PAC-Bayes Un-Expected Bernstein Inequality
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We present a new PAC-Bayesian generalization bound. Standard bounds contain a √(L_n ·KL/n) complexity term which dominates unless L_n, the empirical error of the learning algorithm's randomized predictions, vanishes. We manage to replace L_n by a term which vanishes in many more situations, essentially whenever the employed learning algorithm is sufficiently stable on the dataset at hand. Our new bound consistently beats state-of-the-art bounds both on a toy example and on UCI datasets (with large enough n). Theoretically, unlike existing bounds, our new bound can be expected to converge to 0 faster whenever a Bernstein/Tsybakov condition holds, thus connecting PAC-Bayesian generalization and excess risk bounds --- for the latter it has long been known that faster convergence can be obtained under Bernstein conditions. Our main technical tool is a new concentration inequality which is like Bernstein's but with X^2 taken outside its expectation.
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