Nest Your Adaptive Algorithm for Parameter-Agnostic Nonconvex Minimax Optimization

TL;DR

This paper introduces NeAda, a nested adaptive algorithm for nonconvex minimax optimization that automatically balances primal and dual updates, achieving near-optimal convergence without prior parameter knowledge.

Contribution

The paper proposes NeAda, a novel nested adaptive framework that ensures parameter-agnostic convergence in nonconvex minimax problems, overcoming limitations of direct adaptive extensions.

Findings

NeAda achieves near-optimal gradient complexities in theory.

NeAda is robust and automatically balances primal and dual updates.

First algorithm with near-optimal rates and parameter-agnostic adaptation in this setting.

Abstract

Adaptive algorithms like AdaGrad and AMSGrad are successful in nonconvex optimization owing to their parameter-agnostic ability -- requiring no a priori knowledge about problem-specific parameters nor tuning of learning rates. However, when it comes to nonconvex minimax optimization, direct extensions of such adaptive optimizers without proper time-scale separation may fail to work in practice. We provide such an example proving that the simple combination of Gradient Descent Ascent (GDA) with adaptive stepsizes can diverge if the primal-dual stepsize ratio is not carefully chosen; hence, a fortiori, such adaptive extensions are not parameter-agnostic. To address the issue, we formally introduce a Nested Adaptive framework, NeAda for short, that carries an inner loop for adaptively maximizing the dual variable with controllable stopping criteria and an outer loop for adaptively minimizing the primal variable. Such mechanism can be equipped with off-the-shelf adaptive optimizers and automatically balance the progress in the primal and dual variables. Theoretically, for nonconvex-strongly-concave minimax problems, we show that NeAda can achieve the near-optimal $\tilde{O}(\epsilon^{-2})$ and $\tilde{O}(\epsilon^{-4})$ gradient complexities respectively in the deterministic and stochastic settings, without prior information on the problem's smoothness and strong concavity parameters. To the best of our knowledge, this is the first algorithm that simultaneously achieves near-optimal convergence rates and parameter-agnostic adaptation in the nonconvex minimax setting. Numerically, we further illustrate the robustness of the NeAda family with experiments on simple test functions and a real-world application.