Dynamic estimation of slowly varying sequences

We consider the problem of sequentially approximating functions of each element in a slowly-varying sequence, i.e. one where the magnitude αi of the difference between the elements at positions i and i-1 is small. Recent work on implicit trace estimation shows that when αt 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 αt, obtaining sharper path-length-style variation bounds of form mathcal O(sumi=1mαi) on the cost of estimating a sequence of length m. This improves upon the previous implicit trace estimation bound of mathcal O(m·maxiαi) [Dharangutte & Musco, NeurIPS~2021], which is achieved by fixing the query budget using the worst-case αi and is thus inefficient for stable sequences with rare bursts. Lastly, while all past work assumes a known bound on αi, 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.