Distributed Variable Sample-Size Gradient-Response and Best-Response Schemes for Stochastic Nash Equilibrium Problems

Abstract

This paper considers an $n$-player stochastic Nash equilibrium problem (NEP) in which the $i$th player minimizes a composite objective $f_i( \bullet ,x_{-i}) + r_i( \bullet )$, where $f_i$ is an expectation-valued smooth function, $x_{-i}$ is a tuple of rival decisions, and $r_i$ is a nonsmooth convex function with an efficient prox-evaluation. In this context, we make the following contributions. (I) Under suitable monotonicity assumptions on the \redpseudogradient map, we derive optimal rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate with degree $v > 0$, the mean-squared error of the iterates decays at a corresponding polynomial rate; in particular, we prove that the iteration and oracle complexities to obtain an $\epsilon$-Nash equilibrium ($\epsilon$-NE) are $\mathcal{O}(1/\epsilon^{1/v})$ and $\mathcal{O}(1/\epsilon^{1+1/v})$, respectively. \redWhen the sample-size is held constant, the iterates converge geometrically to a neighborhood of the Nash equilibrium in an expected-value sense. (II) We then overlay \bf VS-PGR with a consensus phase with a view towards developing distributed protocols for aggregative stochastic NEPs. In the resulting \bf d-VS-PGR scheme, when the sample-size at each iteration grows at a geometric rate while the communication rounds per iteration grow at the rate of $ k+1 $, computing an $\epsilon$-NE requires similar iteration and oracle complexities to \bf VS-PGR with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$. Notably, (I) and (II) rely on weaker oracle assumptions in that the conditionally unbiasedness assumption is relaxed while the bound on the conditional second moment may be state-dependent. (III) Under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where each player solves a sample-average BR problem. When the sample-size increases at a suitable geometric rate, the resulting iterates converge at a geometric rate while the iteration and oracle complexity are, respectively, $\mathcal{O}(\ln(1/\epsilon))$ and $\mathcal{O}(1/\epsilon)$. If the sample-size increases at a polynomial rate with degree $v$, the mean-squared error decays at a corresponding polynomial rate while the iteration and oracle complexities are $\mathcal{O}(1/\epsilon^{1/v})$ and $\mathcal{O}(1/\epsilon^{1+1/v})$, respectively. (IV) Akin to (II), the distributed variant \bf d-VS-PBR achieves similar iteration and oracle complexities to the centralized VS-PBR with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$ when the communication rounds per iteration increase at the rate of $ k+1 $. Finally, we present preliminary numerics to provide empirical support for the rate and complexity statements.