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Continuous-time Discounted Mirror-Descent Dynamics in Monotone Concave Games

Bolin Gao, Lacra Pavel

TL;DR

Two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer are introduced and it is shown that these new dynamics can converge asymptotically in concave games with merely monotone (negative) pseudogradient.

Abstract

In this paper, we consider concave continuous-kernel games characterized by monotonicity properties and propose discounted mirror descent-type dynamics. We introduce two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer. Depending on the properties of the regularizer we show that these new dynamics can converge asymptotically in concave games with monotone (negative) pseudo-gradient. Furthermore, we show that when the regularizer enjoys strong convexity, the resulting dynamics can converge even in games with hypo-monotone (negative) pseudo-gradient, which corresponds to a shortage of monotonicity.

Continuous-time Discounted Mirror-Descent Dynamics in Monotone Concave Games

TL;DR

Two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer are introduced and it is shown that these new dynamics can converge asymptotically in concave games with merely monotone (negative) pseudogradient.

Abstract

In this paper, we consider concave continuous-kernel games characterized by monotonicity properties and propose discounted mirror descent-type dynamics. We introduce two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer. Depending on the properties of the regularizer we show that these new dynamics can converge asymptotically in concave games with monotone (negative) pseudo-gradient. Furthermore, we show that when the regularizer enjoys strong convexity, the resulting dynamics can converge even in games with hypo-monotone (negative) pseudo-gradient, which corresponds to a shortage of monotonicity.

Paper Structure

This paper contains 11 sections, 8 theorems, 40 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Convex Sets, Fenchel Duality and Monotone Operators
  4. $N$-Player Concave Games
  5. Problem Setup
  6. A General Framework for Designing Discounted Mirror Descent Dynamics
  7. Properties of Induced Mirror Maps
  8. Convergence of DMD under Induced Mirror Maps
  9. Examples of DMD
  10. Simulation Results
  11. Conclusions

Key Result

Proposition 1

Let $\vartheta^p\!:\! \mathbb{R}^{n_p} \!\to \!\mathbb{R}\!\cup\!\{\infty\}$ be a closed, proper, convex. Then, the following hold: (i) If $\!\vartheta^p\!\!$ is steep, then $\! \!\mathop{\mathrm{rbd}}\nolimits(\mathop{\mathrm{dom}}\nolimits(\vartheta^p\!)\!) \!\not\subset \! \!\mathop{\mathrm{dom}

Figures (6)

  • Figure 1: Monotone game, $\epsilon\!\!=\!\!0.5$
  • Figure 2: Hypo-monotone game, $\epsilon\!\!=\!\!5.1$
  • Figure 3: Comparison between discrete-time PDMD and ITR
  • Figure 4: Learning the Mean, $\epsilon = 0.1$
  • Figure 5: Solution trajectories
  • ...and 1 more figures

Theorems & Definitions (25)

  • Remark 1
  • Definition 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof
  • ...and 15 more