Dynamic estimation of slowly varying sequences

arXiv:2606.23655v1 Announce Type: new Abstract: We consider the problem of sequentially approximating functions of each element in a slowly-varying sequence, i.e. one where the magnitude alpha_i of the difference between the elements at positions i and i-1 is small. Recent work on implicit trace estimation shows that when alpha_t is small, reusing queries to past sequence elements can reduce the overall cost [Dharangutte & Musco, NeurIPS2021; Woodruff et al., NeurIPS2022]. 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 alpha_t, obtaining sharper path-length-style variation bounds of form mathcal O(sum_{i=1}^malpha_i) on the cost of estimating a sequence of length m. This improves upon the previous implicit trace estimation bound of mathcal O(mdotmax_ialpha_i) [Dharangutte & Musco, NeurIPS~2021], which is achieved by fixing the query budget using the worst-case alpha_i and is thus inefficient for stable sequences with rare bursts. Lastly, while all past work assumes a known bound on alpha_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.

Source: arXiv cs.LG | 2026-06-23