Probability in Banach Spaces: Isoperimetry and Processes

By Michel Ledoux, Michel Talagrand. 1991

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Fair Regression: Quantitative Definitions and Reduction-based Algorithms (2019, background): “For a ﬁxed tuple ( s 1 , u 1 ) , . . . , ( s n , u n ) , we have where the ﬁrst inequality follows from Theorem 12(5) of Bartlett & Mendelson (2002) and the last inequality follows from the contraction principle of Ledoux & Talagrand (1991), speciﬁcally their Theorem 4.12.”
Balanced Policy Evaluation and Learning (2017, methodology): “Therefore, by the Rademacher comparison lemma [22, Thm. 4.12], we have”
Optimal Rates for Random Fourier Features (2015, background): “Due to this, [5, 27] study the convergence behavior of Am when the diameter of S grows with m and show that almost sure convergence of Am is guaranteed as long as the diameter of S is e.”
Rademacher and Gaussian Complexities: Risk Bounds and Structural Results (2003, background): “The inequality of part 4 is due to Ledoux and Talagrand (1991, Corollary 3.17).”
A Reductions Approach to Fair Classification (2018, background): “…Theorem 12(5) of Bartlett & Mendelson (2002) for bounding function classes shifted by an offset, in our case g ( x, a, y, 0) , and Theorem 4.4 of Ledoux & Talagrand (1991) for bounding function classes under contraction, in our case g ( x, a, y, 1) − g ( x, a, y, 0) , yielding Together with the…”
Domain Adaptation: Learning Bounds and Algorithms (2009, positive): “Thus, by Talagrand’s contraction lemma (Ledoux & Talagrand, 1991),̂R(LH) is bounded by2qR̂(H ′) with H ′ = {x 7→ (h′(x) − h(x)) : h, h′ ∈ H}.”
Interpolation Consistency Training for Semi-Supervised Learning (2019, background): “Lemma 5 (Talagrand’s Contraction Lemma by Ledoux & Talagrand, 2013; Mohri & Medina, 2014).”
Local Rademacher complexities (2005, background): “Recallthefollowing im provem entofRio’s[29]version ofTalagrand’sconcentration inequality,which isdueto Bousquet[7,8].”
The Price of Selection in Differential Privacy (2017, background): “-valued random variables, let Z= P n i=1 Z i, and let = E[Z]. Then 1.(Upper Tail) 8>0 P[Z(1+)] e 2 2+ , and 2.(Lower Tail) 82(0;1) P[Z(1 )] e 12 2: Theorem 2.8 (Anticoncentration [LT13]). Let Z1;:::;Z nbe independent and uniform in f1g, and let Z= 1 n P n i=1 Z i. Then for every >0, there ex”
Sparse additive models (2007, background): “By Gaussian comparison results (Ledoux and Talagrand, 1991), we have then that E.”
User-Friendly Tail Bounds for Sums of Random Matrices (2010, methodology): “We use a different type of proof that is inspired by methods from probability in Banach spaces [33].”
The landscape of empirical risk for nonconvex losses (2016, methodology): “Now we apply the Rademacher contraction inequality [LT13] to bound this quantity, which says that for any set T ⊂ Rn and any family of L-Lipschitz functions {φi}i∈[n], φi : R → R, with φi(0) = 0, E [ sup t∈T n ∑”
The Benefit of Multitask Representation Learning (2015, background): “One key result in the latter direction is known as Slepian’s Lemma (Slepian (1962), Ledoux and Talagrand (1991)).”
Policy Evaluation and Optimization with Continuous Treatments (2018, positive): “The result follows from the Rademacher generalization bound [1, Thm. 8] concentrating E ˆ v τ near E ˆ v τ , Thm. 1 relating V τ to E ˆ v τ , and the Rademacher comparison lemma [14, Thm. 4.12].”
Robust Lasso With Missing and Grossly Corrupted Observations (2011, methodology): “This can be estimated via Sudakov–Fernique inequality [41].”
Excess risk bounds for multitask learning with trace norm regularization (2012, background): “( A standard symmetrization argument see [19], Lemma 6.3) shows that where the σ k are Rademacher variables and E σ is the expectation conditional on the A 1 , . . . , A N .”
High-dimensional additive modeling (2008, background): “− Therefore, by the contraction inequality of Ledoux and Talagrand [18], we have Corollary 8.”
Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞ (2011, unknown): “2) on page 57 of [22], give Pr{‖EY0(EΠn(f |Y0)− f0) +G‖∞ >Mεn} ≤ Pr{‖G‖∞ −E‖G‖∞ >Mεn −‖E(EΠn(f |Y0)− f0)‖∞ −E‖G‖∞} (60) ≤ Pr{‖G‖∞ −E‖G‖∞ > (M − C̄1 − C̄2)εn} ≤ exp ( − − C̄1 − C̄2) (2)ε(2)n C2 32 Jn/n )”
Uncertainty principles and ideal atomic decomposition (2001, background): “This is an exceptionally well-studied quantity; it is the classical example of so-called “concentration of measure phenomena” and “isoperimetry” [21].”