  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.
