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Randhir Kumar
Randhir Kumar

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From Perceptron to Softmax Regression: Demystifying Generalized Linear Models (GLM)

#machinelearning #ai #programming #beginners

From Perceptron to Generalized Linear Models

🔹 1. Introduction

  • Quick recap from Blog 1: “We discussed ML fundamentals...”
  • Why linear models matter in ML (classification, regression, interpretability)
  • The evolution: From Perceptron ➡ Logistic Regression ➡ GLMs ➡ Softmax

🔹 2. The Perceptron: The OG Classifier

🧩 What is a Perceptron?

  • Inspired from biological neurons
  • Takes weighted sum of inputs + bias → passes through a step function (activation)

🧮 Mathematical Representation: 

y = f(W · X + b)
Where f = step function (0 or 1)
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🎯 Limitations:

  • Only works for linearly separable data
  • Can’t output probabilities
  • No probabilistic interpretation

📸 Visual:
LinearlySeperable

.

🔹 3. Exponential Family of Distributions: The Foundation of GLMs

🧪 What is the Exponential Family?

  • A set of probability distributions written in a general form: 
P(y | θ) = h(y) * exp(η(θ)·T(y) - A(θ))
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Where:

  • η(θ) = natural parameter
  • T(y) = sufficient statistic
  • A(θ) = log-partition function

📦 Common Examples in Exponential Family:

Distribution Use Case
Bernoulli Binary classification
Gaussian Linear regression
Poisson Count data
Multinomial Multi-class classification

🔹 4. Generalized Linear Models (GLM)

⚙️ What is a GLM?

A flexible extension of linear regression that models:

E[y | x] = g⁻¹(X · β)
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Where:

  • g⁻¹ = inverse link function
  • X · β = linear predictor
  • y = output variable

🧠 Components of GLM:

  1. Linear predictor: Xβ
  2. Link function: connects predictor to mean of distribution
  3. Distribution: from exponential family

🎯 Examples of GLMs:

GLM Variant Link Function Distribution
Linear Regression Identity g(y)=y Gaussian
Logistic Regression Logit log(p/1-p) Bernoulli
Poisson Regression log(y) Poisson

📸 Visual:
GLM

.

🔹 5. Softmax Regression (Multinomial Logistic Regression)

🔁 What is Softmax?

  • Extension of logistic regression for multi-class classification
  • Uses softmax function to output probabilities across classes

📐 Equation: 

P(y = j | x) = exp(w_j · x) / Σ_k exp(w_k · x)
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🤔 Why use Softmax?

  • Predicts probability distribution over classes
  • Works for mutually exclusive categories (e.g., digit classification 0–9)

📸 Visual:
SOFTMAXPROBALITY

🔹 6. Perceptron vs GLM vs Softmax Regression

Feature Perceptron GLM Softmax Regression
Probabilistic? ❌ ✅ ✅
Activation Step Function Depends on task Softmax
Output Binary (0/1) Real-valued / Prob Probabilities over k classes
Interpretability Low High Medium

🔹 7. Real-World Applications

  • Perceptron: Simple binary classifiers, early neural networks
  • GLMs: Medical stats, econometrics, GLM for insurance risk modeling
  • Softmax: Image classification (e.g., MNIST), NLP classification

🔹 8. Conclusion

  • Perceptron = Starting point
  • GLM = Bridge between linear models and probability theory
  • Softmax = Modern ML essential for multi-class prediction

🧠 "Understanding these models builds the foundation for deep learning and beyond."

  • Want code walkthroughs of perceptron & softmax in Python? Comment below!
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