【Comparison Between SGD and BGD】: Comparison and Selection of Stochastic Gradient Descent and Batch Gradient Descent

# 1. Introduction: Comparing SGD and BGD

In the realm of machine learning, optimization algorithms play a pivotal role in the training and performance of models. Gradient descent is one of the commonly used optimization methods, and among them, Batch Gradient Descent (BGD) and Stochastic Gradient Descent (SGD) are two typical representatives. This chapter will introduce the comparison of these two methods, helping readers to better understand their similarities and differences, and to choose between them in practical applications.

Understanding when to use BGD and when to use SGD is crucial for achieving good training results. Next, we will delve into BGD and SGD, helping readers to fully understand their principles, pros and cons, and applicable scenarios, so that they can be better applied to actual machine learning tasks.

# 2.1 Overview and Principle Analysis of BGD
Batch Gradient Descent (BGD) is an optimization algorithm used to find the minimum value of a function, especially for training machine learning models. In this section, we will explore the overview and principles of BGD in depth.

### 2.1.1 What is Gradient Descent?
Gradient descent is an optimization algorithm that iteratively reduces the numerical value of the objective function. It uses the gradient information of the objective function to guide the search direction, thereby finding the minimum value of the function.

### 2.1.2 Principles of the Batch Gradient Descent Algorithm
The core idea of BGD is to use the gradient of all samples when updating model parameters to calculate the adjustment amount for the parameters. Specifically, for model parameter **θ**, the update formula is as follows:

θ = θ - α * ∇J(θ)

Where, α represents the learning rate, and ∇J(θ) represents the gradient of the loss function J(θ) with respect to θ.

### 2.1.3 The Relationship Between BGD and the Method of Least Squares
BGD is closely related to the method of least squares. In the method of least squares, model parameters are solved by minimizing the sum of squared errors between actual and predicted values. BGD can be considered as a numerical optimization algorithm and is one of the common methods for solving parameters in the method of least squares.

## 2.2 Analysis of Advantages and Disadvantages of BGD
In practice, BGD, as a classic optimization algorithm, has certain advantages and disadvantages. We will analyze them in detail next.

### 2.2.1 Advantages: Guarantee of Global Optimum
Since BGD uses all data samples for gradient computation, it can guarantee convergence to a global optimum under reasonable conditions, especially for optimization problems of convex functions.

### 2.2.2 Disadvantages: Large Computation and Slow Convergence
Although BGD can converge globally, in the case of large datasets, computing the gradient of all samples leads to large computation and slow convergence, especially in high-dimensional feature spaces.

### 2.2.3 The Application of BGD on Large Datasets
On large datasets, the disadvantages of BGD become more pronounced, with long computation times and low efficiency. Therefore, in scenarios with large datasets, optimization algorithms such as Stochastic Gradient Descent (SGD) are usually considered to speed up training.

With the introduction of the above sections, we have a preliminary understanding of the concept, principles, and pros and cons of BGD. In the following sections, we will delve into the Stochastic Gradient Descent (SGD) algorithm to further complete our understanding of different gradient descent algorithms.

# 3. In-depth Understanding of Stochastic Gradient Descent (SGD)

Stochastic Gradient Descent (SGD) is an optimization algorithm that, compared to Batch Gradient Descent (BGD), is more suitable for large datasets. The following will delve into the principles, pros and cons analysis, and comparison with Mini-batch gradient descent of SGD.

### 3.1 Overview and Principle Analysis of SGD

#### 3.1.1 What is Stochastic Gradient Descent?

Stochastic Gradient Descent is an optimization method that updates parameters using only one sample per iteration, estimating the overall gradient descent direction by randomly selecting small batches of data, ultimately finding the optimal solution.

#### 3.1.2 Principles of the Stochastic Gradient Descent Algorithm

- Initialize model parameters
- Randomly select a sample
- Calculate the gradient of the sample
- Update model parameters based on the gradient
- Repeat the above steps until convergence conditions are met

#### 3.1.3 Comparison Between SGD and Mini-batch Gradient Descent

SGD is similar to Mini-batch gradient descent, the difference being that Mini-batch selects a small portion of data for gradient computation, while SGD selects only one data point each time. The advantage of SGD is that each iteration is fast, making it suitable for large datasets.

### 3.2 Analysis of Advantages and Disadvantages of SGD

#### 3.2.1 Advantages: Fast Computation and Suitability for Large Datasets

- Advantage One: Since SGD only needs to compute one sample per iteration, it is faster.
- Advantage Two: For large datasets, SGD is computationally efficient and can find local optima more quickly.

#### 3.2.2 Disadvantages: Unstable Convergence and Likelihood of Local Optima

- Disadvantage One: Due to using only one sample per iteration, SGD has a high degree of randomness in the update directi