Dynamic estimation of slowly varying sequences

Abstract: 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 $α_t$ 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.

• The paper introduces an adaptive meta-algorithm for dynamically estimating functions in slowly varying sequences, scaling sample complexity with the sequence's path-length.
• It updates running estimates by recycling previous values and adaptively estimating residuals using unbiased estimators with sub-exponential concentration.
• Empirical results show significant improvements in dynamic trace, spectral, and Monte Carlo estimations compared to traditional fixed-budget methods.

Dynamic Estimation in Slowly Varying Sequences: An Adaptive Meta-Algorithmic Approach

Problem Setting and Motivation

The paper addresses the general problem of dynamically estimating functions of elements in a slowly varying sequence, where the local change $\alpha_t = \|v_t - v_{t-1}\|$ is small for each timestep. This problem is fundamental in several domains including dynamic trace estimation in matrix analysis, maintaining statistics in drifting data streams, sequential spectral density estimation in time-evolving networks, and other online estimation settings. The key challenge is minimizing the cumulative sample/query complexity required to maintain accurate estimates at each step, particularly when the sequence exhibits low variation except for rare shifts.

Previous work, such as [10], has established query complexity bounds proportional to the worst-case local change ($m \cdot \max_t \alpha_t$), which is statistically suboptimal for sequences dominated by stable intervals. The present work advances the field by proposing an adaptive framework that scales estimation costs with the path-length of the sequence ($\sum_{t=2}^m \alpha_t$), a more nuanced and theoretically optimal measure of sequence variability.

Algorithmic Framework

The core contribution is a general-purpose, sequential stochastic approximation framework for estimating both linear and nonlinear functions over sequences in abstract vector spaces. The framework leverages well-concentrated unbiased estimators (with sub-exponential concentration properties), which covers estimators for trace, spectral statistics, Monte Carlo integrals, and related quantities.

The algorithm operates as follows:

• At each time $t$, instead of recomputing the estimate from scratch, the algorithm updates a running estimate via (i) recycling the previous estimate (with damping), and (ii) estimating the residual (the change in the target function induced by the state change).
• Critically, the query/sample budget for each step is local and adaptive, scaling with the magnitude of the change $\alpha_t$. The algorithm can function in both settings where $\alpha_t$ is known, and where it must be estimated on-the-fly via norm-estimation oracles.
• Theoretical analysis shows that, for a target error $\epsilon$ and failure probability $\delta$, the total sample complexity achieves

$$O\left(\frac{\log(m/\delta)}{\epsilon^2} (1 + \sum_{t=2}^m \alpha_t)\right) + \text{lower-order terms},$$

substantially improving over previous bounds and achieving adaptivity to the sequence's variation.

A significant technical insight is that as long as the static estimator has sub-exponential concentration (as in Hutchinson's estimator for traces), the sequential combination of residual estimation and adaptive budgeting yields uniformly tight error guarantees across all $m$ steps.

Technical Analysis

Two central results underpin the paper's claims:

1. Dynamic Sample Complexity with Known Step Sizes. When the local change $m \cdot \max_t \alpha_t$0 is known, the adaptive algorithm maintains error at each step with high probability, with total sample complexity scaling with $m \cdot \max_t \alpha_t$1 rather than $m \cdot \max_t \alpha_t$2 (Theorem 1).
2. Unknown Step Sizes and Norm Estimation. In practice, $m \cdot \max_t \alpha_t$3 is rarely known exactly (e.g., in the implicit matrix-vector product setting, evaluating $m \cdot \max_t \alpha_t$4 is itself a challenge). The algorithm supports a two-phase update: first, approximate $m \cdot \max_t \alpha_t$5 using a norm-estimation oracle (which can itself rely on Hutchinson's estimator or Monte Carlo samples); second, allocate resources adaptively based on this proxy. The additional complexity overhead for norm estimation is only logarithmic in $m \cdot \max_t \alpha_t$6 and independent of the target error (Theorem 2).

These results rely on fine-grained concentration inequalities for sub-exponential random variables and a careful inductive analysis of error propagation in sequential stochastic updates.

Applications

The framework unifies and extends prior theory and algorithms for several prototypical dynamic estimation problems:

• Dynamic Trace Estimation: For sequences of implicit matrices, the algorithm matches or improves upon previous methods [10, 29] with explicit bounds on MVP (matrix-vector product) cost. Empirically, the adaptive algorithm outperforms the best prior fixed-budget methods, especially in sequences with rare but large changes.
• Matrix Powers and Spectral Statistics: The method extends directly to dynamic estimation of nonlinear matrix functions such as matrix powers and spectral moments, despite their nonlinearity. This is made possible by bounding the change in such functions via telescoping and sub-multiplicativity arguments.
• Dynamic Monte Carlo Integration: The general framework encompasses maintaining sequential estimates of evolving integrals, with guarantees on function evaluation cost scaling with total $m \cdot \max_t \alpha_t$7-path-length of the integrand sequence.
• Dynamic Dirichlet Problem: The work applies to classic PDE settings where boundary data drifts slowly, allowing efficient estimation of the time-evolving solution at a fixed interior point via adaptive random walk sampling.

Empirical results on synthetic and realistic scenarios (such as Hessian trace estimation in neural network training trajectories) demonstrate substantial sample complexity reductions relative to state-of-the-art baselines, confirming the practical impact of the path-length-based adaptation.

Broader Implications and Future Directions

From a theoretical standpoint, the results concretely bridge ideas from online learning (path-length-type regret bounds) and stochastic estimation, and provide a generally applicable template for adaptive resource allocation in sequential statistics. The framework's flexibility in handling both linear and certain nonlinear target functions extends its reach beyond classical trace estimation to dynamic system analysis, high-dimensional integrals, and even PDEs with moving boundary data.

The ability to estimate unknown local sequence variation on-the-fly with negligible additional cost has significant practical implications, as it removes the main tuning barrier of earlier methods and aligns estimator cost more closely with intrinsic difficulty of the problem instance.

Future research may explore extending the framework to:

• Higher-order interactions or more complex nonlinear observables;
• Adaptive estimation when full access to residual generation is not possible (e.g., limited functional evaluation or noisy feedback);
• Structured estimation over sequences with block-wise or hierarchical variation;
• Application to dynamically changing large graphs and real-time data streams with more complicated drift patterns.

Conclusion

This work introduces a general, adaptive framework for dynamic estimation in slowly varying sequences, achieving tight sample complexity bounds scaling with path-length rather than worst-case change. The generality of the estimator requirements ensures broad applicability, covering a range of statistics (trace, spectral moments, integrals, PDE solutions). Empirical results corroborate the theoretical claims, and the introduced methodology constitutes a strong foundation for future advances in adaptive sequential estimation in both theoretical and applied machine learning contexts.