Abstract
Bayesian inference in deep neural networks is challenging due to the high-dimensional, strongly multi-modal parameter posterior density landscape. Markov chain Monte Carlo approaches asymptotically recover the true posterior but are considered prohibitively expensive for large modern architectures. Local methods, which have emerged as a popular alternative, focus on specific parameter regions that can be approximated by functions with tractable integrals. While these often yield satisfactory empirical results, they fail, by definition, to account for the multi-modality of the parameter posterior. Such coarse approximations can be detrimental in practical applications, notably safety-critical ones. In this work, we argue that the dilemma between exact-but-unaffordable and cheap-but-inexact approaches can be mitigated by exploiting symmetries in the posterior landscape. These symmetries, induced by neuron interchangeability and certain activation functions, manifest in different parameter values leading to the same functional output value. We show theoretically that the posterior predictive density in Bayesian neural networks can be restricted to a symmetry-free parameter reference set. By further deriving an upper bound on the number of Monte Carlo chains required to capture the functional diversity, we propose a straightforward approach for feasible Bayesian inference. Our experiments suggest that efficient sampling is indeed possible, opening up a promising path to accurate uncertainty quantification in deep learning.
Keywords
- Uncertainty quantification
- Predictive uncertainty
- Bayesian inference
- Monte Carlo sampling
- Posterior symmetry
J. G. Wiese and L. Wimmer—Equal contribution.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
We assume the likelihood to be parameterized by a single parameter vector. In the case of neural networks (NNs), the parameter contains all weights and biases.
- 2.
- 3.
Recall that the pre-activation of neuron i in layer l is \(o_{li} = \sum _{j = 1}^{M_{l-1}} w_{lij}z_{(l-1)j} + b_{li}\). By the commutative property of sums, any permutation \(\pi : J \rightarrow J\) of elements from the set \(J = \{1, \dots , M_{l - 1} \}\) will lead to the same pre-activation:
\(o_{li} = \sum _{j \in J} w_{lij}z_{(l-1)j} + b_{li} = \sum _{j \in \pi (J)} w_{lij}z_{(l-1)j} + b_{li}.\).
- 4.
- 5.
References
Agrawal, D., Ostrowski, J.: A classification of G-invariant shallow neural networks. In: Advances in Neural Information Processing Systems (2022)
Ainsworth, S., Hayase, J., Srinivasa, S.: Git Re-Basin: merging models modulo permutation symmetries. In: The Eleventh International Conference on Learning Representations (2023)
Bardenet, R., Kégl, B.: An adaptive Monte-Carlo Markov chain algorithm for inference from mixture signals. J. Phys. Conf. Ser. 368, 012044 (2012)
Bona-Pellissier, J., Bachoc, F., Malgouyres, F.: Parameter identifiability of a deep feedforward ReLU neural network (2021)
Van den Broeck, G., Kersting, K., Natarajan, S., Poole, D.: An Introduction to Lifted Probabilistic Inference. MIT Press, Cambridge (2021)
Chen, A.M., Lu, H.M., Hecht-Nielsen, R.: On the geometry of feedforward neural network error surfaces. Neural Comput. 5(6), 910–927 (1993)
Daxberger, E., Kristiadi, A., Immer, A., Eschenhagen, R., Bauer, M., Hennig, P.: Laplace redux - effortless Bayesian deep learning. In: 35th Conference on Neural Information Processing Systems (NeurIPS 2021) (2021)
Draxler, F., Veschgini, K., Salmhofer, M., Hamprecht, F.: Essentially no barriers in neural network energy landscape. In: Proceedings of the 35th International Conference on Machine Learning, pp. 1309–1318. PMLR (2018)
Dua, D., Graff, C.: UCI Machine Learning Repository (2017)
Ensign, D., Neville, S., Paul, A., Venkatasubramanian, S.: The complexity of explaining neural networks through (group) invariants. In: Proceedings of Machine Learning Research, vol. 76 (2017)
Eschenhagen, R., Daxberger, E., Hennig, P., Kristiadi, A.: Mixtures of Laplace approximations for improved post-Hoc uncertainty in deep learning. In: Bayesian Deep Learning Workshop, NeurIPS 2021 (2021)
Garipov, T., Izmailov, P., Podoprikhin, D., Vetrov, D.P., Wilson, A.G.: Loss surfaces, mode connectivity, and fast ensembling of DNNs. In: Proceedings of the 32nd Conference on Neural Information Processing Systems (NeurIPS 2018) (2018)
Gelman, A., Hwang, J., Vehtari, A.: Understanding predictive information criteria for Bayesian models. Stat. Comput. 24(6), 997–1016 (2014)
Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Springer, Heidelberg (2007). https://doi.org/10.1007/BFb0103945
