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Convergence Rate of Learning a Strongly Variationally Stable Equilibrium

Tatiana Tatarenko, Maryam Kamgarpour

TL;DR

This work derives the rate of convergence to the strongly variationally stable Nash equilibrium in a convex game, for a zeroth-order learning algorithm, from the best known rates for strongly monotone games under zeroth-order information.

Abstract

We derive the rate of convergence to the strongly variationally stable Nash equilibrium in a convex game, for a zeroth-order learning algorithm. Though we do not assume strong monotonicity of the game, our rates for the one-point feedback and for the two-point feedback match the best known rates for strongly monotone games under zeroth-order information.

Convergence Rate of Learning a Strongly Variationally Stable Equilibrium

TL;DR

This work derives the rate of convergence to the strongly variationally stable Nash equilibrium in a convex game, for a zeroth-order learning algorithm, from the best known rates for strongly monotone games under zeroth-order information.

Abstract

We derive the rate of convergence to the strongly variationally stable Nash equilibrium in a convex game, for a zeroth-order learning algorithm. Though we do not assume strong monotonicity of the game, our rates for the one-point feedback and for the two-point feedback match the best known rates for strongly monotone games under zeroth-order information.
Paper Structure (16 sections, 6 theorems, 42 equations, 2 figures, 1 algorithm)

This paper contains 16 sections, 6 theorems, 42 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Game setup and Zeroth-Order Algorithm
  3. Payoff-based Learning algorithm
  4. Algorithm iterates
  5. Gradient estimation in one and two-point settings
  6. Properties of the gradient estimators
  7. Convergence Rate of the Algorithm
  8. Discussion on the established rates
  9. Algorithm parameters
  10. Comparison with optimal rates for strongly monotone games
  11. Comparison with optimal rates for strongly convex optimization
  12. Proof of main result
  13. Numerical example
  14. Conclusion
  15. Auxiliary Result
  16. ...and 1 more sections

Key Result

Lemma 1

Given Assumptions assum:convex and assum:infty, for $j=1,2$,

Figures (2)

  • Figure 1: Convergence of the algorithm for the game of Example \ref{['example:3player']} with strategy sets as $[-1,2]$.
  • Figure 2: Convergence of the algorithm for the game of Example \ref{['example:3player']} with strategy sets as $[0,1]$.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Remark 1
  • Example 2
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • ...and 6 more