利用优化获取一个宽度无关、并行、更简单且更快的正SDP求解器
Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver
AI总结：
本文提出一种基于优化框架的简单算法，通过新的矩阵不等式和随机化梯度截断技术，实现与最佳线性规划求解器相同的迭代次数，解决正SDP求解问题。
摘要：
我们研究了设计近似求解打包和覆盖半正定规划（或正SDP）的多项对数深度算法。这实际上是正线性规划问题的自然扩展。尽管正线性规划可以在使用$\tilde{O}(\log^{2} n/\varepsilon^2)$并行迭代的情况下以多项对数深度解决，但Jain和Yao提出的最佳正SDP求解器需要$O(\log^{14} n /\varepsilon^{13})$并行迭代。几种替代求解器已被提出以减少迭代次数的指数。然而，这些工作的收敛性分析的正确性受到质疑，因为它们都依赖于无法推广到矩阵代数的代数单调性性质。在本文中，我们提出了一种基于为线性规划求解器提出的优化框架的非常简单的算法。我们的算法只需要$\tilde{O}(\log^2 n / \varepsilon^2)$次迭代，与最佳线性规划求解器相同。为了克服先前方法遇到的障碍，我们的分析需要一个扩展Lieb-Thirring不等式的新的矩阵不等式，以及一种随机化的梯度截断技术的符号一致变体。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
