On the computation of equilibria in monotone and potential stochastic hierarchical games

Abstract

We consider a class of hierarchical noncooperative N−player games where the ith player solves a parametrized mathematical program with equilibrium constraints (MPEC) with the caveat that the implicit form of the ith player's in MPEC is convex in player strategy, given rival decisions. This represents a challenging class of games that subsumes multi-leader multifollower games in which player-specific problems are convex in an implicit sense, given rival decisions. We consider settings where player playoffs are expectation-valued with lower-level equilibrium constraints imposed in an almost-sure sense; i.e. player problems are parametrized stochastic MPECs. Few, if any, general purpose schemes exist for computing equilibria even for deterministic specializations of such games. We develop computational schemes in two distinct regimes: (a) Monotone regimes. When player-specific implicit problems are convex, then the necessary and sufficient equilibrium conditions are given by a stochastic inclusion. Under a monotonicity assumption on the operator, we develop a variance-reduced stochastic proximalpoint scheme that achieves deterministic rates of convergence in terms of solving proximalpoint problems in both monotone and strongly monotone regimes. In addition, the schemes are characterized by either optimal or near-optimal oracle (or sample) complexity guarantees. Finally, the generated sequences are shown to be convergent to a solution in an almost-sure sense in both monotone and strongly monotone regimes. Notably, the schemes do not impose any smoothness assumptions on player problems and allow for state-dependence in noise; (b) Potentiality. When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework. However, any such avenue is impeded by the need to efficiently compute an approximate solution of an MPEC with a strongly convex implicit objective. To this end, we consider the smoothed counterpart of this game where each player's problem is smoothed via randomized smoothing. Notably, under suitable assumptions, we show that an η-smoothed game admits an η-approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence that converges almost surely to an η-approximate Nash equilibrium in both relaxed and unrelaxed settings. This scheme is reliant on computing the proximal problem, a stochastic MPEC whose implicit form has a strongly convex objective, with increasing accuracy in finite-time. The smoothing framework allows for developing a variance-reduced zeroth-order scheme for such problems that admits a fast rate of convergence. Numerical studies on a class of multi-leader multi-follower games suggest that variance-reduced proximal schemes provide significantly better accuracy with far lower run-times. The relaxed best-response scheme scales well will problem size and generally displays more stability than its unrelaxed counterpart.

Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaces

“…If k = 0 and k = 1 the above scheme reduces to the stochastic forward-backwardforward method developed in [26,27], with important applications in Gaussian communication networks [16] and dynamic user equilibrium problems [28]. However, even more connections to existing methods can be made.…”

Stochastic Relaxed Inertial Forward-Backward-Forward splitting for Monotone Inclusions in Hilbert spaces

“…If α k = 0 and ρ k = 1 the above scheme reduces to the stochastic forward-backward-forward method developed in [15,24], with important applications in Gaussian communication networks [80] and dynamic user equilibrium problems [82]. However, even more connections to existing methods can be made.…”

“…Rate analysis for stochastic extragradient (SEG) have led to optimal rates for Lipschitz and monotone operators [50], as well as extensions to non-Lipschitzian [87] and pseudomonotone settings [45,51]. To alleviate the computational complexity single-projection schemes, such as the stochastic forward-backward-forward (SFBF) method [15,24], as well as subgradient-extragradient and projected reflected algorithms [25] have been studied as well.…”

“…In terms of run-time, improvements in iteration complexities achieved by mini-batch approaches are significant; e.g. in strongly monotone regimes, the iteration complexity improves from O( 1 ) to O(ln( 1 )) [24,25]. Beyond run-time advantages, such avenues provide asymptotic and rate guarantees under possibly weaker assumptions on the problem as well as the oracle; in particular, mini-batch schemes allow for possibly biased oracles and state-dependency of the noise [25].…”

“…In a general structured monotone inclusion setting [73] derive rate statements for cocoercive mean operators in the context of forward-backward splitting methods. More recently, Cui and Shanbhag [24] provide linear and sublinear rates of convergence for a variance-reduced inexact proximal-point scheme for both strongly monotone and monotone inclusion problems. However, to the best of our knowledge, our results are the first published for a stochastic operator splitting algorithm, featuring relaxation and inertial effects.…”