Hecht-Nielsen, R.: On the algebraic structure of feedforward network weight spaces. In: Advanced Neural Computers, pp. 129–135. Elsevier, Amsterdam (1990)
Hoffman, M.D., Gelman, A.: The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res. 15(47), 1593–1623 (2014)
Hüllermeier, E., Waegeman, W.: Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods. Mach. Learn. 110 (2021)
Izmailov, P., Vikram, S., Hoffman, M.D., Wilson, A.G.: What are Bayesian neural network posteriors really like? In: Proceedings of the 38th International Conference on Machine Learning, vol. 139. PMLR (2021)
Kipf, T.N., Welling, M.: Semi-supervised classification with graph convolutional networks. In: ICLR 2017 (2017)
Kůrková, V., Kainen, P.C.: Functionally equivalent feedforward neural networks. Neural Comput. 6(3), 543–558 (1994)
Lakshminarayanan, B., Pritzel, A., Blundell, C.: Simple and scalable predictive uncertainty estimation using deep ensembles. In: Proceedings of the 31st Conference on Neural Information Processing Systems (NIPS 2017) (2017)
MacKay, D.J.C.: Bayesian interpolation. Neural Comput. 4, 415–447 (1992)
Margossian, C.C., Hoffman, M.D., Sountsov, P., Riou-Durand, L., Vehtari, A., Gelman, A.: Nested \(\hat{R}\): assessing the convergence of Markov chain Monte Carlo when running many short chains (2022)
Nalisnick, E.T.: On priors for Bayesian neural networks. Ph.D. thesis, University of California, Irvine (2018)
Niepert, M.: Markov chains on orbits of permutation groups. In: Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence, p. 624–633. UAI’12, AUAI Press, Arlington, Virginia, USA (2012)
Niepert, M.: Symmetry-aware marginal density estimation. In: Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence. AAAI’13, pp. 725–731. AAAI Press (2013)
Papamarkou, T., Hinkle, J., Young, M.T., Womble, D.: Challenges in Markov chain Monte Carlo for Bayesian neural networks. Stat. Sci. 37(3) (2022)
Pearce, T., Leibfried, F., Brintrup, A.: Uncertainty in neural networks: approximately Bayesian ensembling. In: Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 108, pp. 234–244. PMLR, 26–28 August 2020
Petzka, H., Trimmel, M., Sminchisescu, C.: Notes on the symmetries of 2-layer ReLU-networks. In: Northern Lights Deep Learning Workshop, vol. 1 (2020)
Pittorino, F., Ferraro, A., Perugini, G., Feinauer, C., Baldassi, C., Zecchina, R.: Deep networks on toroids: removing symmetries reveals the structure of flat regions in the landscape geometry. In: Proceedings of the 39th International Conference on Machine Learning, vol. 162. PMLR (2022)
Pourzanjani, A.A., Jiang, R.M., Petzold, L.R.: Improving the identifiability of neural networks for Bayesian inference. In: Second Workshop on Bayesian Deep Learning (NIPS) (2017)
Rosenthal, J.S.: Parallel computing and Monte Carlo algorithms. Far East J. Theor. Stat. 4, 207–236 (2000)
Sen, D., Papamarkou, T., Dunson, D.: Bayesian neural networks and dimensionality reduction (2020). arXiv: 2008.08044
Sussmann, H.J.: Uniqueness of the weights for minimal feedforward nets with a given input-output map. Neural Netw. 5(4), 589–593 (1992)
Tatro, N.J., Chen, P.Y., Das, P., Melnyk, I., Sattigeri, P., Lai, R.: Optimizing mode connectivity via neuron alignment. In: Proceedings of the 34th Conference on Neural Information Processing Systems (NeurIPS 2020) (2020)
Vlačić, V., Bölcskei, H.: Affine symmetries and neural network identifiability. Adv. Math. 376, 107485 (2021)
Wilson, A.G., Izmailov, P.: Bayesian deep learning and a probabilistic perspective of generalization. In: Proceedings of the 34th International Conference on Neural Information Processing Systems. NIPS’20. Curran Associates Inc., Red Hook, NY, USA (2020)
Acknowledgments
LW is supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the German Federal Ministry of Education and Research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Wiese, J.G., Wimmer, L., Papamarkou, T., Bischl, B., Günnemann, S., Rügamer, D. (2023). Towards Efficient MCMC Sampling in Bayesian Neural Networks by Exploiting Symmetry. In: Koutra, D., Plant, C., Gomez Rodriguez, M., Baralis, E., Bonchi, F. (eds) Machine Learning and Knowledge Discovery in Databases: Research Track. ECML PKDD 2023. Lecture Notes in Computer Science(), vol 14169. Springer, Cham. https://doi.org/10.1007/978-3-031-43412-9_27
Download citation
DOI: https://doi.org/10.1007/978-3-031-43412-9_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-43411-2
Online ISBN: 978-3-031-43412-9
eBook Packages: Computer ScienceComputer Science (R0)
-
Published in cooperation with
http://www.ecmlpkdd.org/
