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A Fisher-Rao gradient flow for entropic mean-field min-max games

Razvan-Andrei Lascu, Mateusz B. Majka, Łukasz Szpruch

TL;DR

This work examines the convergence in continuous-time of a Fisher-Rao (Mean-Field Birth-Death) gradient flow in the context of solving convex-concave min-max games with entropy regularization and proposes appropriate Lyapunov functions to demonstrate convergence with explicit rates to the unique mixed Nash equilibrium.

Abstract

Gradient flows play a substantial role in addressing many machine learning problems. We examine the convergence in continuous-time of a \textit{Fisher-Rao} (Mean-Field Birth-Death) gradient flow in the context of solving convex-concave min-max games with entropy regularization. We propose appropriate Lyapunov functions to demonstrate convergence with explicit rates to the unique mixed Nash equilibrium.

A Fisher-Rao gradient flow for entropic mean-field min-max games

TL;DR

This work examines the convergence in continuous-time of a Fisher-Rao (Mean-Field Birth-Death) gradient flow in the context of solving convex-concave min-max games with entropy regularization and proposes appropriate Lyapunov functions to demonstrate convergence with explicit rates to the unique mixed Nash equilibrium.

Abstract

Gradient flows play a substantial role in addressing many machine learning problems. We examine the convergence in continuous-time of a \textit{Fisher-Rao} (Mean-Field Birth-Death) gradient flow in the context of solving convex-concave min-max games with entropy regularization. We propose appropriate Lyapunov functions to demonstrate convergence with explicit rates to the unique mixed Nash equilibrium.
Paper Structure (14 sections, 5 theorems, 134 equations)

This paper contains 14 sections, 5 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Notation and setup
  3. Fisher-Rao (mean-field birth-death) gradient flow
  4. Sketch of convergence proof for the FR gradient flow
  5. Our contribution
  6. Related works
  7. Main results
  8. Proof of Theorem \ref{['thm:convergence']}
  9. Technical results and proofs
  10. Proof of Lemma \ref{['lemma:Vinequalities']}.
  11. Existence and uniqueness of the FR flow
  12. Notation and definitions
  13. A formal derivation of the Fisher-Rao gradient flow
  14. Example: Convergence of discrete-time Fisher-Rao dynamics for a non-interactive min-max game

Key Result

Theorem 2.2

Suppose that Assumption assumption: boundedness-first-flat, assump:F and condition eq:assumpSup from Assumption assump:m0 hold. Then for each $(\nu_0, \mu_0) \in \mathcal{P}_{\text{ac}}(\mathcal{X}) \times \mathcal{P}_{\text{ac}}(\mathcal{Y}),$ there exists a unique pair of continuous and differenti and there exist constants $R_{1, \nu}, R_{1, \mu} > 1$ such that for all $t > 0$, Additionally, if

Theorems & Definitions (16)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1: Relative $\sigma$-strong-convexity-concavity to $\operatorname{D_{KL}}$
  • proof : Proof of Theorem \ref{['thm:convergence']}
  • proof : Proof of Lemma \ref{['lemma:Vinequalities']}
  • proof : Proof of Theorem \ref{['thm:existence']}
  • Lemma A.1
  • ...and 6 more