Abstract
We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation, driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system’s controllable parameters are two variable weight sequences, that, respectively, pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle, showing that individual trajectories exhibit exponential concentration around this average.
Highlights
Dynamical systems governed by monotone operators play an important role in the fields of optimization, game theory (Nash equilibrium and generalized Nash equilibrium problems), fixed point theory, partial differential equations and many other fields of applied mathematics
A classical example of this arises in the study of gradient descent dynamics and its connection with Cauchy’s steepest descent algorithm—or, more generally, in the relation between the mirror descent (MD) class of algorithms [3] and dynamical systems derived from Bregman projections and Hessian Riemannian metrics [4,5,6]
We provide below a “large deviations” bound that shows that the ergodic gap process g(X (t)) is exponentially concentrated around its mean value: Theorem 4.3 Suppose (H1)–(H3) hold, and that (SMD) is started from the initial condition (s, y) = (0, 0)
Summary
Dynamical systems governed by monotone operators play an important role in the fields of optimization (convex programming), game theory (Nash equilibrium and generalized Nash equilibrium problems), fixed point theory, partial differential equations and many other fields of applied mathematics. The study of the relationship between continuous- and discrete-time models has given rise to a vigor-. The starting point of much of this literature is that an iterative algorithm can be seen as a discretization of a continuous dynamical system. A classical example of this arises in the study of (projected) gradient descent dynamics and its connection with Cauchy’s steepest descent algorithm—or, more generally, in the relation between the mirror descent (MD) class of algorithms [3] and dynamical systems derived from Bregman projections and Hessian Riemannian metrics [4,5,6]
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