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".

翻译：与标准定义相反,我们不认为样本大小是事先固定的。允许不同样本大小导致时间- 硬质迷轴率和估计值。这可以是强烈对立的,基于所有样本大小的最坏情况,也可以是在所有停顿时间的最坏情况基础上的对立性较弱。我们表明,标准和时间- 硬度率通常以对数因素为最多,对于一些(和我们所推测的)指数型家庭来说,它们因确切的迭代对数因素而有所不同。在许多情况下,时间- 硬度率可以说比较自然,例如,它们使我们同时获得强大的模型选择一致性和最佳估计率,从而避免“ACIC-BIC两难点 ” 。