Generalization Bounds via Information Density and Conditional Information Density
This work provides incremental theoretical improvements for researchers in statistical learning theory, focusing on generalization bounds.
The paper tackles the problem of bounding generalization error in randomized learning algorithms by developing an approach based on exponential inequalities, resulting in novel bounds for sub-Gaussian loss functions and extensions to conditional settings, recovering and extending prior results.
We present a general approach, based on exponential inequalities, to derive bounds on the generalization error of randomized learning algorithms. Using this approach, we provide bounds on the average generalization error as well as bounds on its tail probability, for both the PAC-Bayesian and single-draw scenarios. Specifically, for the case of sub-Gaussian loss functions, we obtain novel bounds that depend on the information density between the training data and the output hypothesis. When suitably weakened, these bounds recover many of the information-theoretic bounds available in the literature. We also extend the proposed exponential-inequality approach to the setting recently introduced by Steinke and Zakynthinou (2020), where the learning algorithm depends on a randomly selected subset of the available training data. For this setup, we present bounds for bounded loss functions in terms of the conditional information density between the output hypothesis and the random variable determining the subset choice, given all training data. Through our approach, we recover the average generalization bound presented by Steinke and Zakynthinou (2020) and extend it to the PAC-Bayesian and single-draw scenarios. For the single-draw scenario, we also obtain novel bounds in terms of the conditional $α$-mutual information and the conditional maximal leakage.