Streaming Semidefinite Programs: O(√(n)) Passes, Small Space and Fast Runtime
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We study the problem of solving semidefinite programs (SDP) in the streaming model. Specifically, m constraint matrices and a target matrix C, all of size n× n together with a vector b∈ℝ^m are streamed to us one-by-one. The goal is to find a matrix X∈ℝ^n× n such that ⟨ C, X⟩ is maximized, subject to ⟨ A_i, X⟩=b_i for all i∈ [m] and X≽ 0. Previous algorithmic studies of SDP primarily focus on time-efficiency, and all of them require a prohibitively large Ω(mn^2) space in order to store all the constraints. Such space consumption is necessary for fast algorithms as it is the size of the input. In this work, we design an interior point method (IPM) that uses O(m^2+n^2) space, which is strictly sublinear in the regime n≫ m. Our algorithm takes O(√(n)log(1/ϵ)) passes, which is standard for IPM. Moreover, when m is much smaller than n, our algorithm also matches the time complexity of the state-of-the-art SDP solvers. To achieve such a sublinear space bound, we design a novel sketching method that enables one to compute a spectral approximation to the Hessian matrix in O(m^2) space. To the best of our knowledge, this is the first method that successfully applies sketching technique to improve SDP algorithm in terms of space (also time).
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