Follow the Compressed Leader: Faster Algorithms for Matrix Multiplicative Weight Updates

Matrix multiplicative weight update (MMWU) is an extremely powerful algorithmic tool for computer science and related fields. However, it comes with a slow running time due to the matrix exponential and eigendecomposition computations. For this reason, many researchers studied the followed-the-perturbed-leader (FTPL) framework which is faster, but a factor $\sqrt{d}$ worse than the optimal regret of MMWU for dimension-$d$ matrices. In this paper, we propose a $\textit{followed-the-compressed-leader}$ framework which, not only matches the optimal regret of MMWU (up to polylog factors), but runs $\textit{even faster}$ than FTPL. Our main idea is to "compress" the matrix exponential computation to dimension 3 in the adversarial setting, or dimension 1 in the stochastic setting. This result resolves an open question regarding how to obtain both (nearly) optimal and efficient algorithms for the online eigenvector problem.