A Gentle Introduction to Gradient-Based Optimization and Variational Inequalities for Machine Learning

TL;DR

This paper introduces gradient-based optimization methods extended to variational inequalities and game-theoretic problems in machine learning, emphasizing intuition and convergence in complex decision-making scenarios.

Contribution

It broadens the scope of gradient-based algorithms to include variational inequalities and game theory, providing motivation and convergence insights.

Findings

Convergence proofs for several algorithms.

Extension of gradient methods to saddle points and monotone games.

Intuitive explanations for complex optimization frameworks.

Abstract

The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradient-based methods remain essential -- given the high dimensionality and large scale of machine-learning problems -- but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variational inequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition.