Distributed Computation of Nash Equilibria for Monotone Aggregative Games via Iterative Regularization

Abstract

We consider a class of N -player nonsmooth aggregative games over networks in stochastic regimes, where the ith player is characterized by a composite cost function f i +d i +r i , f i is a smooth expectationvalued function dependent on its own strategy and an aggregate function of rival strategies, d i is a convex hierarchical term dependent on its strategy, and r i is a nonsmooth convex function of its strategy with an efficient prox-evaluation. Though the aggregate is unknown to any given player, each player can interact with its neighbors to construct an estimate of the aggregate. We design a fully distributed iterative proximal stochastic gradient method overlaid by a Tikhonov regularization, where each player may independently choose its steplengths and regularization parameters while meeting some coordination requirements. Under a monotonicity assumption on the concatenated player-specific gradient mapping, we prove that the generated sequence converges almost surely to the least-norm Nash equilibrium (i.e., a Nash equilibrium with the smallest two-norm). In addition, for the case with each r i being an indicator function of a compact convex set, we establish the convergence rate associated with the expected gap function at the time-averaged sequence. Furthermore, we consider the extension to the private hierarchical regime where each player is a leader with respect to a collection of private followers competing in a strongly monotone game, parametrized by leader decisions. By leveraging a convolutionsmoothing framework, we present amongst the first fully distributed schemes for computing a Nash equilibrium of a game complicated by such a hierarchical structure. Based on this framework, we extend the rate statements to accommodate the computation of a hierarchical stochastic Nash equilibrium by using a Fitzpatrick gap function. Notably, both sets of fully distributed schemes display near-optimal sample-complexities, suggesting that this hierarchical structure does not lead to performance degradation. Finally, we validate the proposed methods on a networked Nash-Cournot equilibrium problem and a hierarchical generalization.

Convergence of sequences: a survey

“…A particular case of Lemma 3.11 is proposed in [44] as a consequence of [95,Theorem 3.3.1]. Let us note that the assumptions in the following result imply those in Lemma 3.11 which is, in turn, more general.…”

“…Let us note that the assumptions in the following result imply those in Lemma 3.11 which is, in turn, more general. Corollary 3.12 (Proposition 3, [44]). Let (v k ) k∈N be a nonnegative sequence such that…”

“…Lemma 3.16 is used also for a variational problem in [49], along with Lemma 3.10. Moving to Nash equilibrium problems, Lemma 3.8 is used in [41,101] while Lemma 3.11 is used in [44].…”

Dynamic coarse‐to‐fine ISAR image blind denoising using active joint prior learning

“…To achieve better denoising effect, the above denoising procedures is iterated for several times. In the th iteration, the residual of the reconstructed image is added to the ()th iteration using the iterative regularization strategy 26 . The noise standard deviation in the th iteration is adjusted as , where is a constant.…”

“…In the tth iteration, the residual of the reconstructed image x t ( −1) is added to the (t − 1)th iteration using the iterative regularization strategy. 26 The noise standard deviation in the tth iteration is adjusted as σ η σ…”