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

TL;DR

This paper demonstrates that gradient-based algorithms for zero-sum games have polynomial smoothed complexity with convergence rates scaling as log(1/epsilon), avoiding exponential dependencies on condition numbers.

Contribution

It provides a smoothed analysis showing polynomial iteration complexity for several gradient algorithms, removing the need for condition number dependence.

Findings

Gradient algorithms have polynomial smoothed complexity in game dimensions.

Convergence rate scales as log(1/epsilon) under smoothing.

Connection established between convergence and perturbation-stability of equilibria.

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 $1/\epsilon$, where $\epsilon > 0$ is the duality gap. While it has been well-documented that linear convergence -- an iteration complexity scaling as $\textsf{log}(1/\epsilon)$ -- 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, $\textsf{log}(1/\epsilon)$, and $1/\sigma$, where $\sigma$ 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.