Gradient Descent Methods

Understanding gradient descent and its variants for optimization

Gradient Descent Methods  

Introduction  

Gradient descent is a fundamental optimization algorithm widely used in machine learning for minimizing loss functions.

Basic Gradient Descent  

The simplest form updates parameters in the direction of the negative gradient:

θ = θ - α∇f(θ)

Where:

• θ: parameters
• α: learning rate
• ∇f(θ): gradient of the objective function

Variants  

Stochastic Gradient Descent (SGD)  

Updates parameters using individual data points or small batches.

Momentum  

Adds momentum to accelerate convergence:

v = βv + α∇f(θ)
θ = θ - v

Adam  

Adaptive learning rates with momentum:

m = β₁m + (1-β₁)∇f(θ)
v = β₂v + (1-β₂)(∇f(θ))²
θ = θ - α * m̂/(√v̂ + ε)

Applications  

• Neural network training
• Linear regression
• Logistic regression
• Support vector machines