Gradient-Based Optimization
In this chapter, you'll be introduced to:
- Gradient Descent Algorithm: Understanding how to use gradient descent for optimizing the mean squared error loss in linear regression.
- Gradient Computation: How to compute gradients for the loss function efficiently.
- Learning Rate Selection: Strategies for choosing appropriate learning rates and their impact on optimization.
- Initialization and Termination Criteria: Best practices for starting and stopping the optimization process.
- Second-Order Methods: An introduction to Newton's method and its comparison with gradient descent.
- Mini-Batch and Stochastic Gradient Descent: Exploring variations of gradient descent suitable for large datasets.
- Practical Considerations: Implementing adaptive learning rates, momentum, and addressing challenges in optimization.
In previous chapters, we discussed linear regression and derived a closed-form solution for minimizing the mean squared error (MSE) loss function. While closed-form solutions are precise and computationally straightforward for small problems, they become impractical for larger, more complex models due to computational inefficiency and numerical instability. Additionally, many advanced models, such as neural networks, do not have closed-form solutions.
This chapter focuses on gradient-based optimization methods, particularly gradient descent, which iteratively adjust model parameters to minimize the loss function. These methods are widely used in machine learning because they scale well with data size and model complexity.
Gradient Descent Algorithm
Basic Idea
Gradient descent is an optimization algorithm that finds the minimum of a function by iteratively moving in the direction of the steepest descent, as defined by the negative of the gradient.
- Gradient: Represents the direction and rate of the steepest increase of the function.
- Steepest Descent: Moving opposite to the gradient decreases the function's value most rapidly.
Mathematical Formulation
Given a differentiable loss function , the update rule for gradient descent is:
- : Parameter vector at iteration .
- : Learning rate (step size).
- : Gradient of the loss function at .
Algorithm Steps
- Initialization:
- Start with an initial guess , which can be zeros or small random values.
- Iteration:
- For each iteration :
- Compute the gradient .
- Update the parameters using the update rule.
- For each iteration :
- Termination:
- Stop when a stopping criterion is met (e.g., a maximum number of iterations, minimal change in loss function, or gradient norm below a threshold).