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\title{Well-Defined Function-Space Variational Inference\\in Bayesian Neural Networks via Regularized KL-Divergence}
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\author[1]{\href{mailto:<tristan.cinquin@uni-tuebingen.de>?Subject=Your UAI 2025 paper}{Tristan Cinquin}{}}
\author[1]{Robert Bamler}
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    University of Tübingen, Germany
}  

\begin{document}
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\begin{abstract}
% 1. Background—usually brief, but including an indication of centrality (1-3 sentences)
Bayesian neural networks (BNN) promise to combine the predictive performance of neural networks with principled uncertainty modeling crucial for safety-critical systems and decision making.
% 2. Real world problem/motivation/gap in knowledge (0.5-1 sentence)
However, posterior uncertainties depend on the choice of prior, and finding informative priors in weight-space has proven difficult.
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This has motivated variational inference (VI) methods that pose priors directly on the function represented by the BNN rather than on weights.
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In this paper, we address a fundamental issue with such function-space VI approaches pointed out by \citet{burt2020understanding}, who showed that the objective function (ELBO) is negative infinite for most priors of interest.
Our solution builds on \emph{generalized} VI with the regularized KL divergence and is, to the best of our knowledge, the first well-defined variational objective for inference in BNNs with Gaussian process (GP) priors.
% 5. Benefits/performance/applications/implications (1-2 sentences)
Experiments show that our method successfully incorporates the properties specified by the GP prior, and that it provides competitive uncertainty estimates for regression, classification and out-of-distribution detection compared to BNN baselines with both function and weight-space priors.
\end{abstract}

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\begin{acknowledgements}
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy~--~EXC number 2064/1~--~Project number 390727645, and by the German Research Foundation (DFG) under project 448588364 of the Emmy Noether Programme.
Additional support was provided by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ:~01IS18039A.
The authors extend their gratitude to the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Tristan Cinquin.
Finally, Tristan Cinquin thanks Marvin Pförtner for very useful discussions on the theory and experiments, and Vincent Fortuin for feedback. 
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\title{Supplementary Material}
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\appendix

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