WederivePAC-Bayesian learning guarantees forheavy-tailed losses, andobtain a novel optimal Gibbs posterior which enjoys finite-sample excess risk bounds atlogarithmic confidence. Ourcoretechnique itselfmakesuseofPAC-Bayesian inequalities in order to derive a robust risk estimator, which by design is easy to compute.

We study the generalisation properties of majority voting on finite ensembles of classifiers, proving margin-based generalisation bounds via the PAC-Bayes theory. These provide state-of-the-art guarantees on a number of classification tasks. Our central results leverage the Dirichlet posteriors studied recently by Zantedeschi et al. (2021) for training voting classifiers; in contrast to that work our bounds apply to non-randomised votes via the use of margins. Our contributions add perspective to the debate on the "margins theory" proposed by Schapire et al. (1998) for the generalisation of ensemble classifiers.

Minimising upper bounds on the population risk or the generalisation gap has been widely used in structural risk minimisation (SRM) -- this is in particular at the core of PAC-Bayesian learning. Despite its successes and unfailing surge of interest in recent years, a limitation of the PAC-Bayesian framework is that most bounds involve a Kullback-Leibler (KL) divergence term (or its variations), which might exhibit erratic behavior and fail to capture the underlying geometric structure of the learning problem -- hence restricting its use in practical applications.As a remedy, recent studies have attempted to replace the KL divergence in the PAC-Bayesian bounds with the Wasserstein distance. Even though these bounds alleviated the aforementioned issues to a certain extent, they either hold in expectation, are for bounded losses, or are nontrivial to minimize in an SRM framework. In this work, we contribute to this line of research and prove novel Wasserstein distance-based PAC-Bayesian generalisation bounds for both batch learning with independent and identically distributed (i.i.d.) data, and online learning with potentially non-i.i.d.

Most PAC-Bayesian bounds hold in the batch learning setting where data is collected at once, prior to inference or prediction. This somewhat departs from many contemporary learning problems where data streams are collected and the algorithms must dynamically adjust. We prove new PAC-Bayesian bounds in this online learning framework, leveraging an updated definition of regret, and we revisit classical PAC-Bayesian results with a batch-to-online conversion, extending their remit to the case of dependent data. Our results hold for bounded losses, potentially \emph{non-convex}, paving the way to promising developments in online learning.

Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on a rotated MNIST dataset with a non-uniform rotation group, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.

Based on the theoretical results, the paper proposes two methods to construct a scoring function for binary classification problems based on sequential Monte Carlo and an expectation-propagation (EP) algorithms, respectively. The performance was evaluated using well-known benchmark data sets and more realistic DNA data set.