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The rapid progress in machine learning in recent years has been based on ahighly productive connection to gradient-based optimization. Further progresshinges in part on a shift in focus from pattern recognition to decision-makingand multi-agent problems. In these broader settings, new mathematicalchallenges emerge that involve equilibria and game theory instead of optima.Gradient-based methods remain essential -- given the high dimensionality andlarge scale of machine-learning problems -- but simple gradient descent is nolonger the point of departure for algorithm design. We provide a gentleintroduction to a broader framework for gradient-based algorithms in machinelearning, beginning with saddle points and monotone games, and proceeding togeneral variational inequalities. While we provide convergence proofs forseveral of the algorithms that we present, our main focus is that of providingmotivation and intuition.

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