Muriel F. Pérez-Ortiz

arXiv2026

Anytime-Valid Tests for Sparse Anomalies

We consider the problem of testing sequentially for the presence of sparse anomalies among a large number of data streams. To this end, we design and analyze Anytime-Valid (AV) tests, which retain type-I error control at arbitrary stopping times. Existing results address exclusively the nonsequential case, which exhibits a subtle phase transition between two regimes where tests are either powerless or powerful. In our sequential setting, we argue, two challenges arise: (1) the standard analysis of AV tests cannot be executed in the relevant sample-size regime; and (2) standard constructions of parameter-adaptive AV tests are either analytically intractable or computationally unfeasible. This work addresses these challenges. Borrowing insights from the nonsequential literature, we propose a framework to analyze AV tests and their shortest possible sample sizes. Under this framework, we show that, in the Gaussian location setting, the oracle AV test has a delicate threshold behavior that is related to -- but not implied by -- the phase transition observed in optimal nonsequential tests. Our main results include a computationally efficient, parameter-adaptive AV test; we show that it achieves the same threshold behavior as the oracle AV test. Numerical simulations illustrate these theoretical findings.

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arXiv2023

Exponential Stochastic Inequality

We develop the concept of exponential stochastic inequality (ESI), a novel notation that simultaneously captures high-probability and in-expectation statements. It is especially well suited to succinctly state, prove, and reason about excess-risk and generalization bounds in statistical learning, specifically, but not restricted to, the PAC-Bayesian type. We show that the ESI satisfies transitivity and other properties which allow us to use it like standard, nonstochastic inequalities. We substantially extend the original definition from Koolen et al. (2016) and show that general ESIs satisfy a host of useful additional properties, including a novel Markov-like inequality. We show how ESIs relate to, and clarify, PAC-Bayesian bounds, subcentered subgamma random variables and *fast-rate conditions* such as the central and Bernstein conditions. We also show how the ideas can be extended to random scaling factors (learning rates).

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