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 $A_1,...,A_m$ 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 $\widetilde{O}\left(m \alpha\sqrt{\log(1/\delta)}/\epsilon + m\log(1/\delta)\right)$, improving the dependence on both $\alpha$ and $\delta$ from Dharangutte and Musco (NeurIPS, 2021). Our procedure works without additional norm bounds on $A_i$ and can be generalized to a bound for the $p$-th Schatten norm for $p \in [1,2]$, giving a complexity of $\widetilde{O}\left(m \alpha\left(\sqrt{\log(1/\delta)}/\epsilon\right)^p +m \log(1/\delta)\right)$. By using novel reductions to communication complexity and information-theoretic analyses of Gaussian matrices, we provide matching lower bounds for static and dynamic trace estimation in all relevant parameters, including the failure probability. Our lower bounds (1) give the first tight bounds for Hutchinson's estimator in the matrix-vector product model with Frobenius norm error even in the static setting, and (2) are the first unconditional lower bounds for dynamic trace estimation, resolving open questions of prior work.

翻译：我们考虑如何在动态环境中将准确跟踪估算所需的矩阵-矢量查询数量减少到最低程度的问题,因为在动态环境中,我们的基础矩阵正在缓慢变化,例如优化进程。具体来说,对于任何美元基质 A_1,.,A_m$,连续的差额以Schatten-$美元标准为单位,我们提供了一个新的双向树总和程序,该程序同时估计所有美元追踪为美元=epsilon差错,以美元=delta$%xlevory(m\alpha\sqrt_log_log(1/delta)}/\\mlglg_right),同时估算所有美元为美元和美元为美元或美元错误,同时估算美元=deletrial_i$的误差,我们的程序无需对美元基质($_i_i_oxlexxxxxxlxxxxxxxxlxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx