# Bayesian posterior inference via optimisation

August 17, 2020 — April 4, 2024

Bayes
estimator distribution
Markov processes
Monte Carlo
probabilistic algorithms
probability

The Bayes-by-backprop terminology seems to come from Blundell et al. (2015).

## 1 Bayesian learning rule

M. E. Khan and Rue (2023):

We show that a wide-range of well-known learning-algorithms from a variety of fields are all specific instances of a single learning algorithm derived from Bayesian principles. The starting point, is the variational formulation by Zellner (1988), which is an extension of [Eq. 1 to optimize over a well-defined candidate distribution $$q(\boldsymbol{\theta})$$, and for which the minimizer $q_*(\boldsymbol{\theta})=\underset{q(\boldsymbol{\theta})}{\arg \min } \quad \mathbb{E}_q\left[\sum_{i=1}^N \ell\left(y_i, f_{\boldsymbol{\theta}}\left(\boldsymbol{x}_i\right)\right)\right]+\mathbb{D}_{K L}[q(\boldsymbol{\theta}) \| p(\boldsymbol{\theta})]$ defines a generalized posterior in lack of a precise likelihood. The prior distribution is related to the regularizer, $$p(\boldsymbol{\theta}) \propto \exp (-R(\boldsymbol{\theta}))$$, and $$\mathbb{D}_{K L}[\cdot \| \cdot]$$ is the Kullback-Leibler Divergence (KLD). In the case where $$\exp \left(-\ell\left(y_i, f_{\boldsymbol{\theta}}\left(\boldsymbol{x}_i\right)\right)\right)$$ is proportional to the likelihood for $$y_i, \forall i$$, then $$q_*(\boldsymbol{\theta})$$ is the posterior distribution for $$\boldsymbol{\theta}$$ .

The result is heavy on natural gradient and exponential families. Also Emti is very charismatic and I defy you to watch his presentation and not feel like this is the One True Way, at least for a few minutes. rPRobably related: Knoblauch, Jewson, and Damoulas (2022).

## 2 SGD as MCMC

Combining Markov Chain Monte Carlo and Stochastic Gradient Descent for in the sense of using SGD to do some cheap approximation to MCMC posterior sampling. Overviews in Ma, Chen, and Fox (2015) and Mandt, Hoffman, and Blei (2017). A lot of probabilistic neural nets leverage this idea.

A related idea is estimating gradients of parameters by Monte Carlo; there is nothing necessarily Bayesian about that per se; in that case we are doing a noisy estimate of a deterministic quantity. In this setting we are interested in the noise itself.

I have a vague memory that this argument is leveraged in Neal (1996)? Should check. For sure the version in Mandt, Hoffman, and Blei (2017) is a highly developed and modern take. Basically, they analyse the distribution near convergence as an autoregressive process:

Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results.

1. We show that constant SGD can be used as an approximate Bayesian posterior inference algorithm. Specifically, we show how to adjust the tuning parameters of constant SGD to best match the stationary distribution to a posterior, minimizing the Kullback-Leibler divergence between these two distributions.
2. We demonstrate that constant SGD gives rise to a new variational EM algorithm that optimizes hyperparameters in complex probabilistic models.
3. We also propose SGD with momentum for sampling and show how to adjust the damping coefficient accordingly.
4. We analyze MCMC algorithms. For Langevin Dynamics and Stochastic Gradient Fisher Scoring, we quantify the approximation errors due to finite learning rates. Finally,
5. we use the stochastic process perspective to give a short proof of why Polyak averaging is optimal. Based on this idea, we propose a scalable approximate MCMC algorithm, the Averaged Stochastic Gradient Sampler.

The article is rather beautiful. Importantly they leverage the assumption that we are sampling from approximately (log-)quadratic posterior modes, which means that we should be suspicious of the method when

1. The posterior is not quadratic, i.e. the distribution is not well approximated by a Gaussian at the mode, and
2. The same for the tails. If there are low-probability but high importance posterior configurations such that they are not Gaussian in the tails, we should be skeptical that they will be sampled well; I have an intuition that this is a more stringent requirement, but TBH I am not sure of the exact relationship of these two conditions.

The analysis leverages gradient flow, which is a continuous limit of stochastic gradient descent.

