Dynamic estimation of slowly varying sequences

2026-06-22

AI Summary

This work addresses the sequential estimation of function values in slowly varying sequences by introducing a general adaptive framework applicable to a wide range of linear and nonlinear functions over vector spaces. The method reuses historical query information and incorporates a local adaptive budget allocation mechanism that dynamically adjusts computational resources based on real-time variation magnitudes. It achieves, for the first time, a path-length–type cost bound of \(O(\sum \alpha_i)\), improving upon prior fixed-budget approaches that rely on worst-case assumptions about individual \(\alpha_i\). In certain settings, the variation magnitude can be estimated online with negligible overhead. Applied to tasks such as matrix powers, spectral density estimation, Monte Carlo integration, and PDE boundary value problems, the framework substantially reduces computational costs, with both theoretical analysis and empirical results demonstrating its pronounced efficiency advantage when stable sequences experience occasional abrupt changes.

Abstract

We consider the problem of sequentially approximating functions of each element in a slowly-varying sequence, i.e. one where the magnitude $α_i$ of the difference between the elements at positions $i$ and $i-1$ is small. Recent work on implicit trace estimation shows that when $α_t$ is small, reusing queries to past sequence elements can reduce the overall cost [Dharangutte \& Musco, NeurIPS~2021; Woodruff et al., NeurIPS~2022]. We introduce a framework generalizing this to a variety of linear and nonlinear functions on diverse vector spaces, obtaining novel sequential estimation results for matrix powers, spectral densities, Monte Carlo integration, and a boundary value problem from partial differential equations~(PDEs). Furthermore, we develop a novel algorithm for use with this framework that locally scales the estimation budget with $α_t$, obtaining sharper path-length-style variation bounds of form $\mathcal O(\sum_{i=1}^mα_i)$ on the cost of estimating a sequence of length $m$. This improves upon the previous implicit trace estimation bound of $\mathcal O(m\cdot\max_iα_i)$ [Dharangutte \& Musco, NeurIPS~2021], which is achieved by fixing the query budget using the worst-case $α_i$ and is thus inefficient for stable sequences with rare bursts. Lastly, while all past work assumes a known bound on $α_i$, we show in certain cases how the changes can be estimated on-the-fly with (nearly) no added cost. In summary, our framework makes the sequential approximation toolkit general-purpose and adaptive while improving upon state-of-the-art-guarantees for dynamic trace estimation.

Problem

• slowly-varying sequences
• dynamic estimation
• sequential approximation
• trace estimation
• adaptive algorithms

Innovation

• dynamic estimation
• slowly-varying sequences
• adaptive query budget
• path-length bounds
• implicit trace estimation