Optimal Sketching for Trace Estimation

Matrix trace estimation is ubiquitous in machine learning applications and has traditionally relied on Hutchinson's method, which requires O(\log(1/\delta)/\epsilon 2) matrix-vector product queries to achieve a (1 \pm \epsilon) -multiplicative approximation to \text{trace}(A) with failure probability \delta on positive-semidefinite input matrices A . Recently, the Hutch algorithm was proposed, which reduces the number of matrix-vector queries from O(1/\epsilon 2) to the optimal O(1/\epsilon), and the algorithm succeeds with constant probability. However, in the high probability setting, the non-adaptive Hutch algorithm suffers an extra O(\sqrt{\log(1/\delta)}) multiplicative factor in its query complexity. Non-adaptive methods are important, as they correspond to sketching algorithms, which are mergeable, highly parallelizable, and provide low-memory streaming algorithms as well as low-communication distributed protocols. In this work, we close the gap between non-adaptive and adaptive algorithms, showing that even non-adaptive algorithms can achieve O(\sqrt{\log(1/\delta)}/\epsilon \log(1/\delta)) matrix-vector products. In addition, we prove matching lower bounds demonstrating that, up to a \log \log(1/\delta) factor, no further improvement in the dependence on \delta or \epsilon is possible by any non-adaptive algorithm.