## 3 Stochastic Weight Averaging

A popular recent development is the Stochastic Weight Averaging family of methods . See Andrew G Wilson’s web page for a brief description of the sub methods, since he seems to have been involved in all of them.

## 5 Stein Variational GD

Perhaps related? An ensemble method. See Stein VGD.

## 6 SG Hamiltonian Monte Carlo

This, surprisingly, works, I am told? T. Chen, Fox, and Guestrin (2014).

## 7 SG thermostats

Some kind of variance control using auxiliary variables? See Ding et al. (2014).

## 8 SG Fisher scoring

See Ahn, Korattikara, and Welling (2012). I assume there is a connection to MC gradients via the score trick?.

## 9 Incoming

Knoblauch, Jewson, and Damoulas (2019):

We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes’ rule as infinite-dimensional optimization problem . First, we use it to prove an optimality result of standard Variational Inference (VI): Under the proposed view, the standard Evidence Lower Bound (ELBO) maximizing VI posterior is preferable to alternative approximations of the Bayesian posterior. Next, we argue for generalizing standard Bayesian inference. The need for this arises in situations of severe misalignment between reality and three assumptions underlying standard Bayesian inference: (1) Well-specified priors, (2) well-specified likelihoods, (3) the availability of infinite computing power. Our generalization addresses these shortcomings with three arguments and is called the Rule of Three (RoT). We derive it axiomatically and recover existing posteriors as special cases, including the Bayesian posterior and its approximation by standard VI. In contrast, approximations based on alternative ELBO-like objectives violate the axioms. Finally, we study a special case of the RoT that we call Generalized Variational Inference (GVI). GVI posteriors are a large and tractable family of belief distributions specified by three arguments: A loss, a divergence and a variational family. GVI posteriors have appealing properties, including consistency and an interpretation as approximate ELBO. The last part of the paper explores some attractive applications of GVI in popular machine learning models, including robustness and more appropriate marginals. After deriving black box inference schemes for GVI posteriors, their predictive performance is investigated on Bayesian Neural Networks and Deep Gaussian Processes, where GVI can comprehensively improve upon existing methods.

