The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent.

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecué and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the flatness of the empirical risk surface on which the posterior concentrates. Our analysis establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields new insights on PAC-Bayesian theory.

Neural Information Processing Systems http://nips.cc/

We study the problem of sequential probability assignment under logarithmic loss, both with and without side information. Our objective is to analyze the minimax regret -- a notion extensively studied in the literature -- in terms of geometric quantities, such as covering numbers and scale-sensitive dimensions. We show that the minimax regret for the case of no side information (equivalently, the Shtarkov sum) can be upper bounded in terms of sequential square-root entropy, a notion closely related to Hellinger distance. For the problem of sequential probability assignment with side information, we develop both upper and lower bounds based on the aforementioned entropy. The lower bound matches the upper bound, up to log factors, for classes in the Donsker regime (according to our definition of entropy).

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecué and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates.

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecué and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates.

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecu\'{e} and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates. Our analysis establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields new insights on PAC-Bayesian theory.

Determining the finite-sample behavior of risk in the problem of regression is arguably one of the most basic problems of Learning Theory and Statistics. This behavior can be studied in substantial generality with the tools of empirical process theory. When functions in a given convex class are uniformly bounded, one may verify the socalled "Bernstein condition." The condition--which relates the variance of the increments of the empirical process to their expectation--implies a certain localization phenomenon around the optimum and forms the basis of the analysis via local Rademacher complexities. The technique has been developed in [9, 8, 5, 2, 4], among others, based on Talagrand's celebrated concentration inequality for the supremum of an empirical process. In a recent pathbreaking paper, [14] showed that a large part of this heavy machinery is not necessary for obtaining tight upper bounds on excess loss, even--and especially--if functions are unbounded. Mendelson observed that only one-sided control of the tail is required in the deviation inequality, and, thankfully, it is the tail that can be controlled under very mild assumptions. In a parallel line of work, the search within the online learning setting for an analogue of "localization" has led to a notion of an "offset" Rademacher process [17], yielding--in a rather clean manner--optimal rates for minimax regret in online supervised learning. It was also shown that the supremum of the offset process is a lower bound on the minimax value, thus establishing its intrinsic nature.