基于梯度的三种优化方法及比较
Three Gradient-Based Optimization Methods and Their Comparison

近年来，高速发展的科学技术将人工智能带入大众的视野。机器学习和深度学习作为人工智能的核心技术，引起了众多学者的关注。在机器学习领域，由于目标模型损失函数的复杂性，使得无法快速有效地得到参数估计的表达式，因此，运用基于梯度的优化方法求解该类优化问题很受欢迎，到目前为止最主流的一个算法就是梯度下降法，但在实际应用中，随着数据规模越来越大，传统的梯度下降法训练的过程及其缓慢，已不能够快速有效的解决大规模机器学习问题。所以，在梯度下降方法的基础上进行了改进，提出了随机梯度下降算法。随机方法因其良好的标度特性在大规模应用问题中受到青睐，本文首先详细介绍了梯度下降法、随机梯度下降法及小批量随机梯度下降法三个方法基本思想及其求解最优化问题的具体过程，然后设计数值例子进行模拟实验，并比较三种方法的优劣性，最后通过实验结果得出结论：梯度下降法收敛性较好，但计算效率低，随机梯度下降法计算效率高，而小批量梯度下降法则是介于二者之间。因此在计算大规模的问题时，随机梯度下降法相较于另外两种方法更为有效。
In recent years, the rapid development of science and technology has brought artificial intelligence into the public eye. Machine learning and deep learning, as the core technologies of artificial intel-ligence, have attracted considerable attention from numerous scholars. In the field of machine learning, the complexity of the loss function for target models makes it challenging to obtain a rapid and effective expression for parameter estimation. Therefore, the application of gradient-based op-timization methods to solve such optimization problems has become popular. Up to now, the most mainstream algorithm is the gradient descent method. However, in practical applications, as data scales increase, the traditional gradient descent method and its slow training process become inef-ficient in addressing large-scale machine learning problems. Therefore, improvements have been made based on the gradient descent method, leading to the introduction of the stochastic gradient descent algorithm. Stochastic methods, due to their favorable scaling characteristics, have gained popularity in large-scale application problems. This paper first provides a detailed introduction to the basic ideas and specific processes of three methods: gradient descent, stochastic gradient de-scent, and mini-batch stochastic gradient descent, for solving optimization problems. Subsequently, numerical examples are designed for simulation experiments, comparing the strengths and weak-nesses of the three methods. The conclusions drawn from the experimental results are as follows: gradient descent exhibits good convergence but low computational efficiency, stochastic gradient descent has high computational efficiency, and mini-batch gradient descent falls between the two. Therefore, when dealing with large-scale computational problems, stochastic gradient descent is more effective compared to the other two methods.