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：近年来机器学习的快速发展基于其与基于梯度的优化之间高度有效的联系。进一步的进展部分取决于关注点从模式识别转向决策制定和多智能体问题。在这些更广泛的背景下，出现了涉及均衡和博弈论而非最优性的新数学挑战。考虑到机器学习问题的高维度和大规模特性，基于梯度的方法仍然至关重要——但简单的梯度下降已不再是算法设计的出发点。我们提供了机器学习中基于梯度算法的更广泛框架的简明导论，从鞍点和单调博弈开始，进而推广到一般变分不等式。虽然我们给出了所介绍的几种算法的收敛性证明，但我们的主要侧重点是提供动机和直观理解。