Prompt:
logistic regression ke baare me sab mathamtical denotation likho aur uska explaination do aur derivatives bhi nikaalo
Response:
नीचे Logistic Regression का पूरा mathematical formulation , notation , intuition , और derivatives (gradient) step-by-step समझाया गया है — बिल्कुल exam + ML interview level तक।
1. Logistic Regression क्या है? (Idea)
Logistic Regression एक binary classification algorithm है
उदाहरण:
Email → Spam (1) / Not Spam (0)
Disease → Yes (1) / No (0)
यह probability predict करता है:
P ( y = 1 ∣ x ) P(y=1 \mid x) P ( y = 1 ∣ x )
2. Notations (Mathematical Symbols)
Symbol
Meaning
x ∈ R n x \in \mathbb{R}^n x ∈ R n
Input feature vector
y ∈ 0 , 1 y \in {0,1} y ∈ 0 , 1
True label
w ∈ R n w \in \mathbb{R}^n w ∈ R n
Weight vector
b ∈ R b \in \mathbb{R} b ∈ R
Bias
z z z
Linear combination
y ^ \hat{y} y ^
Predicted probability
m m m
Number of training examples
3. Linear Model
पहले हम linear combination निकालते हैं:
z = w T x + b z = w^T x + b z = w T x + b
लेकिन
z z z
की value
− ∞ -\infty − ∞
से
+ ∞ +\infty + ∞
तक जा सकती है
[ 0 , 1 ] [0,1] [ 0 , 1 ]
में।
4. Sigmoid Function (Logistic Function)
इसलिए हम Sigmoid लगाते हैं:
σ ( z ) = 1 1 + e − z \sigma(z) = \frac{1}{1 + e^{-z}} σ ( z ) = 1 + e − z 1
अब prediction:
y ^ = P ( y = 1 ∣ x ) = σ ( w T x + b ) \hat{y} = P(y=1 \mid x) = \sigma(w^T x + b) y ^ = P ( y = 1 ∣ x ) = σ ( w T x + b )
Sigmoid Properties:
z → + ∞ ⇒ σ ( z ) → 1 z \to +\infty \Rightarrow \sigma(z) \to 1 z → + ∞ ⇒ σ ( z ) → 1
z → − ∞ ⇒ σ ( z ) → 0 z \to -\infty \Rightarrow \sigma(z) \to 0 z → − ∞ ⇒ σ ( z ) → 0
5. Hypothesis Function
h θ ( x ) = y ^ = 1 1 + e − ( w T x + b ) h_\theta(x) = \hat{y} = \frac{1}{1 + e^{-(w^T x + b)}} h θ ( x ) = y ^ = 1 + e − ( w T x + b ) 1
6. Cost Function (Log Loss / Binary Cross Entropy)
Simple MSE यहाँ काम नहीं करता, इसलिए हम Log Loss use करते हैं:
Single example loss:
L ( y ^ , y ) = − [ y log ( y ^ ) + ( 1 − y ) log ( 1 − y ^ ) ] L(\hat{y}, y) = - \left[ y \log(\hat{y}) + (1-y)\log(1-\hat{y}) \right] L ( y ^ , y ) = − [ y log ( y ^ ) + ( 1 − y ) log ( 1 − y ^ ) ]
पूरे dataset के लिए cost:
J ( w , b ) = 1 m ∑ i = 1 m [ − y ( i ) log ( y ^ ( i ) ) − ( 1 − y ( i ) ) log ( 1 − y ^ ( i ) ) ] J(w,b) = \frac{1}{m} \sum_{i=1}^{m} \left[ - y^{(i)} \log(\hat{y}^{(i)}) - (1-y^{(i)}) \log(1-\hat{y}^{(i)}) \right] J ( w , b ) = m 1 i = 1 ∑ m [ − y ( i ) log ( y ^ ( i ) ) − ( 1 − y ( i ) ) log ( 1 − y ^ ( i ) ) ]
7. Optimization Goal
हमें find करना है:
min w , b J ( w , b ) \min_{w,b} J(w,b) w , b min J ( w , b )
इसके लिए हम Gradient Descent use करते हैं।
8. Derivatives (सबसे ज़रूरी हिस्सा 🔥)
Step 1: Sigmoid derivative
σ ( z ) = 1 1 + e − z \sigma(z) = \frac{1}{1+e^{-z}} σ ( z ) = 1 + e − z 1
d σ d z = σ ( z ) ( 1 − σ ( z ) ) \frac{d\sigma}{dz} = \sigma(z)(1-\sigma(z)) d z d σ = σ ( z ) ( 1 − σ ( z ))
Step 2: Cost function derivative
Final result (after chain rule):
Weight के respect में gradient:
