Open Access
May 2017 Empirical entropy, minimax regret and minimax risk
Alexander Rakhlin, Karthik Sridharan, Alexandre B. Tsybakov
Bernoulli 23(2): 789-824 (May 2017). DOI: 10.3150/14-BEJ679


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.


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Alexander Rakhlin. Karthik Sridharan. Alexandre B. Tsybakov. "Empirical entropy, minimax regret and minimax risk." Bernoulli 23 (2) 789 - 824, May 2017.


Received: 1 March 2014; Revised: 1 July 2014; Published: May 2017
First available in Project Euclid: 4 February 2017

zbMATH: 1380.62176
MathSciNet: MR3606751
Digital Object Identifier: 10.3150/14-BEJ679

Keywords: Aggregation , empirical risk minimization , Entropy , minimax regret , minimax risk

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 2 • May 2017
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