Inexact best-response schemes for stochastic Nash games: Linear convergence and Iteration complexity analysis

2016 ◽  
Author(s):  
Uday V. Shanbhag ◽  
Jong-Shi Pang ◽  
Suvrajeet Sen
Keyword(s):  
Complexity Analysis ◽  
Linear Convergence ◽  
Iteration Complexity ◽  
Nash Games ◽  
Best Response ◽  
Stochastic Nash Games

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

2020 ◽  
Vol 45 (1) ◽  
pp. 157-190 ◽  
Author(s):  
Jinlong Lei ◽  
Uday V. Shanbhag ◽  
Jong-Shi Pang ◽  
Suvrajeet Sen
Keyword(s):  
Optimization Problems ◽  
Almost Sure Convergence ◽  
Iteration Complexity ◽  
Two Stage ◽  
Nash Game ◽  
Nash Games ◽  
Mean Convergence ◽  
Best Response ◽  
Update Frequency ◽  
Stochastic Nash Games

In this paper, we consider a stochastic Nash game in which each player minimizes a parameterized expectation-valued convex objective function. In deterministic regimes, proximal best-response (BR) schemes have been shown to be convergent under a suitable spectral property associated with the proximal BR map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at each iteration. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the BR problem is computed in an expected-value sense via a stochastic approximation (SA) scheme. On the basis of this framework, we present three inexact BR schemes: (i) First, we propose a synchronous inexact BR scheme where all players simultaneously update their strategies. (ii) Second, we extend this to a randomized setting where a subset of players is randomly chosen to update their strategies while the other players keep their strategies invariant. (iii) Third, we propose an asynchronous scheme, where each player chooses its update frequency while using outdated rival-specific data in updating its strategy. Under a suitable contractive property on the proximal BR map, we proceed to derive almost sure convergence of the iterates to the Nash equilibrium (NE) for (i) and (ii) and mean convergence for (i)–(iii). In addition, we show that for (i)–(iii), the generated iterates converge to the unique equilibrium in mean at a linear rate with a prescribed constant rather than a sublinear rate. Finally, we establish the overall iteration complexity of the scheme in terms of projected stochastic gradient (SG) steps for computing an [Formula: see text]-NE2 (or [Formula: see text]-NE∞) and note that in all settings, the iteration complexity is [Formula: see text], where [Formula: see text] in the context of (i), and c > 0 represents the positive cost of randomization in (ii) and asynchronicity and delay in (iii). Notably, in the synchronous regime, we achieve a near-optimal rate from the standpoint of solving stochastic convex optimization problems by SA schemes. The schemes are further extended to settings where players solve two-stage stochastic Nash games with linear and quadratic recourse. Finally, preliminary numerics developed on a multiportfolio investment problem and a two-stage capacity expansion game support the rate and complexity statements.

scholarly journals A Fast Alternating Minimization Algorithm for Nonlocal Vectorial Total Variational Multichannel Image Denoising

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Rubing Xi ◽  
Zhengming Wang ◽  
Xia Zhao ◽  
Meihua Xie
Keyword(s):  
Complexity Analysis ◽  
Iterative Algorithms ◽  
Linear Convergence ◽  
Convergence Properties ◽  
Theoretical Derivation ◽  
Variational Models ◽  
Fixed Point Algorithm ◽  
Alternating Minimization Algorithm ◽  
Uniqueness Of The Solution ◽  
Nonlocal Regularization

The variational models with nonlocal regularization offer superior image restoration quality over traditional method. But the processing speed remains a bottleneck due to the calculation quantity brought by the recent iterative algorithms. In this paper, a fast algorithm is proposed to restore the multichannel image in the presence of additive Gaussian noise by minimizing an energy function consisting of anl2-norm fidelity term and a nonlocal vectorial total variational regularization term. This algorithm is based on the variable splitting and penalty techniques in optimization. Following our previous work on the proof of the existence and the uniqueness of the solution of the model, we establish and prove the convergence properties of this algorithm, which are the finite convergence for some variables and theq-linear convergence for the rest. Experiments show that this model has a fabulous texture-preserving property in restoring color images. Both the theoretical derivation of the computation complexity analysis and the experimental results show that the proposed algorithm performs favorably in comparison to the widely used fixed point algorithm.

Accelerated Iterations for Finding the Soft-constrained Stochastic Nash Games Equilibrium

2010 ◽  
Author(s):  
I. G. Ivanov ◽  
R. I. Rusinova ◽  
B. M. Lomev ◽  
Michail D. Todorov ◽  
Christo I. Christov
Keyword(s):  
Nash Games ◽  
Stochastic Nash Games
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