Gradient Descent Method

Overview

Gradient Descent is an optimization algorithm used to minimize a function by iteratively adjusting the model parameters. It$$s widely employed in machine learning, particularly for training models like linear regression, neural networks, and more.

Mathematical Formulation

Given a function $($J$(\backslash$theta$))($often referred to as the cost function or loss function$),$ where $(\backslash$theta$)$ represents the model parameters $($coefficients$),$ the goal is to find the optimal $(\backslash$theta$)$ that minimizes $($J$(\backslash$theta$)).$ The update rule for Gradient Descent is:

$[\backslash$theta$\_\{\backslash$text$\{$new$\}\}=\backslash$theta$\_\{\backslash$text$\{$old$\}\}-\backslash$alpha

abla J$(\backslash$theta$\_\{\backslash$text$\{$old$\}\})]$

Where:

$(\backslash$alpha$)($alpha$)$ is the learning rate, a hyperparameter that controls the step size during each iteration.

$($

abla J$(\backslash$theta$\_\{\backslash$text$\{$old$\}\}))$ is the gradient of the cost function with respect to $(\backslash$theta$)$ at the current parameter values.

Effect of Learning Rate

Small Learning Rate $((\backslash$alpha$))$:

Pros: Smaller steps lead to more precise convergence.

Cons: Convergence can be slow, especially in deep learning models. It might get stuck in local minima.

Recommendation: Use a small learning rate when you have time for slow convergence and want precision.

Large Learning Rate $((\backslash$alpha$))$:

Pros: Faster convergence.

Cons: Risk of overshooting the minimum $($diverging$).$ Might miss the optimal solution.

Recommendation: Use a large learning rate when you want faster convergence but monitor for divergence.

Convex vs$.$ Non$-$Convex Cost Functions

Convex Cost Function:

Pros: Has a single global minimum. Gradient Descent is guaranteed to converge to this minimum.

Example: Quadratic functions.

Recommendation: Any reasonable learning rate works well.

Non$-$Convex Cost Function:

Pros: Multiple local minima, saddle points, and plateaus.

Cons: Gradient Descent can get stuck in local minima.

Recommendation:

Initialize from different starting points.

Use a learning rate that balances exploration and exploitation.

Consider using advanced optimization techniques $($e$.$g$.,$ Adam, RMSProp$).$

Choosing a Good Learning Rate

Grid Search:

Try a range of learning rates and evaluate performance on a validation set.

Time$-$consuming but effective.

Learning Rate Schedules:

Start with a large learning rate and gradually reduce it during training.

Common schedules: Step decay, exponential decay, or $1/$t decay.

Adaptive Methods:

Algorithms like Adam and RMSProp adaptively adjust the learning rate based on past gradients.

Often perform well without manual tuning.