Stochastic mirror descent dynamics and their convergence in monotone variational inequalities

Panayotis Mertikopoulos (1) , Mathias Staudigl (2)

1 POLARIS - Performance analysis and optimization of LARge Infrastructures and Systems
2 Maastricht University [Maastricht]

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 sto- chastic 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.

Domaines: Optimisation et contrôle [math.OC]

Panayotis Mertikopoulos, Mathias Staudigl. Stochastic mirror descent dynamics and their convergence in monotone variational inequalities. Journal of Optimization Theory and Applications, 2018, 179 (3), pp.838-867. ⟨10.1007/s10957-018-1346-x⟩. ⟨hal-01643343⟩