aiqbench analysis
This is statistical theory rather than a practical ML system. It gives a way to estimate dependence in a single time-series trajectory and turn that estimate into data-driven confidence bounds, without knowing the model's constants. For ML readers, the main value is in bounds that could justify learning criteria or generalization guarantees for sequential data with long memory, though the paper's assumptions (Wasserstein contraction, regularity of kernel estimates) limit where it applies.
Empirical concentration for chains with infinite memory
arXiv:2610.32265v1 Announce Type: cross Abstract: Concentration inequalities for time series usually involve unknown quantities measuring dependence. This prevents their direct use as confidence bounds, or as criteria for learning algorithms. Recent works proved empirical concentration inequalities for stationary Markov chains, where the unknown quantities are estimated. We extend this approach beyond the Markov case, by considering stationary chains with infinite memory. We first establish a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. We then estimate its dependence coefficients from a single trajectory, using either transition counts when the state space is finite, or derivatives of kernel conditional-density estimates for a continuous state space. We give general error bounds and illustrate their dependence on the memory tail. Finally, a fixed multiplicative safety factor turns these estimators into empirical concentration bounds. While some regularity assumptions are required to make the bounds empirical, our procedure remains fully adaptive in the sense that it does not require the knowledge of the constants involved in these assumptions. The empirical confidence bounds derived from our results are asymptotically as tight as the theoretical ones.
Abstract:Concentration inequalities for time series usually involve unknown quantities measuring dependence. This prevents their direct use as confidence bounds, or as criteria for learning algorithms. Recent works proved empirical concentration inequalities for stationary Markov chains, where the unknown quantities are estimated. We extend this approach beyond the Markov case, by considering stationary chains with infinite memory. We first establish a bounded-differences inequality on bounded Polish metric spaces under a summable Wasserstein contraction condition. We then estimate its dependence coefficients from a single trajectory, using either transition counts when the state space is finite, or derivatives of kernel conditional-density estimates for a continuous state space. We give general error bounds and illustrate their dependence on the memory tail. Finally, a fixed multiplicative safety factor turns these estimators into empirical concentration bounds. While some regularity assumptions are required to make the bounds empirical, our procedure remains fully adaptive in the sense that it does not require the knowledge of the constants involved in these assumptions. The empirical confidence bounds derived from our results are asymptotically as tight as the theoretical ones.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2610.32265 [stat.ML]
(or arXiv:2610.32265v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.32265 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 276 words
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