Bayesian deep learning

2017-01-11 – 2020-12-10
Maybe related 🤷:

Probably approximately a horse

WARNING: more than usually chaotic notes here

Bayesian inference for massively parameterised networks.

To learn:

  • marginal likelihood in model selection: how does it work with many optima?

Closely related: Generative models where we train a process to generate the phenomenon of interest.

Backgrounders

Radford Neal’s thesis (Neal 1996) is a foundational asymptotically-Bayesian use of neural networks. Yarin Gal’s PhD Thesis (Gal 2016) summarizes some implicit approximate approaches (e.g. the Bayesian interpretation of dropout. Diederik P. Kingma’s thesis is a blockbuster in this tradition.

Alex Graves did a poster of his paper (Graves 2011) of a simplest prior uncertainty thing for recurrent nets - (diagonal Gaussian weight uncertainty) There is a 3rd party quick and dirty implementation.

One could refer to the 2019 NeurIPS Bayes deep learning workshop site which will have some more modern positioning. There was a tutorial in 2020: by Dustin Tran, Jasper Snoek, Balaji Lakshminarayanan: Practical Uncertainty Estimation & Out-of-Distribution Robustness in Deep Learning.

Generative methods are useful here, e.g. the variational autoencoder and affiliated reparameterization trick. Likelihood free methods seems to be in the air too.

Sampling from

TBD

Stochastic Gradient Descent

I think this argument is leveraged in Neal (1996). But see this version Mandt, Hoffman, and Blei (2017):

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.

Ensemble methods

Deep learning has its own twists on model averaging and bagging: Neural ensembles.

Practicalities

The computational toolsets for “neural” probabilistic programming and vanilla probabilistic programming are converging. See the tool listing under probabilistic programming.

References

Baydin, Atılım Güneş, Lei Shao, Wahid Bhimji, Lukas Heinrich, Lawrence Meadows, Jialin Liu, Andreas Munk, et al. 2019. “Etalumis: Bringing Probabilistic Programming to Scientific Simulators at Scale.” In. http://arxiv.org/abs/1907.03382.
Doerr, Andreas, Christian Daniel, Martin Schiegg, Duy Nguyen-Tuong, Stefan Schaal, Marc Toussaint, and Sebastian Trimpe. 2018. “Probabilistic Recurrent State-Space Models.” January 31, 2018. http://arxiv.org/abs/1801.10395.
Domingos, Pedro. 2020. “Every Model Learned by Gradient Descent Is Approximately a Kernel Machine.” November 30, 2020. http://arxiv.org/abs/2012.00152.
Dupont, Emilien, Arnaud Doucet, and Yee Whye Teh. 2019. “Augmented Neural ODEs.” April 2, 2019. http://arxiv.org/abs/1904.01681.
Eleftheriadis, Stefanos, Tom Nicholson, Marc Deisenroth, and James Hensman. 2017. “Identification of Gaussian Process State Space Models.” In Advances in Neural Information Processing Systems 30, edited by I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, 5309–19. Curran Associates, Inc. http://papers.nips.cc/paper/7115-identification-of-gaussian-process-state-space-models.pdf.
Fabius, Otto, and Joost R. van Amersfoort. 2014. “Variational Recurrent Auto-Encoders.” In Proceedings of ICLR. http://arxiv.org/abs/1412.6581.
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———. 2016. “Uncertainty in Deep Learning.” University of Cambridge.
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———. 2015b. “Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning.” In Proceedings of the 33rd International Conference on Machine Learning (ICML-16). http://arxiv.org/abs/1506.02142.
———. 2016. “Bayesian Convolutional Neural Networks with Bernoulli Approximate Variational Inference.” In 4th International Conference on Learning Representations (ICLR) Workshop Track. http://arxiv.org/abs/1506.02158.
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