A Rank-1 Sketch for Matrix Multiplicative Weights

We show that a simple randomized sketch of the matrix multiplicative weight (MMW) update enjoys (in expectation) the same regret bounds as MMW, up to a small constant factor. Unlike MMW, where every step requires full matrix exponentiation, our steps require only a single product of the form $e^A b$, which the Lanczos method approximates efficiently. Our key technique is to view the sketch as a \emph{randomized mirror projection}, and perform mirror descent analysis on the \emph{expected projection}. Our sketch solves the online eigenvector problem, improving the best known complexity bounds by $\Omega(\log^5 n)$. We also apply this sketch to semidefinite programming in saddle-point form, yielding a simple primal-dual scheme with guarantees matching the best in the literature.