TY - GEN
T1 - Single timescale regularized stochastic approximation schemes for monotone nash games under uncertainty
AU - Koshal, Jayash
AU - Nedić, Angelia
AU - Shanbhag, Uday V.
PY - 2010
Y1 - 2010
N2 - In this paper, we consider the distributed computation of equilibria arising in monotone stochastic Nash games over continuous strategy sets. Such games arise in settings when the gradient map of the player objectives is a monotone mapping over the cartesian product of strategy sets, leading to a monotone stochastic variational inequality. We consider the application of projection-based stochastic approximation (SA) schemes. However, such techniques are characterized by a key shortcoming: in their traditional form, they can only accommodate strongly monotone mappings. In fact, standard extensions of SA schemes for merely monotone mappings require the solution of a sequence of related strongly monotone problems, a natively two-timescale scheme. Accordingly, we consider the development of single timescale techniques for computing equilibria when the associated gradient map does not admit strong monotonicity. We first show that, under suitable assumptions, standard projection schemes can indeed be extended to allow for strict, rather than strong monotonicity. Furthermore, we introduce a class of regularized SA schemes, in which the regularization parameter is updated at every step, leading to a single timescale method. The scheme is a stochastic extension of an iterative Tikhonov regularization method and its global convergence is established. To aid in networked implementations, we consider an extension to this result where players are allowed to choose their steplengths independently and show the convergence of the scheme if the deviation across their choices is suitably constrained.
AB - In this paper, we consider the distributed computation of equilibria arising in monotone stochastic Nash games over continuous strategy sets. Such games arise in settings when the gradient map of the player objectives is a monotone mapping over the cartesian product of strategy sets, leading to a monotone stochastic variational inequality. We consider the application of projection-based stochastic approximation (SA) schemes. However, such techniques are characterized by a key shortcoming: in their traditional form, they can only accommodate strongly monotone mappings. In fact, standard extensions of SA schemes for merely monotone mappings require the solution of a sequence of related strongly monotone problems, a natively two-timescale scheme. Accordingly, we consider the development of single timescale techniques for computing equilibria when the associated gradient map does not admit strong monotonicity. We first show that, under suitable assumptions, standard projection schemes can indeed be extended to allow for strict, rather than strong monotonicity. Furthermore, we introduce a class of regularized SA schemes, in which the regularization parameter is updated at every step, leading to a single timescale method. The scheme is a stochastic extension of an iterative Tikhonov regularization method and its global convergence is established. To aid in networked implementations, we consider an extension to this result where players are allowed to choose their steplengths independently and show the convergence of the scheme if the deviation across their choices is suitably constrained.
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U2 - 10.1109/CDC.2010.5717489
DO - 10.1109/CDC.2010.5717489
M3 - Conference contribution
AN - SCOPUS:79953138707
SN - 9781424477456
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 231
EP - 236
BT - 2010 49th IEEE Conference on Decision and Control, CDC 2010
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 49th IEEE Conference on Decision and Control, CDC 2010
Y2 - 15 December 2010 through 17 December 2010
ER -