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Convergence of $\text{log}(1/ε)$ for Gradient-Based Algorithms in Zero-Sum Games without the Condition Number: A Smoothed Analysis

Ioannis Anagnostides, Tuomas Sandholm

TL;DR

It is shown that several gradient-based algorithms in the celebrated framework of smoothed analysis have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, $\textsf{log}(1/\epsilon)$ and $1/\sigma$, where $\sigma$ measures the magnitude of the smoothing perturbation.

Abstract

Gradient-based algorithms have shown great promise in solving large (two-player) zero-sum games. However, their success has been mostly confined to the low-precision regime since the number of iterations grows polynomially in $1/ε$, where $ε> 0$ is the duality gap. While it has been well-documented that linear convergence -- an iteration complexity scaling as $\textsf{log}(1/ε)$ -- can be attained even with gradient-based algorithms, that comes at the cost of introducing a dependency on certain condition number-like quantities which can be exponentially large in the description of the game. To address this shortcoming, we examine the iteration complexity of several gradient-based algorithms in the celebrated framework of smoothed analysis, and we show that they have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, $\textsf{log}(1/ε)$, and $1/σ$, where $σ$ measures the magnitude of the smoothing perturbation. Our result applies to optimistic gradient and extra-gradient descent/ascent, as well as a certain iterative variant of Nesterov's smoothing technique. From a technical standpoint, the proof proceeds by characterizing and performing a smoothed analysis of a certain error bound, the key ingredient driving linear convergence in zero-sum games. En route, our characterization also makes a natural connection between the convergence rate of such algorithms and perturbation-stability properties of the equilibrium, which is of interest beyond the model of smoothed complexity.

Convergence of $\text{log}(1/ε)$ for Gradient-Based Algorithms in Zero-Sum Games without the Condition Number: A Smoothed Analysis

TL;DR

It is shown that several gradient-based algorithms in the celebrated framework of smoothed analysis have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, and , where measures the magnitude of the smoothing perturbation.

Abstract

Gradient-based algorithms have shown great promise in solving large (two-player) zero-sum games. However, their success has been mostly confined to the low-precision regime since the number of iterations grows polynomially in , where is the duality gap. While it has been well-documented that linear convergence -- an iteration complexity scaling as -- can be attained even with gradient-based algorithms, that comes at the cost of introducing a dependency on certain condition number-like quantities which can be exponentially large in the description of the game. To address this shortcoming, we examine the iteration complexity of several gradient-based algorithms in the celebrated framework of smoothed analysis, and we show that they have polynomial smoothed complexity, in that their number of iterations grows as a polynomial in the dimensions of the game, , and , where measures the magnitude of the smoothing perturbation. Our result applies to optimistic gradient and extra-gradient descent/ascent, as well as a certain iterative variant of Nesterov's smoothing technique. From a technical standpoint, the proof proceeds by characterizing and performing a smoothed analysis of a certain error bound, the key ingredient driving linear convergence in zero-sum games. En route, our characterization also makes a natural connection between the convergence rate of such algorithms and perturbation-stability properties of the equilibrium, which is of interest beyond the model of smoothed complexity.

Paper Structure

This paper contains 30 sections, 29 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. The error bound
  4. Further implications
  5. Notation
  6. Smoothed analysis of the error bound
  7. Overview
  8. Characterization of the error bound
  9. Smoothed analysis
  10. Parameterized results for perturbation-stable games
  11. Conclusions and future research
  12. Further related work
  13. Preliminaries
  14. Further notation
  15. Smoothed complexity
  16. ...and 15 more sections

Key Result

Theorem 1.2

With high probability over the randomness of $\mathbf{A} \in {\mathbb{R}}^{n \times m}$ (def:Gauss-perturb), $\texttt{OGDA}$, $\texttt{EGDA}$ and $\texttt{IterSmooth}$ converge to an $\epsilon$-equilibrium after $\poly(n, m, 1/\sigma) \cdot \mathsf{log}(1/\epsilon)$ iterations.

Theorems & Definitions (48)

  • Definition 1.1: Zero-sum games under Gaussian perturbations
  • Theorem 1.2
  • Definition 1.3: Error bound
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Proposition 3.1
  • Definition 3.2: Non-degenerate game
  • Definition 3.3
  • Lemma 3.3
  • ...and 38 more