An Inexact Primal-Dual Smoothing Framework for Large-Scale Non-Bilinear Saddle Point Problems

TL;DR

This paper introduces an inexact primal-dual smoothing framework for large-scale non-bilinear saddle point problems with primal strong convexity, improving primal oracle complexity and efficiently handling many component functions via a randomized approach.

Contribution

It proposes a novel inexact primal-dual smoothing framework with a randomized variant for large-scale problems, achieving improved oracle complexities over existing methods.

Findings

Significant primal oracle complexity improvement.

Effective handling of many component functions with randomized algorithms.

Achieves best-known oracle complexities for convex problems with constraints.

Abstract

We develop an inexact primal-dual first-order smoothing framework to solve a class of non-bilinear saddle point problems with primal strong convexity. Compared with existing methods, our framework yields a significant improvement over the primal oracle complexity, while it has competitive dual oracle complexity. In addition, we consider the situation where the primal-dual coupling term has a large number of component functions. To efficiently handle this situation, we develop a randomized version of our smoothing framework, which allows the primal and dual sub-problems in each iteration to be inexactly solved by randomized algorithms in expectation. The convergence of this framework is analyzed both in expectation and with high probability. In terms of the primal and dual oracle complexities, this framework significantly improves over its deterministic counterpart. As an important application, we adapt both frameworks for solving convex optimization problems with many functional constraints. To obtain an $\varepsilon$-optimal and $\varepsilon$-feasible solution, both frameworks achieve the best-known oracle complexities.