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- W3212813711 abstract "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 $text{tr}(A)$ 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. Finally, our experiments demonstrate the superior performance of our sketch over the adaptive Hutch++ algorithm, which is less parallelizable, as well as over the non-adaptive Hutchinson's method." @default.
- W3212813711 created "2021-11-22" @default.
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- W3212813711 date "2021-10-31" @default.
- W3212813711 modified "2023-09-23" @default.
- W3212813711 title "Optimal Sketching for Trace Estimation" @default.
- W3212813711 doi "https://doi.org/10.48550/arxiv.2111.00664" @default.
- W3212813711 hasPublicationYear "2021" @default.
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