Concentration inequalities and model selection

By Pascal Massart, Jean Picard, École d'été de probabilités de Saint-Flour. 2007

993 citations recorded by the source corpus.

Subfields: Applied Mathematics, Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Hybrid Deterministic-Stochastic Methods for Data Fitting (2011, background): “If the individual ∇ f i ( x ) are concentrated around ∇ f ( x ), this might allow us to use concentration inequalities [20] to show that the convergence rates hold with high probability.”
Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning (2017, background): “Such bounds have sparked recent interest in sequential decision making [18, 19, 20, 21, 22] but to the best of our knowledge we are the first to leverage them for RL.”
Oracle Inequalities for High Dimensional Vector Autoregressions (2012, background): “Since c = µg ( x ) + (1 − µ ) g ( y ) for some 0 The Borell-Cirelson-Sudakov inequality (see e.g. Massart (2007), Theorem 3.4) then yields that the ﬁrst probability in (21) can be be bounded by 2 exp Regarding the second probability in (21) note that by the union bound and standard tail…”
Risk minimization by median-of-means tournaments (2016, background): “A sample of the rich literature includes Gy¨orﬁ, Kohler, Krzyzak, Walk [17], van de Geer [39], Bartlett, Bousquet, and Mendelson [6], Koltchinskii [23], Massart [28].”
Convex Formulations for Fair Principal Component Analysis (2018, methodology): “Consequently, applying Pinsker’s inequality (Massart 2007) gives we have (Kullback 1997): where The key observation here is that (4) is minimized when s + = s − and m + = m − , and we will use this insight to propose constraints for FPCA.”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, methodology): “Now, Cirelson’s inequality [see, e.g., Massart (2007)] P (√ U > √ kn + √ 2x )≤ exp(−x)(26) used with x = ( √ Mn−2√kn)2 8 implies Lemma 5.”
Finite-Dimensional Projection for Classification and Statistical Learning (2008, neutral): “If we draw a parallel to what can be shown in the least squares regression case [5], we must however consider several caveats .”
On the asymptotic properties of the group lasso estimator for linear models (2008, methodology): “The lasso, originally put forward by Tibshirani (1996) for linear regression models, is a regularization procedure in which the penalty for model complexity is the l1 norm of the estimated coefficients.”
Anti-concentration and honest, adaptive confidence bands (2013, methodology): “There exist several techniques in the literature to construct l̂n so that n,f (l̂n) can be controlled and the confidence band Cn is adaptive; see, for example, [35] for a thorough introduction.”
Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality (2010, methodology): “The functional classes Ns,d(α,L) were considered in approximation theory by Nikol’skii; see, e.g., Nikol’skii (1969). Minimax estimation of densities from the class Ns,d(α,L) was considered in Ibragimov and Khasminskii (1981).”
Statistical performance of support vector machines (2008, background): “4) [with the square loss l(x, y) = (x− y)2 replacing the hinge loss] is completely elucidated (see [24], Section 4.”
Global risk bounds and adaptation in univariate convex regression (2013, background): “Convex regression is an important problem in the general area of nonparametric estimation under shape constraints.”
A More Powerful Two-Sample Test in High Dimensions using Random Projection (2011, background): “By the Gaussian isoperimetric inequality [30], we have for any s > 0,”