Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver

TL;DR

This paper introduces a simple, parallelizable algorithm for positive semidefinite programs that achieves polylogarithmic depth and matches the iteration complexity of LP solvers, overcoming previous challenges in SDP solving.

Contribution

The paper presents a novel, width-independent, parallel algorithm for positive SDPs with a simplified analysis and iteration complexity comparable to LP solvers, using new matrix inequalities.

Findings

Achieves polylogarithmic depth for positive SDP solving.

Requires only O(log^2 n / ^2) iterations, matching LP solvers.

Introduces a new matrix inequality extending Lieb-Thirring's inequality.

Abstract

We study the design of polylogarithmic depth algorithms for approximately solving packing and covering semidefinite programs (or positive SDPs for short). This is a natural SDP generalization of the well-studied positive LP problem. Although positive LPs can be solved in polylogarithmic depth while using only $\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best known positive SDP solvers due to Jain and Yao require $O(\log^{14} n /\varepsilon^{13})$ parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations. However, the correctness of the convergence analyses in these works has been called into question, as they both rely on algebraic monotonicity properties that do not generalize to matrix algebra. In this paper, we propose a very simple algorithm based on the optimization framework proposed for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n / \varepsilon^2)$ iterations, matching that of the best LP solver. To surmount the obstacles encountered by previous approaches, our analysis requires a new matrix inequality that extends Lieb-Thirring's inequality, and a sign-consistent, randomized variant of the gradient truncation technique proposed in.