Streaming Semidefinite Programs: $O(\sqrt{n})$ Passes, Small Space and Fast Runtime

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 $n\times n$ together with a vector $b\in \mathbb{R}^m$ are streamed to us one-by-one. The goal is to find a matrix $X\in \mathbb{R}^{n\times n}$ such that $\langle C, X\rangle$ is maximized, subject to $\langle A_i, X\rangle=b_i$ for all $i\in [m]$ and $X\succeq 0$. Previous algorithmic studies of SDP primarily focus on \emph{time-efficiency}, and all of them require a prohibitively large $\Omega(mn^2)$ space in order to store \emph{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 $\widetilde O(m^2+n^2)$ space, which is strictly sublinear in the regime $n\gg m$. Our algorithm takes $O(\sqrt n\log(1/\epsilon))$ 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).