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.

翻译：近年来机器学习领域的快速发展，在很大程度上得益于其与基于梯度的优化方法之间卓有成效的联系。该领域的进一步发展，部分取决于研究重点从模式识别向决策制定及多智能体问题的转变。在这些更广泛的场景中，涉及均衡与博弈论而非最优性的新数学挑战随之出现。考虑到机器学习问题的高维性与大规模特性，基于梯度的方法仍至关重要——但简单的梯度下降法已不再是算法设计的出发点。我们提供了关于机器学习中基于梯度的算法更广泛框架的入门介绍，从鞍点与单调博弈开始，逐步过渡到一般变分不等式。虽然我们为所介绍的若干算法提供了收敛性证明，但主要侧重点在于提供动机与直观理解。