Gradient Descent, Stochastic Optimization, and Other Tales
Jun Lu
TL;DR
The paper demystifies gradient-based optimization by grounding it in precise mathematics and geometry. It surveys vanilla gradient descent, stochastic variants, and line-search strategies, deriving update rules and convergence properties for convex and quadratic settings, and then extends to learning-rate schedules, momentum, and modern adaptive methods (AdaGrad, RMSProp, AdaDelta, AdaSmooth, Adam, Nadam). It culminates with second-order approaches (Newton, damped Newton, Levenberg–Marquardt) and conjugate-gradient methods, including preconditioning to address ill-conditioning. Collectively, the work provides a self-contained, rigorous toolkit for understanding when and how to apply these methods in ML tasks, with practical guidance and illustrative analyses of convergence rates and saddle-point behavior. The inclusion of appendices on Taylor expansion and Cholesky decomposition reinforces the theoretical foundations and practical computational strategies for quadratic and nonlinear optimization.
Abstract
The goal of this paper is to debunk and dispel the magic behind black-box optimizers and stochastic optimizers. It aims to build a solid foundation on how and why the techniques work. This manuscript crystallizes this knowledge by deriving from simple intuitions, the mathematics behind the strategies. This tutorial doesn't shy away from addressing both the formal and informal aspects of gradient descent and stochastic optimization methods. By doing so, it hopes to provide readers with a deeper understanding of these techniques as well as the when, the how and the why of applying these algorithms. Gradient descent is one of the most popular algorithms to perform optimization and by far the most common way to optimize machine learning tasks. Its stochastic version receives attention in recent years, and this is particularly true for optimizing deep neural networks. In deep neural networks, the gradient followed by a single sample or a batch of samples is employed to save computational resources and escape from saddle points. In 1951, Robbins and Monro published \textit{A stochastic approximation method}, one of the first modern treatments on stochastic optimization that estimates local gradients with a new batch of samples. And now, stochastic optimization has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this article is to give a self-contained introduction to concepts and mathematical tools in gradient descent and stochastic optimization.