Sharper Bounds for Gaussian and Empirical Processes

By Michel Talagrand. 1994

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Additive models with trend filtering (2017, background): “[2005] (see also Talagrand [1994]). Lemma 14. Let z, i = 1, . . . , n denote an i.i.d. sample from a distribution Λ on [0, 1]. Write 〈·, ·〉2 for the L2(Λ) inner product, and 〈·, ·〉n for the L2(Λn) inner product. Let J satisfy Assumption C3, and let ρ > 0 be arbitrary. Then there are constants c0, n0”
Local Rademacher complexities (2005, background): “2, is a more or less direct consequence of Talagrand’s inequality for empirical processes [30].”
Generalizing from a Few Examples (2019, background): “s that can be shattered (split in all possible ways) by H. The sample complexity S is tightly bounded as S = Θ VC(H) ϵ2 + log(1/δ) ϵ2 , (4) where the lower and upper bound are proven in [123] and [112] respectively. As shown, sample complexityS increases with more complicated Hchosen by the model,”
Quasi-oracle estimation of heterogeneous treatment effects (2017, methodology): “Below we state Talagrand’s Concentration Inequality for an empirical process indexed by a class of uniformly bounded functions (Talagrand, 1994, 1996).”
Size-free generalization bounds for convolutional neural networks (2019, methodology): “Its proof, which uses standard techniques [see Pollard, 1984, Talagrand, 1994, 1996, Barron et al., 1999, Giné and Guillou, 2001, Mohri et al., 2018], is in Appendix A. Lemma 2.4.”
Offline Multi-Action Policy Learning: Generalization and Optimization (2018, unknown): “Step 2: High probability bound on maximum deviation via Talagrand inequality. From the previous step, it remains to bound the difference between supπ∈Π ∣∣∣∆̃(π1, π2)−∆(π1, π2)∣∣∣ and E [ supπ∈Π ∣∣∣∆̃(π1, π2) − ∆(π1, π2)∣∣∣]. We establish a high-probability bound on this difference by applying the Be”
Private Hypothesis Selection (2019, unknown): “Theorem 4.4 (Uniform Convergence [Tal94]) .”
Differentially Private M-Estimators (2011, methodology): “First (Lemma 10) by empirical process theory [9, 10] we have, under conditions A(1) and A(2), the sampling error term in (8) is of order OP (1/ √ n), uniformly on Θ0.”
Block thresholding for density estimation: local and global adaptivity (2005, methodology): “whereC1 andC2 are the universal constants fromTalagrand [27].”
Direct importance estimation for covariate shift adaptation (2008, unknown): “Then by Talagrand (1994), EQ sup f ∈F ‖Qn f 2‖ ≤ sup f ∈F Q f 2 + 8Rn(F).”
Asymptotics for panel quantile regression models with individual effects (2012, background): “By the exponential β-mixing property (condition (D1)) and Theorem 3 of Yoshihara (1978), we now deduce that max1≤i≤n E[|zni|(3)] = O(1), which leads to that ∑n i=1 E[|zni|(3)] = O(n) = o(n).”
Uniform limit theorems for wavelet density estimators (2008, unknown): “Finally, for term (IV), using (6) and ( 37 ) we have”
Concentration inequalities and asymptotic results for ratio type empirical processes (2006, background): “A more interesting bound that follows from randomization together with a contraction principle for Rademacher processes is the following, given by Talagrand ([42], Corollary 3.”
Learning without Concentration (2014, unknown): “Therefore, L f (X, Y ) = ( f − f ∗)2(X) + 2( f − f ∗)(X)( f ∗(X) − Y ) = ( f − f ∗)2(X) + 2ξ ( f − f ∗)(X), (3.1) where, as always, f ∗ is the unique minimizer in F of E( f (X) − Y )2 and ξ (X, Y ) = f ∗(X) − Y .”