Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization

TL;DR

This paper introduces the first optimal primal-dual algorithm for bilinearly coupled smooth minimax problems, achieving the lower complexity bound with a simple, single-loop method that handles both smooth and bilinear terms efficiently.

Contribution

The paper develops the Lifted Primal-Dual (LPD) method, the first optimal algorithm for this class of problems, with a unified primal-dual framework and single-gradient oracle iteration.

Findings

Achieves the lower complexity bound for bilinear minimax problems.

Provides a simple, single-loop algorithm with one gradient call per iteration.

Demonstrates fast convergence in numerical experiments on quadratic and policy evaluation problems.

Abstract

We study the bilinearly coupled minimax problem: $\min_{x} \max_{y} f(x) + y^\top A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions and admit first-order gradient oracles. Surprisingly, no known first-order algorithms have hitherto achieved the lower complexity bound of $\Omega((\sqrt{\frac{L_x}{\mu_x}} + \frac{\|A\|}{\sqrt{\mu_x \mu_y}} + \sqrt{\frac{L_y}{\mu_y}}) \log(\frac1{\varepsilon}))$ for solving this problem up to an $\varepsilon$ primal-dual gap in the general parameter regime, where $L_x, L_y,\mu_x,\mu_y$ are the corresponding smoothness and strongly convexity constants. We close this gap by devising the first optimal algorithm, the Lifted Primal-Dual (LPD) method. Our method lifts the objective into an extended form that allows both the smooth terms and the bilinear term to be handled optimally and seamlessly with the same primal-dual framework. Besides optimality, our method yields a desirably simple single-loop algorithm that uses only one gradient oracle call per iteration. Moreover, when $f$ is just convex, the same algorithm applied to a smoothed objective achieves the nearly optimal iteration complexity. We also provide a direct single-loop algorithm, using the LPD method, that achieves the iteration complexity of $O(\sqrt{\frac{L_x}{\varepsilon}} + \frac{\|A\|}{\sqrt{\mu_y \varepsilon}} + \sqrt{\frac{L_y}{\varepsilon}})$. Numerical experiments on quadratic minimax problems and policy evaluation problems further demonstrate the fast convergence of our algorithm in practice.