【Fundamentals】 Detailed Explanation of Gradient Descent Algorithm and MATLAB Code

# 1. Gradient Descent Algorithm Overview**

The gradient descent algorithm is an iterative optimization technique used to find the local minimum of a function. It updates parameters iteratively by moving along the direction of the negative gradient of the function, thereby gradually approaching the optimal solution. Gradient descent is widely applied in machine learning and deep learning because it effectively optimizes complex nonlinear functions.

# 2. Principles of the Gradient Descent Algorithm

### 2.1 Concept and Calculation of Gradient

**Concept of Gradient**

The gradient is a vector that represents the rate of change of a function at a certain point. For a multivariate function `f(x1, x2, ..., xn)`, its gradient at the point `(x1, x2, ..., xn)` is:

∇f(x1, x2, ..., xn) = [∂f/∂x1, ∂f/∂x2, ..., ∂f/∂xn]

Where `∂f/∂xi` is the partial derivative of function `f` with respect to variable `xi`.

**Calculation of Gradient**

The gradient can be calculated using the following methods:

- **Analytical Method:** Directly compute the partial derivatives of the function.
- **Numerical Method:** Approximate the partial derivatives using finite differences or other numerical methods.

### 2.2 Mathematical Principles of Gradient Descent Algorithm

The gradient descent algorithm is an iterative algorithm for finding the local minimum of a function. It starts from an initial point and then iteratively updates the position of the point along the negative direction of the function's gradient until it reaches the local minimum.

**Mathematical Principle**

The mathematical principle of the gradient descent algorithm is as follows:

x_new = x_old - α * ∇f(x_old)

Where:

- `x_old` is the current point.
- `x_new` is the updated point.
- `α` is the learning rate, which controls the step size.
- `∇f(x_old)` is the gradient of the current point.

**Learning Rate**

The learning rate `α` is an important parameter in the gradient descent algorithm. It controls the step size and affects the convergence speed and accuracy of the algorithm. Too large a learning rate can cause instability in the algorithm, while too small a rate can result in slow convergence.

### 2.3 Variants of the Gradient Descent Algorithm

The standard gradient descent algorithm has some drawbacks, such as slow convergence and the tendency to get stuck in local minima. To address these issues, several variants of the gradient descent algorithm have been proposed:

**Momentum Gradient Descent Algorithm**

The momentum gradient descent algorithm accelerates convergence by introducing a momentum term. The momentum term records the historical changes of the gradient and adds it to the current gradient, thus allowing the algorithm to take larger steps in the direction of convergence.

**RMSprop Algorithm**

The RMSprop algorithm improves convergence speed and stability by adaptively adjusting the learning rate. It calculates the root mean square (RMS) of the gradients and uses it to adjust the learning rate.

**Adam Algorithm**

The Adam algorithm combines the advantages of momentum and RMSprop, making it an efficient and robust variant of the gradient descent algorithm. It uses momentum and adaptive learning rates and performs well in various machine learning tasks.

# 3. Implementing Gradient Descent in MATLAB

### 3.1 MATLAB Functions for Gradient Descent Algorithm

MATLAB provides various functions to implement the gradient descent algorithm, the most common being the `fminunc` function. The `fminunc` function is an unconstrained optimization function that minimizes a scalar function using quasi-Newton methods.

The syntax for `fminunc` is:

x = fminunc(fun, x0, options)

Where:

* `fun` is the handle to the scalar function to be minimized.