Randomized Feasibility-Update Algorithms for Stochastic Variational Inequality Problems
Abhishek Chakraborty, Angelia Nedić
TL;DR
The paper tackles stochastic monotone variational inequalities with constraint sets defined as infinite intersections of convex level sets, where full projection is impractical. It develops randomized feasibility updates and two modified projection-free schemes, Korpelevich and Popov, to solve SVIs while analyzing convergence via a modified dual gap and feasible-set progress. The authors prove an O(1/√T) convergence rate for the averaged iterates under standard assumptions, with the infeasibility gap decaying geometrically as the inner feasibility sample size grows. They also demonstrate favorable runtime performance against primal–dual and ADMM-based methods in a zero-sum matrix game, highlighting practical viability for large-scale or infinitely constrained problems. The work advances projection-free, scalable methods for SVIs with potentially infinite constraint sets and provides insights into parameter-free step-size choices and averaging schemes.
Abstract
This paper considers stochastic monotone variational inequalities whose feasible region is the intersection of a (possibly infinite) number of convex functional level sets. A projection-based approach or direct Lagrangian-based techniques for such problems can be computationally expensive if not impossible to implement. To deal with the problem, we consider randomized methods that avoid the projection step on the whole constraint set by employing random feasibility updates. In particular, we propose and analyze modified stochastic Korpelevich and Popov methods for solving monotone stochastic VIs. We introduce a modified dual gap function and prove the convergence rates with respect to this function. We illustrate the performance of the methods in simulations on a zero-sum matrix game.