Optimal Algorithm with Complexity Separation for Strongly Convex-Strongly Concave Composite Saddle Point Problems

TL;DR

This paper introduces a new optimal algorithm for strongly convex-strongly concave saddle point problems that achieves complexity separation for different parts of the problem, especially when the strong convexity and concavity parameters differ.

Contribution

The paper presents the first optimal algorithm with complexity separation for saddle point problems where the convexity and concavity parameters are unequal, improving efficiency for composite problems.

Findings

Achieves optimal overall complexity with logarithmic dependence on accuracy.

Requires fewer oracle calls for gradient evaluations of p, q, and R.

Demonstrates optimal complexity for bilinear saddle point problems.

Abstract

In this work, we focuses on the following saddle point problem $\min_x \max_y p(x) + R(x,y) - q(y)$ where $R(x,y)$ is $L_R$-smooth, $\mu_x$-strongly convex, $\mu_y$-strongly concave and $p(x), q(y)$ are convex and $L_p, L_q$-smooth respectively. We present a new algorithm with optimal overall complexity $\mathcal{O}\left(\left(\sqrt{\frac{L_p}{\mu_x}} + \frac{L_R}{\sqrt{\mu_x \mu_y}} + \sqrt{\frac{L_q}{\mu_y}}\right)\log \frac{1}{\varepsilon}\right)$ and separation of oracle calls in the composite and saddle part. This algorithm requires $\mathcal{O}\left(\left(\sqrt{\frac{L_p}{\mu_x}} + \sqrt{\frac{L_q}{\mu_y}}\right) \log \frac{1}{\varepsilon}\right)$ oracle calls for $\nabla p(x)$ and $\nabla q(y)$ and $\mathcal{O} \left( \max\left\{\sqrt{\frac{L_p}{\mu_x}}, \sqrt{\frac{L_q}{\mu_y}}, \frac{L_R}{\sqrt{\mu_x \mu_y}} \right\}\log \frac{1}{\varepsilon}\right)$ oracle calls for $\nabla R(x,y)$ to find an $\varepsilon$-solution of the problem. To the best of our knowledge, we are the first to develop optimal algorithm with complexity separation in the case $\mu_x \not = \mu_y$. Also, we apply this algorithm to a bilinear saddle point problem and obtain the optimal complexity for this class of problems.