Randomized Primal-Dual Methods with Adaptive Step Sizes

TL;DR

This paper introduces a class of randomized primal-dual algorithms with adaptive step sizes for large-scale saddle point problems, achieving optimal convergence rates under various convexity assumptions and demonstrating effectiveness in kernel matrix learning.

Contribution

The paper develops a novel randomized primal-dual framework with adaptive step sizes, providing convergence guarantees and practical implementation for large-scale saddle point problems.

Findings

Achieves $ ext{O}(M/k)$ convergence rate in expected primal-dual gap.

Attains $ ext{O}(M/k^2)$ rate under strong convexity assumptions.

Demonstrates competitive performance in kernel matrix learning tasks.

Abstract

In this paper we propose a class of randomized primal-dual methods to contend with large-scale saddle point problems defined by a convex-concave function $\mathcal{L}(\mathbf{x},y)\triangleq\sum_{i=1}^m f_i(x_i)+\Phi(\mathbf{x},y)-h(y)$. We analyze the convergence rate of the proposed method under mere convexity and strong convexity assumptions of $\mathcal{L}$ in $\mathbf{x}$-variable. In particular, assuming $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz and $\nabla_\mathbf{x}\Phi(\cdot,y)$ is coordinate-wise Lipschitz for any fixed $y$, the ergodic sequence generated by the algorithm achieves the convergence rate of $\mathcal{O}(M/k)$ in the expected primal-dual gap. Furthermore, assuming that $\mathcal{L}(\cdot,y)$ is strongly convex for any $y$, and that $\Phi(\mathbf{x},\cdot)$ is affine for any $\mathbf{x}$, the scheme enjoys a faster rate of $\mathcal{O}(M/k^2)$ in terms of primal solution suboptimality. We implemented the proposed algorithmic framework to solve kernel matrix learning problem, and tested it against other state-of-the-art first-order methods