2013/08/31 by Alexander Rakhlin, Karthik Sridharan, Alexandre B. Tsybakov · 2 citations
Computer Science · Decision Sciences · Mathematics · #Applied mathematics #Combinatorics #Convexity #Dimension (graph theory) #Econometrics #Economics #Empirical risk minimization #Machine Learning and Algorithms #Mathematical optimization #Mathematics #Minimax #Regret #Risk and Portfolio Optimization #Sample size determination #Statistical Methods and Inference #Statistics #cs.LG #math.ST #stat.TH
paper · pdf · doi:10.3150/14-bej679
published as Bernoulli 2017, Vol. 23, No. 2, 789-824 · Published at http://dx.doi.org/10.3150/14-BEJ679 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex publication_date 2017/02/04 · arxiv created 2017/07/03 · arxiv updated 2017/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 ε-p growth of the empirical ε-entropy, the excess risk of the proposed method attains the rate n-2/(2+p) for p∈(0,2) and n-1/p for p>2 where n is the sample size. Furthermore, for p∈(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∈(0,2). In other words, for p∈(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 vlog(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.