Probability in Banach spaces

By Michel Ledoux, Michel Talagrand. 1991

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Citations in the corpus

Variance-based Regularization with Convex Objectives (2016, neutral): “suﬀered by making prediction hθ,xi when the label is y. By taking the function class F = {(x,y) → ℓ(yhx,θi) − ℓ(0) | θ∈ Θ}, in this case, an application of the Ledoux-Talagrand contraction inequality [28] implies for any y1,x1,...,yn,xn that E " sup θ∈Θ Xn i=1 εi [ℓ(yihθ,xii) −ℓ(0)] # ≤ E &quot”
Convexity, Classification, and Risk Bounds (2006, unknown): “The Ledoux and Talagrand (1991) contraction inequality and Lemma 7 imply that ξgF ( ) ≤ 4LE sup { 1 n n∑ i=1 σidφ ( Yi f (Xi),Yi f ∗(Xi) ) : f ∈F ,Egf = } ≤ 4LE sup { 1 n n∑ i=1 σidφ ( Yi f (Xi),Yi f ∗(Xi) ) : f ∈F , d̃φ( f , f ∗)2 ≤ 2aB } = 4LE sup { 1 n n∑ i=1 σi f (Xi,Yi) : f ∈ Fφ,Ef 2 ≤…”
Multilevel Monte Carlo Methods (2001, methodology): “valued Monte Carlo method needs the theory of scalar-valued random variables, that is, standard probability theory, we require for our analysis the respective vector-valued tools, which are provided by Banach space probability theory (see [ 9 ]).”
Aggregation via empirical risk minimization (2009, background): “Clearly, for every 1 ≤ i ≤ n, ‖Wi‖2 = M , and by the Kahane–Khintchine inequality [14]”
Donsker-type theorems for nonparametric maximum likelihood estimators (2007, background): “We give su¢ cient conditions on such sieves in a subsequent paper, see Nickl and Pötscher (2005). 2.”
Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators (2013, background): “ill be given in the forthcoming sections. 4.1. Theprobabilistictools. Let us recall the probabilistic tools that we shall use. They all concern the Orlicz norm of a random process and can be found in [17]. A Young function ψis a convex increasing function on R+ such that limt→∞ ψ(t) = ∞ and ψ(0) = 0”