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.

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).

We study a dynamic version of the implicit trace estimation problem. Given access to an oracle for computing matrix-vector multiplications with a dynamically changing matrix A, our goal is to maintain an accurate approximation to A's trace using as few multiplications as possible. We present a practical algorithm for solving this problem and prove that, in a natural setting, its complexity is quadratically better than the standard solution of repeatedly applying Hutchinson's stochastic trace estimator. We also provide an improved algorithm assuming additional common assumptions on A's dynamic updates. We support our theory with empirical results, showing significant computational improvements on three applications in machine learning and network science: tracking moments of the Hessian spectral density during neural network optimization, counting triangles and estimating natural connectivity in a dynamically changing graph.

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.

We study a dynamic version of the implicit trace estimation problem. Given access to an oracle for computing matrix-vector multiplications with a dynamically changing matrix A, our goal is to maintain an accurate approximation to A's trace using as few multiplications as possible. We present a practical algorithm for solving this problem and prove that, in a natural setting, its complexity is quadratically better than the standard solution of repeatedly applying Hutchinson's stochastic trace estimator. We also provide an improved algorithm assuming additional common assumptions on A's dynamic updates. We support our theory with empirical results, showing significant computational improvements on three applications in machine learning and network science: tracking moments of the Hessian spectral density during neural network optimization, counting triangles and estimating natural connectivity in a dynamically changing graph.

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). 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 {\it even in the static setting}, and (2) are the first unconditional lower bounds for dynamic trace estimation, resolving open questions of prior work.

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.