Generalization Bounds via Information Density and Conditional Information Density

TL;DR

This paper introduces a unified exponential inequalities framework to derive new and existing bounds on the generalization error of randomized learning algorithms, incorporating information density and conditional information measures.

Contribution

It develops a general approach using exponential inequalities to obtain novel and existing generalization bounds based on information density and extends to subset-dependent algorithms.

Findings

Bounds depend on information density for sub-Gaussian losses.

Extended bounds for algorithms based on random data subsets.

Introduces bounds involving conditional $eta$-mutual information and maximal leakage.

Abstract

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 $\alpha$-mutual information and the conditional maximal leakage.