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).

翻译：我们研究在流式模型下求解半定规划（SDP）的问题。具体而言，我们在流中逐次接收$m$个约束矩阵和一个目标矩阵$C$（所有矩阵的规模均为$n\times n$）以及一个向量$b\in \mathbb{R}^m$。目标是找到一个矩阵$X\in \mathbb{R}^{n\times n}$，使得在约束条件$\langle A_i, X\rangle=b_i$（对所有$i\in [m]$）和$X\succeq 0$下，最大化$\langle C, X\rangle$。现有SDP算法研究主要关注时间效率，且所有算法均需存储所有约束条件，导致存储空间高达$\Omega(mn^2)$。这一空间消耗对快速算法而言是必要的，因为它相当于输入规模。本文设计了一种内点法（IPM），其空间复杂度为$\widetilde O(m^2+n^2)$，在$n\gg m$的情形下严格亚线性。该算法需要$O(\sqrt n\log(1/\epsilon))$轮遍历，这一复杂度对IPM而言是标准水平。此外，当$m$远小于$n$时，本文算法的时间复杂度与现有最优SDP求解器相当。为实现这一亚线性空间界，我们设计了一种新型的草图化方法，能够在$O(m^2)$空间内计算Hessian矩阵的谱近似。据我们所知，这是首个成功应用草图化技术从空间（以及时间）上改进SDP算法的方法。