A nonasymptotic law of iterated logarithm for general M-estimators
This work addresses the need for reliable, anytime statistical guarantees in machine learning, particularly for robust estimators and bandit problems, though it is incremental in extending existing asymptotic results to non-asymptotic settings.
The authors derived the first non-asymptotic deviation bounds for general M-estimators, providing anytime guarantees under conditions like Lipschitz loss and risk curvature, applicable to robust and heavy-tailed cases. They applied this to best arm identification in bandits, developing a provably optimal algorithm supported by numerical experiments.
M-estimators are ubiquitous in machine learning and statistical learning theory. They are used both for defining prediction strategies and for evaluating their precision. In this paper, we propose the first non-asymptotic "any-time" deviation bounds for general M-estimators, where "any-time" means that the bound holds with a prescribed probability for every sample size. These bounds are nonasymptotic versions of the law of iterated logarithm. They are established under general assumptions such as Lipschitz continuity of the loss function and (local) curvature of the population risk. These conditions are satisfied for most examples used in machine learning, including those ensuring robustness to outliers and to heavy tailed distributions. As an example of application, we consider the problem of best arm identification in a parametric stochastic multi-arm bandit setting. We show that the established bound can be converted into a new algorithm, with provably optimal theoretical guarantees. Numerical experiments illustrating the validity of the algorithm are reported.