Empirical entropy, minimax regret and minimax risk

TL;DR

This paper introduces a new aggregation method for regression that achieves optimal rates in well-specified models and explores the relationship between minimax regret and risk across different entropy growth regimes.

Contribution

It proposes a novel aggregation approach that attains sharp oracle inequalities and establishes the equivalence of minimax regret and risk rates for certain entropy conditions.

Findings

Achieves the rate $n^{-2/(2+p)}$ for $p extless 2$

Matches minimax risk behavior in well-specified models for $p extless 2$

Provides optimal rates for Vapnik-Chervonenkis classes without convexity

Abstract

We consider the random design regression model with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish sharp oracle inequalities for its risk. We show that, under the $\varepsilon^{-p}$ growth of the empirical $\varepsilon$-entropy, the excess risk of the proposed method 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 risk rate matches the behavior of the minimax risk of function estimation in regression problems under the well-specified model. This yields a conclusion that the rates of statistical estimation in well-specified models (minimax risk) 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 statistical learning 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, in general, slower than for the minimax risk. Our oracle inequalities also imply the $v\log(n/v)/n$ rates for Vapnik-Chervonenkis type classes of dimension $v$ without the usual convexity assumption on the class; we show that these rates are optimal. Finally, for a slightly modified method, we derive a bound on the excess risk of $s$-sparse convex aggregation improving that of Lounici [Math. Methods Statist. 16 (2007) 246-259] and providing the optimal rate.