## 10 References

Ahn, Korattikara, and Welling. 2012. In Proceedings of the 29th International Coference on International Conference on Machine Learning. ICML’12.
Alexos, Boyd, and Mandt. 2022. In Proceedings of the 39th International Conference on Machine Learning.
Bissiri, Holmes, and Walker. 2016. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Blundell, Cornebise, Kavukcuoglu, et al. 2015. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37. ICML’15.
Bradley, Gomez-Uribe, and Vuyyuru. 2022. Machine Learning: Science and Technology.
Brosse, Moulines, and Durmus. 2018. In Proceedings of the 32nd International Conference on Neural Information Processing Systems. NIPS’18.
Catoni. 2007. IMS Lecture Notes Monograph Series.
Chada, and Tong. 2022. Mathematics of Computation.
Chandramoorthy, Loukas, Gatmiry, et al. 2022.
Chaudhari, Choromanska, Soatto, et al. 2017.
Chaudhari, and Soatto. 2018. In 2018 Information Theory and Applications Workshop (ITA).
Chen, Tianqi, Fox, and Guestrin. 2014. In Proceedings of the 31st International Conference on Machine Learning.
Chen, Zaiwei, Mou, and Maguluri. 2021.
Choi, Jang, and Alemi. 2019.
Csiszár. 1975. The Annals of Probability.
Dehaene. 2016. arXiv:1612.05053 [Stat].
Detommaso, Cui, Spantini, et al. 2018. In Proceedings of the 32nd International Conference on Neural Information Processing Systems. NIPS’18.
Dieuleveut, Durmus, and Bach. 2018.
Ding, Fang, Babbush, et al. 2014. In Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2. NIPS’14.
Donsker, and Varadhan. 1975. Communications on Pure and Applied Mathematics.
Durmus, and Moulines. 2016. arXiv:1605.01559 [Math, Stat].
Dutordoir, Hensman, van der Wilk, et al. 2021. In arXiv:2105.04504 [Cs, Stat].
Feng, and Tu. 2021. Proceedings of the National Academy of Sciences.
Futami, Sato, and Sugiyama. 2017. arXiv:1710.06595 [Stat].
Ge, Lee, and Risteski. 2020. arXiv:1812.00793 [Cs, Math, Stat].
Girolami, and Calderhead. 2011. Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Goldt, and Seifert. 2017. Physical Review Letters.
Grenander, and Miller. 1994. Journal of the Royal Statistical Society: Series B (Methodological).
Gu, Levine, Sutskever, et al. 2016. In Proceedings of ICLR.
Hodgkinson, Salomone, and Roosta. 2019. arXiv:1903.12322 [Cs, Stat].
Honkela, Tornio, Raiko, et al. 2008. In Neural Information Processing. Lecture Notes in Computer Science.
Immer, Korzepa, and Bauer. 2021. In International Conference on Artificial Intelligence and Statistics.
Izmailov, Maddox, Kirichenko, et al. 2020. In Proceedings of The 35th Uncertainty in Artificial Intelligence Conference.
Izmailov, Podoprikhin, Garipov, et al. 2018.
Khan, Mohammad Emtiyaz, Immer, Abedi, et al. 2020. arXiv:1906.01930 [Cs, Stat].
Khan, Mohammad Emtiyaz, and Lin. 2017. In Artificial Intelligence and Statistics.
Khan, Mohammad, Nielsen, Tangkaratt, et al. 2018. In Proceedings of the 35th International Conference on Machine Learning.
Khan, Mohammad Emtiyaz, and Rue. 2023.
Knoblauch, Jewson, and Damoulas. 2019.
———. 2022. “An Optimization-Centric View on Bayes’ Rule: Reviewing and Generalizing Variational Inference.” Journal of Machine Learning Research.
Kristiadi, Hein, and Hennig. 2021. In Uncertainty in Artificial Intelligence.
Liu, and Wang. 2019. In Advances In Neural Information Processing Systems.
Ma, Chen, and Fox. 2015. In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2. NIPS’15.
Maclaurin, Duvenaud, and Adams. 2015. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics.
Maddox, Garipov, Izmailov, et al. 2019.
Mandt, Hoffman, and Blei. 2017. JMLR.
Margossian, Vehtari, Simpson, et al. 2020. arXiv:2004.12550 [Stat].
Martens. 2020. Journal of Machine Learning Research.
Neal. 1996.
Norton, and Fox. 2016. arXiv:1610.00781 [Math, Stat].
Osawa, Swaroop, Khan, et al. 2019. In Advances in Neural Information Processing Systems.
Papamarkou, Skoularidou, Palla, et al. 2024.
Parisi. 1981. Nuclear Physics B.
Rásonyi, and Tikosi. 2022. Statistics & Probability Letters.
Ritter, Kukla, Zhang, et al. 2021. arXiv:2105.14594 [Cs, Stat].
Ruiz, Titsias, and Blei. 2016. In Advances In Neural Information Processing Systems.
Sato. 2001. Neural Computation.
Shang, Zhu, Leimkuhler, et al. 2015. In Advances in Neural Information Processing Systems. NIPS’15.
Smith, Dherin, Barrett, et al. 2020. In.
Sun, Yang, Xun, et al. 2023. ACM Transactions on Knowledge Discovery from Data.
Wainwright, and Jordan. 2008. Graphical Models, Exponential Families, and Variational Inference. Foundations and Trends® in Machine Learning.
Welling, and Teh. 2011. In Proceedings of the 28th International Conference on International Conference on Machine Learning. ICML’11.
Wenzel, Roth, Veeling, et al. 2020. In Proceedings of the 37th International Conference on Machine Learning.
Wilson, and Izmailov. 2020.
Xifara, Sherlock, Livingstone, et al. 2014. Statistics & Probability Letters.
Zellner. 1988. The American Statistician.
———. 2002. Journal of Econometrics, Information and Entropy Econometrics,.
Zhang, Tong. 1999. In Proceedings of the Twelfth Annual Conference on Computational Learning Theory. COLT ’99.
Zhang, Xinhua. 2013.
Zhang, Yao, Saxe, Advani, et al. 2018. Molecular Physics.
Zhang, Guodong, Sun, Duvenaud, et al. 2018. In Proceedings of the 35th International Conference on Machine Learning.