Optimal Query Complexities for Dynamic Trace Estimation

• URL: http://arxiv.org/abs/2209.15219v1
• Date: Fri, 30 Sep 2022 04:15:44 GMT
• Title: Optimal Query Complexities for Dynamic Trace Estimation
• Authors: David P. Woodruff, Fred Zhang, Qiuyi Zhang
• Abstract summary: 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. We provide a novel binary tree summation procedure that simultaneously estimates all $m$ traces up to $epsilon$ error with $delta$ 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.
• Score: 59.032228008383484
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Quantum linear system algorithm with optimal queries to initial state preparation [0.0]
  Quantum algorithms for linear systems produce the solution state $A-1|brangle$ by querying two oracles. We present a quantum linear system algorithm making $mathbfThetaleft (1/sqrtpright)$ queries to $O_b$, which is optimal in the success probability. In various applications such as solving differential equations, we can further improve the dependence on $p$ using a block preconditioning scheme.
  arXiv  Detail & Related papers  (2024-10-23T18:00:01Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Towards Characterizing the First-order Query Complexity of Learning (Approximate) Nash Equilibria in Zero-sum Matrix Games [0.0]
  We show that exact equilibria can be computed efficiently from $O(fracln Kepsilon)$ instead of $O(fracln Kepsilon2)$ queries. We introduce a new technique for lower bounds, which allows us to obtain lower bounds of order $tildeOmega(frac1Kepsilon)$ for any $epsilon leq 1 / (cK4)$.
  arXiv  Detail & Related papers  (2023-04-25T12:42:59Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• The Complexity of Dynamic Least-Squares Regression [11.815510373329337]
  complexity of dynamic least-squares regression. Goal is to maintain an $epsilon-approximate solution to $min_mathbfx(t)| mathbfA(t) mathbfb(t) |$ for all $tin.
  arXiv  Detail & Related papers  (2022-01-01T18:36:17Z)
• Active Sampling for Linear Regression Beyond the $\ell_2$ Norm [70.49273459706546]
  We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector. We show that this dependence on $d$ is optimal, up to logarithmic factors. We also provide the first total sensitivity upper bound $O(dmax1,p/2log2 n)$ for loss functions with at most degree $p$ growth.
  arXiv  Detail & Related papers  (2021-11-09T00:20:01Z)
• Chi-square and normal inference in high-dimensional multi-task regression [7.310043452300736]
  The paper proposes chi-square and normal methodologies for the unknown coefficient matrix $B*$ of size $ptimes T$ in a Multi-Task (MT) linear model.
  arXiv  Detail & Related papers  (2021-07-16T11:19:49Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.