Concentration Inequalities for Sums and Martingales

By Bernard Bercu, Bernard Delyon, E. Rio. 2015

152 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Numerical Analysis

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Citations in the corpus

Stochastic Primal-Dual Q-Learning (2018, positive): “To do so, ﬁrst deﬁne and E (2) Then, G T is written by where The next step is to use the Berstein inequality for Martingales (Freedman, 1975; Fan et al., 2012; Bercu et al., 2015) to prove the convergence of G T /T , in (33).”
Time-uniform Chernoff bounds via nonnegative supermartingales (2018, methodology): “For the square-integrable martingale cases (f, g), we achieve control for a broad class of processes by incorporating the conditional variance and the observed squared deviations, as known for d = 1 (Delyon, 2009, Theorem 4; Bercu et al., 2015).”
Time-uniform, nonparametric, nonasymptotic confidence sequences (2018, methodology): “First, in the method of mixtures, one replaces the likelihood ratio with a mixture ∫ ∏ i[fλ(Xi)/f0(Xi)]dF(λ), which is still a martingale [5, 7, 14, 16, 38, 41, 55, 57, 58, 72, 73].”
Minimax optimality of permutation tests (2020, result): “In contrast to the previous studies (e.g. Chatterjee, 2007; Bercu et al., 2015; Albert, 2019) that are restricted to simple linear statistics, the proposed combinatorial inequalities are broadly applicable to the class of degenerate U -statistics.”
Uniform, nonparametric, non-asymptotic confidence sequences (2018, methodology): “In the method of mixtures (Ville, 1939; Wald, 1945; Darling and Robbins, 1968; Robbins and Siegmund, 1969, 1970; Robbins, 1970; de la Peña et al., 2007; Balsubramani, 2014; Bercu et al., 2015), we replace the simple likelihood ratio with a mixture ∫ ∏ i fλ(Xi)/f0(Xi) dF (λ), which is still a marting”
Policy Optimization via Importance Sampling (2018, background): “4 Even Bernstein inequalities [4], are hardly applicable since, for instance, in the case of univariate Gaussian distributions, the importance weights display a fat tail behavior (see Appendix C).”
Nonasymptotic upper bounds for the reconstruction error of PCA (2016, result): “me from showing that this condition can be replaced by the localised version in (3.22). 3.16. Remark. Concentration inequalities for eigenvalues using tools from measure concentration can be found in [23, 26, 24, 1, 9, 6]. The main dierence to our deviation inequalities is that we take into account ”