Weight Uncertainty in Neural Network

We introduce a new, efficient, principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. It regularises the weights by minimising a compression cost, known as the variational free energy or the expected lower bound on the marginal likelihood. We show that this principled kind of regularisation yields comparable performance to dropout on MNIST classification. We then demonstrate how the learnt uncertainty in the weights can be used to improve generalisation in non-linear regression problems, and how this weight uncertainty can be used to drive the exploration-exploitation trade-off in reinforcement learning.

[1]  W. R. Thompson.  ON THE LIKELIHOOD THAT ONE UNKNOWN PROBABILITY EXCEEDS ANOTHER IN VIEW OF THE EVIDENCE OF TWO SAMPLES , 1933 .

[2]  Yann LeCun,et al.  Une procedure d'apprentissage pour reseau a seuil asymmetrique (A learning scheme for asymmetric threshold networks) , 1985 .

[3]  Geoffrey E. Hinton,et al.  Learning representations by back-propagating errors , 1986, Nature.

[4]  T. J. Mitchell,et al.  Bayesian Variable Selection in Linear Regression , 1988 .

[5]  Wray L. Buntine,et al.  Bayesian Back-Propagation , 1991, Complex Syst..

[6]  David J. C. MacKay,et al.  A Practical Bayesian Framework for Backpropagation Networks , 1992, Neural Computation.

[7]  Geoffrey E. Hinton,et al.  Keeping the neural networks simple by minimizing the description length of the weights , 1993, COLT '93.

[8]  E. George,et al.  Journal of the American Statistical Association is currently published by American Statistical Association. , 2007 .

[9]  David Mackay,et al.  Probable networks and plausible predictions - a review of practical Bayesian methods for supervised neural networks , 1995 .

[10]  Hugh Chipman,et al.  Bayesian variable selection with related predictors , 1995, bayes-an/9510001.

[11]  Michael I. Jordan,et al.  Mean Field Theory for Sigmoid Belief Networks , 1996, J. Artif. Intell. Res..

[12]  Geoffrey E. Hinton,et al.  A View of the Em Algorithm that Justifies Incremental, Sparse, and other Variants , 1998, Learning in Graphical Models.

[13]  W. Freeman,et al.  Generalized Belief Propagation , 2000, NIPS.

[14]  Michael I. Jordan,et al.  Bayesian parameter estimation via variational methods , 2000, Stat. Comput..

[15]  Tom Minka,et al.  A family of algorithms for approximate Bayesian inference , 2001 .

[16]  Patrice Y. Simard,et al.  Best practices for convolutional neural networks applied to visual document analysis , 2003, Seventh International Conference on Document Analysis and Recognition, 2003. Proceedings..

[17]  Thomas P. Minka,et al.  Divergence measures and message passing , 2005 .

[18]  Karl J. Friston,et al.  Variational free energy and the Laplace approximation , 2007, NeuroImage.

[19]  A. Gelman.  Objections to Bayesian statistics , 2008 .

[20]  Manfred Opper,et al.  The Variational Gaussian Approximation Revisited , 2009, Neural Computation.

[21]  Aurélien Garivier,et al.  Parametric Bandits: The Generalized Linear Case , 2010, NIPS.

[22]  Geoffrey E. Hinton,et al.  Rectified Linear Units Improve Restricted Boltzmann Machines , 2010, ICML.

[23]  Wei Chu,et al.  A contextual-bandit approach to personalized news article recommendation , 2010, WWW '10.

[24]  Alex Graves,et al.  Practical Variational Inference for Neural Networks , 2011, NIPS.

[25]  Yoshua Bengio,et al.  Deep Sparse Rectifier Neural Networks , 2011, AISTATS.

[26]  Lihong Li,et al.  An Empirical Evaluation of Thompson Sampling , 2011, NIPS.

[27]  Nitish Srivastava,et al.  Improving neural networks by preventing co-adaptation of feature detectors , 2012, ArXiv.

[28]  Rémi Munos,et al.  Thompson Sampling: An Asymptotically Optimal Finite-Time Analysis , 2012, ALT.

[29]  Marc'Aurelio Ranzato,et al.  Large Scale Distributed Deep Networks , 2012, NIPS.

[30]  David S. Leslie,et al.  Optimistic Bayesian Sampling in Contextual-Bandit Problems , 2012, J. Mach. Learn. Res..

[31]  Shipra Agrawal,et al.  Analysis of Thompson Sampling for the Multi-armed Bandit Problem , 2011, COLT.

[32]  Yann LeCun,et al.  Regularization of Neural Networks using DropConnect , 2013, ICML.

[33]  Shipra Agrawal,et al.  Further Optimal Regret Bounds for Thompson Sampling , 2012, AISTATS.

[34]  Miguel Lázaro-Gredilla,et al.  Doubly Stochastic Variational Bayes for non-Conjugate Inference , 2014, ICML.

[35]  Daan Wierstra,et al.  Stochastic Backpropagation and Approximate Inference in Deep Generative Models , 2014, ICML.

[36]  Max Welling,et al.  Auto-Encoding Variational Bayes , 2013, ICLR.

[37]  Daan Wierstra,et al.  Deep AutoRegressive Networks , 2013, ICML.

[38]  Geoffrey E. Hinton,et al.  Distilling the Knowledge in a Neural Network , 2015, ArXiv.

[39]  A. Guez.  Sample-based Search Methods for Bayes-Adaptive Planning , 2015 .