… guaranteed coverage, ie, it holds with probability at least 1 − α … supporting lemmas and then
the main result in Theorem 2.7. … Our bound outperforms Hoeffding and Maurer and Pontil’s …

… ities called empirical Bernstein bounds (EBBs) [Maurer and Pontil, 2009], which are … We
now state lemmas which we use to derive our stratified empirical … and Massimiliano Pontil. …

… As a consequence of Lemma 2, we can use classic notions … recite Theorem 11 from Maurer
and Pontil (2009) that gives a … empirical Bernstein bound (6) and the confidence bounds on …

… upper bound is superior, we conclude with empirical results … we prove that our high-probability
upper bound is a strict … present a lemma that is used in the proofs of both theorems. …

… inequality of the following form, which holds with probability at … PAC-Bayes empirical
Bernstein’s inequality by relaxing our … of m we consider a lower tail Bernstein’s inequality (Maurer…

… noncommutative Bernstein inequalities in (Maurer and Pontil… we give a proof of Lemma 2
and in Section 3 we give applications … is appropriate, to achieve empirical error η we will likely …

… While for Bernstein’s inequality we want the variance to be … Our preliminary survey provides
a small probabilistic toolbox … Lemma 4.4 in Chatterjee (2008) for the second inequality. So …

… ], Maurer and Pontil [2018] introduced Bernstein-type … Semi-empirical bounds proposed
in this paper offer … , where the proof crucially relies on Lemma 1, whose proof deferred to the …

… We start with the most basic question in probability theory and … Theorem 2 (Empirical Bernstein
Inequality [Maurer and Pontil… , which is based on the Unexpected Bernstein Lemma [Fan …

… estimation, empirical Bernstein bounds and BerryEsseen type bounds of … ) with probability
at least 1 − δ in X we have for every … representation of f (x) − f (x ), as given by the next lemma. …