Minimax rates without the fixed sample size assumption

Abstract: We generalize the notion of minimax convergence rate. In contrast to the standard definition, we do not assume that the sample size is fixed in advance. Allowing for varying sample size results in time-robust minimax rates and estimators. These can be either strongly adversarial, based on the worst-case over all sample sizes, or weakly adversarial, based on the worst-case over all stopping times. We show that standard and time-robust rates usually differ by at most a logarithmic factor, and that for some (and we conjecture for all) exponential families, they differ by exactly an iterated logarithmic factor. In many situations, time-robust rates are arguably more natural to consider. For example, they allow us to simultaneously obtain strong model selection consistency and optimal estimation rates, thus avoiding the "AIC-BIC dilemma".