Empirical entropy, minimax regret and minimax risk

摘要

We consider the random design regression model with square loss. We propose amethod that aggregates empirical minimizers (ERM) over appropriately chosenrandom subsets and reduces to ERM in the extreme case, and we establish sharporacle inequalities for its risk. We show that, under the $\varepsilon^{-p}$growth of the empirical $\varepsilon$-entropy, the excess risk of the proposedmethod attains the rate $n^{-2/(2+p)}$ for $p\in(0,2)$ and $n^{-1/p}$ for $p>2$where $n$ is the sample size. Furthermore, for $p\in(0,2)$, the excess riskrate matches the behavior of the minimax risk of function estimation inregression problems under the well-specified model. This yields a conclusionthat the rates of statistical estimation in well-specified models (minimaxrisk) and in misspecified models (minimax regret) are equivalent in the regime$p\in(0,2)$. In other words, for $p\in(0,2)$ the problem of statisticallearning enjoys the same minimax rate as the problem of statistical estimation.On the contrary, for $p>2$ we show that the rates of the minimax regret are, ingeneral, slower than for the minimax risk. Our oracle inequalities also implythe $v\log(n/v)/n$ rates for Vapnik-Chervonenkis type classes of dimension $v$without the usual convexity assumption on the class; we show that these ratesare optimal. Finally, for a slightly modified method, we derive a bound on theexcess risk of $s$-sparse convex aggregation improving that of Lounici [Math.Methods Statist. 16 (2007) 246-259] and providing the optimal rate.