;

&lt;h3&gt;
  
  
  1. Client generates Secret and Challenge
&lt;/h3&gt;

&lt;p&gt;In this steps the client create a Secret and a Challenge. Secret is any random string and the Challenge is hashed version of the Secret using any of the cryptographic algorithms (SHA3-256 in our case).&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;code_verifier = randomString(43-128 chars) 
code_challenge = base64url(SHA3-256(code_verifier))
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h3&gt;
  
  
  2. Client send authorize request with &lt;code&gt;code_challange&lt;/code&gt;.
&lt;/h3&gt;

&lt;p&gt;The client initiates authorization.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;GET /authorize?
  response_type=code&amp;amp;
  client_id=your-client-id&amp;amp;
  redirect_uri=https://yourapp.com/callback&amp;amp;
  code_challenge=f591f035a31f320015177cadb73e2056939b5ddbd30e902c5f274bc092b29545&amp;amp;
  code_challenge_method=SHA3-256
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;So there are two methods to pass &lt;code&gt;code_challange_method&lt;/code&gt;. First we can save it in the database of our authorization server as part of client's config, secondly we can pass it in the URL as a parameter, to add an extra layer of security in second method we can keep changing the method in each request.&lt;/p&gt;

&lt;h3&gt;
  
  
  3. User approval and server's redirect
&lt;/h3&gt;

&lt;p&gt;The user logs in and approves apps access to its resources, and the authorization server redirects to client's callback url with &lt;code&gt;authorization_code&lt;/code&gt;.&lt;/p&gt;

&lt;p&gt;Once the user is authenticated the server stores the &lt;code&gt;code_challenge&lt;/code&gt; and &lt;code&gt;code_challenge_method&lt;/code&gt; against &lt;code&gt;authorization_code&lt;/code&gt; that it issued.&lt;/p&gt;

&lt;h3&gt;
  
  
  4. Client initiates token exchange
&lt;/h3&gt;

&lt;p&gt;Now once redirected, the client then uses &lt;code&gt;authorization_code&lt;/code&gt; to get tokens in exchange. Here the client sends &lt;code&gt;code_secret&lt;/code&gt; as parameter in request. This is the original Secret generated in the first step.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;POST /token
Content-Type: application/x-www-form-urlencoded

grant_type=authorization_code&amp;amp;
code=XYZ&amp;amp;
code_secret=some_code
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The server compares the hash of &lt;code&gt;code_secret&lt;/code&gt; with the earlier &lt;code&gt;code_challenge&lt;/code&gt;. If they match, it issues tokens. Here while generating the hash from &lt;code&gt;code_secret&lt;/code&gt; one may introduce hash rounds a client's config, which is stored over database and only known to authorization server and the client. Adding hash rounds makes apps more secure and even less vulnerable to attacks. Even if someone mimics the PKCE flow, creating their own secret and challenge, they might not be aware of exact number of hash rounds.&lt;/p&gt;

&lt;h1&gt;
  
  
  When to use PKCE?
&lt;/h1&gt;

&lt;p&gt;Use PKCE whenever your app is a public client:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Native mobile apps (iOS, Android)&lt;/li&gt;
&lt;li&gt;Single Page Applications (React, Angular, Vue)&lt;/li&gt;
&lt;li&gt;Desktop apps (Electron, etc.)&lt;/li&gt;
&lt;/ul&gt;

&lt;h1&gt;
  
  
  Implementation Tips
&lt;/h1&gt;

&lt;ul&gt;
&lt;li&gt;Use &lt;code&gt;SHA-256&lt;/code&gt; as cryptographic method, or keep changing for each request&lt;/li&gt;
&lt;li&gt;Never re-use &lt;code&gt;code_secret&lt;/code&gt;, always generate new Secret for new request&lt;/li&gt;
&lt;li&gt;Keep the PKCE logic in secure memory, avoid writing to local storage in browsers&lt;/li&gt;
&lt;/ul&gt;

&lt;h1&gt;
  
  
  Conclusion
&lt;/h1&gt;

&lt;p&gt;So, at last we can conclude that OAuth2.0 can be a great choice to authorize users if you are building a public client, but with PKCE it adds more security with minimal complexity.&lt;/p&gt;

&lt;p&gt;Thanks for you time! &lt;br&gt;
Have a great day ahead!&lt;/p&gt;

</description>
      <category>oauth</category>
      <category>pkce</category>
      <category>openid</category>
      <category>authentication</category>
    </item>
  </channel>
</rss>
