Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes

Abstract

This paper considers an N -player stochastic Nash game in which the ith player minimizes a composite objective fi(x) + ri(xi), where fi is expectation-valued and ri has an efficient prox-evaluation. In this context, we make the following contributions. (i) Under a strong monotonicity assumption on the concatenated gradient map, we derive (optimal) rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. Notably, when the samplesize and the number of consensus steps at each iteration grow at a suitable rate, a linear rate of convergence can be achieved; (iii) Finally, 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 the proximal BR is computed by solving a sample-average problem. If the batch-size for computing the sample-average is raised at a suitable rate, we show that the resulting iterates converge at a linear rate and derive the oracle complexity.

A fully-distributed proximal-point algorithm for Nash equilibrium seeking with linear convergence rate

“…Similarly, the proximal best-response dynamics proposed in [21] for stochastic games require an increasing number of data transmissions per iteration. Contribution: Motivated by the above, in this paper we further exploit the restricted monotonicity property used in [16]- [18] to solve Nash equilibrium problems under partialdecision information.…”

Distributed Variable Sample-Size Gradient-response and Best-response Schemes for Stochastic Nash Equilibrium Problems over Graphs

“…This implies (27) by the definition of C 3 in (28) and β ∈ (0, 1). 2 Based on the recursion (27) in Prop.…”

“…This implies (27) by the definition of C 3 in (28) and β ∈ (0, 1). 2 Based on the recursion (27) in Prop. 2 and by using Lemma 3, we may obtain the linear convergence of Alg.…”

Stability Analysis of Nash Equilibrium in Loss-Aversion-Based Noncooperative Dynamical Systems

“…These authors extended this work to the partial-information scenario by modifying the pseudo-gradient dynamics with an additional stabilizing component that essentially ensures innerloop passivity from input to a predicted state [20]. In terms of the the scenario with stochastic payoff functions, Lei and Shanbhag showed the convergence to a Nash equilibrium in [21]. The applications of game theoretic approach inspired by the pseudo-gradient dynamics are found in numerous fields, e.g., communication networks [22], [23], incentive schemes [24], pricing mechanisms [25], [26], to name but a few.…”