PAC-Bayesian Generalisation Error Bounds for Gaussian Process Classification - Simplified Proof and Tight Distribution-Free Bounds

Aug 3, 2026

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Approximate Bayesian Gaussian process (GP) classification offers powerful nonparametric learning with interpretable probabilistic models and compatibility with Bayesian model selection and feature selection. This paper applies the PAC-Bayesian theorem of McAllester to establish distribution-free generalisation error bounds for a broad class of approximate Bayesian GP classifiers. A new simplified proof is provided using convex duality. The bounds are instantiated and evaluated for Laplace GP classification and sparse greedy GP classification, and experiments on real-world data show tight guarantees for moderate training sample sizes, supporting their learning-theoretic justification.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages37

Reading Time1 h 33 min

AccessFree to download

Introduction

Bayesian learning and generalisation

PAC-Bayesian theorem and distribution-free bounds

Main results overview

Application to approximate Bayesian GP classifiers

Instantiations and experiments

Laplace GPC and sparse greedy GPC

Empirical evaluation and comparison

Discussion

earning PAC-Bayesian Generalisation Error Bounds for Gaussian Process Classi􀀌cation Matthias Seeger [seeger@dai.ed.ac.uk](seeger@dai.ed.ac.uk) Institute for Adaptive and Neural Computation University of Edinburgh 5 Forrest Hill, Edinburgh EH1 2QL, UK Editor: Peter Bartlett Abstract Approximate Bayesian Gaussian process (GP) classi􀀌cation techniques are powerful nonparametric learning methods, similar in appearance and performance to support vector machines. Based on simple probabilistic models, they render interpretable results and can be embedded in Bayesian frameworks for model selection, feature selection, etc. In this paper, by applying the PAC-Bayesian theorem of McAllester (1999a), we prove distributionfree generalisation error bounds for a wide range of approximate Bayesian GP classi􀀌cation techniques. We also provide a new and much simpli􀀌ed proof for this powerful theorem, making use of the concept of convex duality which is a backbone of many machine learning techniques. We instantiate and test our bounds for two particular GPC techniques, including a recent sparse method which circumvents the unfavourable scaling of standard GP algorithms. As is shown in experiments on a real-world task, the bounds can be very tight for moderate training sample sizes. To the best of our knowledge, these results provide the tightest known distribution-free error bounds for approximate Bayesian GPC methods, giving a strong learning-theoretical justi􀀌cation for the use of these techniques. Keywords: Gaussian Processes, Generalisation Error Bounds, PAC-Bayesian Framework, Bayesian Learning, Sparse Approximations, Gibbs Classi􀀌er, Kernel Machines, Convex Duality. 1. Introduction The Bayesian framework for probabilistic inference is widely used all over the statistics and machine learning communities, due to its high 􀀍exibility, its ability to render interpretable results and its conceptual simplicity. Within the framework, essential and di􀀎cult tasks like model and feature selection have canonical solutions. Complex models for real-world situations can be combined from simple, well-understood components in a structured way. Last, but not least, pitfalls hindering successful generalisation from 􀀌nite data, such as over-􀀌tting, can be tackled in a clear and principled way, so that Bayesian or approximate Bayesian solutions are typically among the top performers on di􀀎cult learning tasks. It is therefore of high theoretical and practical importance to analyse and understand the generalisation capability of (approximate) Bayesian methods. Many analyses so far have concentrated on the case where the true data distribution (stable aspects of which we try to learn) comes from a known family, which is either exactly the model family that the Bayesian method is using, or one which is close in some sense (e.g. , Haussler and 􀀍c2002 Matthias Seeger. Opper, 1997, Haussler et al. , 1994, Sollich, 1999) . Such analyses are important because they show up the principal limitations of the model family and the induction method, and because they often render close approximations to the true generalisation error we observe on independent test samples. However, they cannot give a guaranteed upper bound on the generalisation error (or other expectations of the true data distribution), because the validity of the whole analysis depends on assumptions that may not hold for the data distribution. PAC analyses 1 of the generalisation capability of a learning technique provide such guaranteed bounds, in the sense that the probability of observing a violation of the bound is shown to be smaller than some a priori 􀀌xed 􀀎 > 0, where the probability is over random draws of the training sample from the true data distribution. We can hope to 􀀌nd such non-trivial bounds for 􀀌nite training sample sizes, because we constrain the sampling process which generates the training set.2 Recently, a general result was obtained by McAllester (1999a) which

FAQs

What does the paper prove about approximate Bayesian Gaussian process classification?

It proves distribution-free generalisation error bounds for a wide range of approximate Bayesian GP classification techniques using the PAC-Bayesian theorem.

How is the PAC-Bayesian theorem applied in this work?

The theorem is specialized to approximate Bayesian GP classifiers to produce data-dependent PAC bounds, with a stated simplified proof strategy in the appendix.

Which GP classification methods are tested experimentally, and what do the results indicate?

The paper evaluates Laplace GPC and sparse greedy GPC, including a sparse method with favorable scaling. Experiments on a real-world task show the bounds can be very tight for moderate training sample sizes.

Aug 3, 2026

Upload by Ophelia