On the Initialization for Convex-Concave Min-max Problems

TL;DR

This paper investigates how the initialization affects convergence rates in convex-concave min-max problems, showing that strict convexity and concavity enable initialization-dependent rates and proposing algorithms that leverage this for faster convergence.

Contribution

It demonstrates that strict convexity and concavity enable initialization-dependent convergence rates and introduces parameter-free algorithms that achieve improved rates without tuning.

Findings

Initialization-dependent convergence rates are achievable under strict convexity and concavity.

Parameter-free algorithms can attain improved asymptotic rates without learning rate tuning.

The proposed algorithms demonstrate faster convergence in experiments.

Abstract

Convex-concave min-max problems are ubiquitous in machine learning, and people usually utilize first-order methods (e.g., gradient descent ascent) to find the optimal solution. One feature which separates convex-concave min-max problems from convex minimization problems is that the best known convergence rates for min-max problems have an explicit dependence on the size of the domain, rather than on the distance between initial point and the optimal solution. This means that the convergence speed does not have any improvement even if the algorithm starts from the optimal solution, and hence, is oblivious to the initialization. Here, we show that strict-convexity-strict-concavity is sufficient to get the convergence rate to depend on the initialization. We also show how different algorithms can asymptotically achieve initialization-dependent convergence rates on this class of functions. Furthermore, we show that the so-called "parameter-free" algorithms allow to achieve improved initialization-dependent asymptotic rates without any learning rate to tune. In addition, we utilize this particular parameter-free algorithm as a subroutine to design a new algorithm, which achieves a novel non-asymptotic fast rate for strictly-convex-strictly-concave min-max problems with a growth condition and H{\"o}lder continuous solution mapping. Experiments are conducted to verify our theoretical findings and demonstrate the effectiveness of the proposed algorithms.