Optimal Query Complexities for Dynamic Trace Estimation

We consider the problem of minimizing the number of matrix-vector queries needed for accurate trace estimation in the dynamic setting where our underlying matrix is changing slowly, such as during an optimization process. Specifically, for any $m$ matrices $\mathbf{A}_1,...,\mathbf{A}_m$ with consecutive differences bounded in Schatten-$1$ norm by $\alpha$, we provide a novel binary tree summation procedure that simultaneously estimates all $m$ traces up to $\epsilon$ error with $\delta$ failure probability with an optimal query complexity of $\widetilde{O}(m \alpha\sqrt{\log(1/\delta)}/\epsilon + m\log(1/\delta))$, improving the dependence on both $\alpha$ and $\delta$ from Dharangutte and Musco (NeurIPS, 2021).