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

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 Õ(log n/ε) parallelizable iterations [4, 33], the best known positive SDP solvers due to Jain and Yao [18] require O(log n/ε) parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations [19, 30]. However, the correctness of the convergence analyses in these works has been called into question [30], 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 in [4] for LP solvers. Our algorithm only needs Õ(log n/ε) 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 [3, 4]. ∗The abstract of a previous version of this paper has appeared in the proceedings of SODA 2016. [1] ar X iv :1 50 7. 02 25 9v 2 [ cs .D S] 1 1 Ja n 20 16