Some applications of concentration inequalities to statistics

By Michel Talagrand. 2005

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On the empirical estimation of integral probability metrics (2012, methodology): “The following result is a simple modification of Massart’s finite class lemma [26], which will be used to prove Proposition 3.”
Concentration inequalities using the entropy method (2003, background): “In this section we simply present the main result, the reader can consult the background in the recent work of Koltchinskii Panchenko [15], Massart [25], and Bartlett, Boucheron, and Lugosi [2].”
Large-Scale Machine Learning (2020, background): “N ] | {z } Eopt Approximation error E app : reduced by choosing a larger model Estimation error E est : reduced by 1 Increasing N, or 2 Choosing smaller model [2, 5, 7] Optimization error E opt : reduced by 1 Running optimization alg.”
Moment inequalities for functions of independent random variables (2005, methodology): “Ledoux’s method relies on abstract functional inequalities known as logarithmic Sobolev inequalities and provides a powerful tool for deriving exponential inequalities for functions of independent random variables; see [6, 7, 8, 14, 30, 31, 36] for various applications.”
Oracle inequalities in empirical risk minimization and sparse recovery problems (2011, methodology): “The trick was previously used in Massart [73] and in Ph.”
Consistency and Convergence Rates of One-Class SVM and Related Algorithms (2005, background): “The control of such quantities has been the topic of much research recently, including for example Tsybakov (1997); Mammen and Tsybakov (1999); Massart (2000); Bartlett et al.”
Augmented minimax linear estimation (2017, unknown): “…Rn(Hn) = O(n−1/2); the Rademacher complexity of H?n = Hn − [0, 1]γψn is bounded by Rn(Hn) + Rn([0, 1]γψn) and the latter is equal to the Rademacher complexity of its extreme points Rn({0, γψn}), which bounded by √ 2 log(2)E‖γψn‖L2(Pn) by Massart’s finite class lemma (Massart, 2000, Lemma 5.2).”
Testing the Manifold Hypothesis (2013, background): “(159) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use We use Massart’s lemma to bound the second term, E [ sup f∈F 1 s s∑ i=1 σi(f̂j(xi)− f̂j−1(xi)) ] = E [ sup f∈F 〈σ, (f̂j − f̂j−1)〉L2(μs) ] (160) ≤ 3αj √ 2 ln(|Tj | · |Tj−1”
From Dependence to Causation (2016, unknown): “Massart (2000) formalizes the third condition as inf x∈X |2P(y = 1 |x = x)− 1| > 0.”
Statistical properties of kernel principal component analysis (2004, methodology): “Our approach builds on ideas pioneered by Massart [13], on the fact that Talagrand’s concentration inequality can be used to obtain sharp oracle inequalities for empirical risk minimization on a collection of function classes when the variance of the relative error can be related to the expected rel”
A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables (2009, background): “For beautiful lectures on the topic, we refer the dear reader to (5; 6).”
Concentration Inequalities (2008, unknown): “For the background we refer to the the recent work of Koltchinskii and Panchenko [43], Massart [44], Bartlett, Boucheron, and Lugosi [45], Lugosi and Wegkamp [46], Bousquet [47].”