### Abstract

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.

Original language | English (US) |
---|---|

Title of host publication | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |

Pages | 231-236 |

Number of pages | 6 |

DOIs | |

State | Published - 2010 |

Externally published | Yes |

Event | 2010 49th IEEE Conference on Decision and Control, CDC 2010 - Atlanta, GA, United States Duration: Dec 15 2010 → Dec 17 2010 |

### Other

Other | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |
---|---|

Country | United States |

City | Atlanta, GA |

Period | 12/15/10 → 12/17/10 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*2010 49th IEEE Conference on Decision and Control, CDC 2010*(pp. 231-236). [5717489] https://doi.org/10.1109/CDC.2010.5717489

**Single timescale regularized stochastic approximation schemes for monotone nash games under uncertainty.** / Koshal, Jayash; Nedich, Angelia; Shanbhag, Uday V.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2010 49th IEEE Conference on Decision and Control, CDC 2010.*, 5717489, pp. 231-236, 2010 49th IEEE Conference on Decision and Control, CDC 2010, Atlanta, GA, United States, 12/15/10. https://doi.org/10.1109/CDC.2010.5717489

}

TY - GEN

T1 - Single timescale regularized stochastic approximation schemes for monotone nash games under uncertainty

AU - Koshal, Jayash

AU - Nedich, 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.

UR - http://www.scopus.com/inward/record.url?scp=79953138707&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953138707&partnerID=8YFLogxK

U2 - 10.1109/CDC.2010.5717489

DO - 10.1109/CDC.2010.5717489

M3 - Conference contribution

AN - SCOPUS:79953138707

SN - 9781424477456

SP - 231

EP - 236

BT - 2010 49th IEEE Conference on Decision and Control, CDC 2010

ER -