Using optimization to obtain a width-independent, parallel, simpler, and faster positive SDP solver
Lee, Yin Tat
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Citation (published version)Zeyuan Allen-Zhu, Yin Tat Lee, Lorenzo Orecchia. 2016. "Using optimization to obtain a width-independent, parallel, simpler, and faster positive SDP solver." Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms. Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, https://doi.org/10.1137/1.9781611974331.ch127
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 log2 n/∊3 parallelizable iterations , the best known positive SDP solvers due to Jain and Yao  require log14 n/∊13 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 , 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  for LP solvers. Our algorithm only needs log2 n/∊3 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].