∂ J ∂ w = 1 m ∑ i = 1 m ( y ^ ( i ) − y ( i ) ) x ( i ) \frac{\partial J}{\partial w} = \frac{1}{m} \sum_{i=1}^{m} (\hat{y}^{(i)} - y^{(i)}) x^{(i)} ∂ w ∂ J = m 1 i = 1 ∑ m ( y ^ ( i ) − y ( i ) ) x ( i )
Bias के respect में gradient:
∂ J ∂ b = 1 m ∑ i = 1 m ( y ^ ( i ) − y ( i ) ) \frac{\partial J}{\partial b} = \frac{1}{m} \sum_{i=1}^{m} (\hat{y}^{(i)} - y^{(i)}) ∂ b ∂ J = m 1 i = 1 ∑ m ( y ^ ( i ) − y ( i ) )
👉 Important insight :
y ^ − y \hat{y} - y y ^ − y
9. Gradient Descent Update Rules
w : = w − α ∂ J ∂ w w := w - \alpha \frac{\partial J}{\partial w} w := w − α ∂ w ∂ J
b : = b − α ∂ J ∂ b b := b - \alpha \frac{\partial J}{\partial b} b := b − α ∂ b ∂ J
जहाँ:
10. Vectorized Form (Deep Learning Friendly)
अगर पूरा data matrix
X ∈ R m × n X \in \mathbb{R}^{m \times n} X ∈ R m × n
:
y ^ = σ ( X w + b ) \hat{y} = \sigma(Xw + b) y ^ = σ ( Xw + b )
∇ w J = 1 m X T ( y ^ − y ) \nabla_w J = \frac{1}{m} X^T (\hat{y} - y) ∇ w J = m 1 X T ( y ^ − y )
∇ b J = 1 m ∑ ( y ^ − y ) \nabla_b J = \frac{1}{m} \sum (\hat{y} - y) ∇ b J = m 1 ∑ ( y ^ − y )
11. Decision Boundary
y ^ ≥ 0.5 ⇒ y = 1 \hat{y} \ge 0.5 \Rightarrow y = 1 y ^ ≥ 0.5 ⇒ y = 1
y ^ < 0.5 ⇒ y = 0 \hat{y} < 0.5 \Rightarrow y = 0 y ^ < 0.5 ⇒ y = 0
Decision boundary:
w T x + b = 0 w^T x + b = 0 w T x + b = 0
12. Intuition Summary 🧠
Logistic regression probability model है
Sigmoid → linear output को probability में बदलता है
Log loss → confident गलत prediction को ज़्यादा punish करता है
Gradient descent → error
( y ^ − y ) (\hat{y}-y) ( y ^ − y )
को minimize करता है
अगर चाहो तो मैं:
Derivation step-by-step chain rule से
Maximum Likelihood Estimation (MLE) proof
Regularization (L1/L2) with derivatives
Geometric interpretation
भी समझा सकता हूँ 😊
Prompt:
chain rule hi samjaao ki derivatives kaise nikaala aek aek rules ke saath
Response:
ठीक है — अब हम Logistic Regression ke derivatives को pure chain rule से, ek-ek step + rule के साथ निकालेंगे।
1. Overall Computational Graph (Flow samjho)
एक data point के लिए:
x → w , b z → sigmoid y ^ → loss L x \;\xrightarrow[]{w,b}\; z \;\xrightarrow[]{\text{sigmoid}}\; \hat y \;\xrightarrow[]{\text{loss}}\; L x w , b z sigmoid y ^ loss L
जहाँ:
z = w T x + b z = w^T x + b z = w T x + b
y ^ = σ ( z ) \hat y = \sigma(z) y ^ = σ ( z )
L = − [ y log y ^ + ( 1 − y ) log ( 1 − y ^ ) ] L = -[y\log \hat y + (1-y)\log(1-\hat y)] L = − [ y log y ^ + ( 1 − y ) log ( 1 − y ^ )]
हमें निकालना है:
∂ L ∂ w , ∂ L ∂ b \frac{\partial L}{\partial w}, \quad \frac{\partial L}{\partial b} ∂ w ∂ L , ∂ b ∂ L
Chain rule:
∂ L ∂ w = ∂ L ∂ y ^ ⋅ ∂ y ^ ∂ z ⋅ ∂ z ∂ w \frac{\partial L}{\partial w} = \frac{\partial L}{\partial \hat y} \cdot \frac{\partial \hat y}{\partial z} \cdot \frac{\partial z}{\partial w} ∂ w ∂ L = ∂ y ^ ∂ L ⋅ ∂ z ∂ y ^ ⋅ ∂ w ∂ z
2. Step-1: Loss ka derivative wrt
y ^ \hat y y ^
Loss function:
L = − [ y log ( y ^ ) + ( 1 − y ) log ( 1 − y ^ ) ] L = -\left[y\log(\hat y) + (1-y)\log(1-\hat y)\right] L = − [ y log ( y ^ ) + ( 1 − y ) log ( 1 − y ^ ) ]
Rule used:
d d x ( log x ) = 1 x \frac{d}{dx}(\log x) = \frac{1}{x} d x d ( log x ) = x 1
Constant multiple rule
Derivative:
∂ L ∂ y ^ = − [ y y ^ − 1 − y 1 − y ^ ] \frac{\partial L}{\partial \hat y} = -\left[ \frac{y}{\hat y} - \frac{1-y}{1-\hat y} \right] ∂ y ^ ∂ L = − [ y ^ y − 1 − y ^ 1 − y ]
3. Step-2: Sigmoid ka derivative wrt
z z z
Sigmoid:
y ^ = σ ( z ) = 1 1 + e − z \hat y = \sigma(z) = \frac{1}{1+e^{-z}} y ^ = σ ( z ) = 1 + e − z 1
Rule used:
Quotient rule
Exponential rule
Result (important):
∂ y ^ ∂ z = y ^ ( 1 − y ^ ) \frac{\partial \hat y}{\partial z} = \hat y(1-\hat y) ∂ z ∂ y ^ = y ^ ( 1 − y ^ )
4. Step-3: Linear part ka derivative
z = w T x + b z = w^T x + b z = w T x + b
Derivatives:
wrt weight
w j w_j w j
:
∂ z ∂ w j = x j \frac{\partial z}{\partial w_j} = x_j ∂ w j ∂ z = x j
Vector form:
∂ z ∂ w = x \frac{\partial z}{\partial w} = x ∂ w ∂ z = x
wrt bias:
∂ z ∂ b = 1 \frac{\partial z}{\partial b} = 1 ∂ b ∂ z = 1
5. Chain Rule Apply karo (Main Magic ✨)
अब सब multiply करो:
∂ L ∂ w = ∂ L ∂ y ^ ⋅ ∂ y ^ ∂ z ⋅ ∂ z ∂ w \frac{\partial L}{\partial w} = \frac{\partial L}{\partial \hat y} \cdot \frac{\partial \hat y}{\partial z} \cdot \frac{\partial z}{\partial w} ∂ w ∂ L = ∂ y ^ ∂ L ⋅ ∂ z ∂ y ^ ⋅ ∂ w ∂ z
Substitute:
= ( − [ y y ^ − 1 − y 1 − y ^ ] ) ⋅ y ^ ( 1 − y ^ ) ⋅ x = \left( -\left[ \frac{y}{\hat y} - \frac{1-y}{1-\hat y} \right] \right) \cdot \hat y(1-\hat y) \cdot x = ( − [ y ^ y − 1 − y ^ 1 − y ] ) ⋅ y ^ ( 1 − y ^ ) ⋅ x
6. Simplification (Very Important Step)
Bracket simplify करो:
− [ y y ^ − 1 − y 1 − y ^ ] ⋅ y ^ ( 1 − y ^ ) -\left[ \frac{y}{\hat y} - \frac{1-y}{1-\hat y} \right] \cdot \hat y(1-\hat y) − [ y ^ y − 1 − y ^ 1 − y ] ⋅ y ^ ( 1 − y ^ )
Common denominator से:
= ( y ^ − y ) = (\hat y - y) = ( y ^ − y )
🔥 यही सबसे beautiful result है
7. Final Gradient (Single Example)
Weight:
∂ L ∂ w = ( y ^ − y ) x \boxed{ \frac{\partial L}{\partial w} = (\hat y - y)x } ∂ w ∂ L = ( y ^ − y ) x
Bias:
∂ L ∂ b = ( y ^ − y ) \boxed{ \frac{\partial L}{\partial b} = (\hat y - y) } ∂ b ∂ L = ( y ^ − y )
8. Multiple Data Points (Averaging)
∂ J ∂ w = 1 m ∑ i = 1 m ( y ^ ( i ) − y ( i ) ) x ( i ) \frac{\partial J}{\partial w} = \frac{1}{m} \sum_{i=1}^{m} (\hat y^{(i)} - y^{(i)})x^{(i)} ∂ w ∂ J = m 1 i = 1 ∑ m ( y ^ ( i ) − y ( i ) ) x ( i )
∂ J ∂ b = 1 m ∑ i = 1 m ( y ^ ( i ) − y ( i ) ) \frac{\partial J}{\partial b} = \frac{1}{m} \sum_{i=1}^{m} (\hat y^{(i)} - y^{(i)}) ∂ b ∂ J = m 1 i = 1 ∑ m ( y ^ ( i ) − y ( i ) )
9. Chain Rule Intuition 🧠
Loss बदलता है → prediction बदलने से
Prediction बदलता है → z बदलने से
z बदलता है → weight बदलने से
इसलिए:
Error = ( y ^ − y ) \text{Error} = (\hat y - y) Error = ( y ^ − y )
अगर चाहो तो मैं:
हर derivative को computational graph पर arrows के साथ
Backpropagation perspective
Same derivation with MLE
Regularization ke saath chain rule
भी करा सकता हूँ 🔥
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