Adversarial Variational Optimization of Non-Differentiable Simulators

Complex computer simulators are increasingly used across fields of science as generative models tying parameters of an underlying theory to experimental observations. Inference in this setup is often difficult, as simulators rarely admit a tractable density or likelihood function. We introduce Adversarial Variational Optimization (AVO), a likelihood-free inference algorithm for fitting a non-differentiable generative model incorporating ideas from generative adversarial networks, variational optimization and empirical Bayes. We adapt the training procedure of generative adversarial networks by replacing the differentiable generative network with a domain-specific simulator. We solve the resulting non-differentiable minimax problem by minimizing variational upper bounds of the two adversarial objectives. Effectively, the procedure results in learning a proposal distribution over simulator parameters, such that the JS divergence between the marginal distribution of the synthetic data and the empirical distribution of observed data is minimized. We evaluate and compare the method with simulators producing both discrete and continuous data.

[1]  Léon Bottou,et al.  Towards Principled Methods for Training Generative Adversarial Networks , 2017, ICLR.

[2]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[3]  Dustin Tran,et al.  Operator Variational Inference , 2016, NIPS.

[4]  F. Maltoni,et al.  MadGraph 5: going beyond , 2011, 1106.0522.

[5]  Sean Gerrish,et al.  Black Box Variational Inference , 2013, AISTATS.

[6]  Michael P. H. Stumpf,et al.  Simulation-based model selection for dynamical systems in systems and population biology , 2009, Bioinform..

[7]  Ferenc Huszár,et al.  Variational Inference using Implicit Distributions , 2017, ArXiv.

[8]  Yoshua Bengio,et al.  Generative Adversarial Nets , 2014, NIPS.

[9]  Klaus-Robert Müller,et al.  Wasserstein Training of Restricted Boltzmann Machines , 2016, NIPS.

[10]  Paul Marjoram,et al.  Markov chain Monte Carlo without likelihoods , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Shakir Mohamed,et al.  Variational Approaches for Auto-Encoding Generative Adversarial Networks , 2017, ArXiv.

[12]  Aaron C. Courville,et al.  Improved Training of Wasserstein GANs , 2017, NIPS.

[13]  Peter Skands,et al.  A brief introduction to PYTHIA 8.1 , 2007, Comput. Phys. Commun..

[14]  Léon Bottou,et al.  Wasserstein GAN , 2017, ArXiv.

[15]  David Pfau,et al.  Unrolled Generative Adversarial Networks , 2016, ICLR.

[16]  C. Robert,et al.  Inference in generative models using the Wasserstein distance , 2017, 1701.05146.

[17]  David Duvenaud,et al.  Backpropagation through the Void: Optimizing control variates for black-box gradient estimation , 2017, ICLR.

[18]  Marco Cuturi,et al.  Sinkhorn Distances: Lightspeed Computation of Optimal Transport , 2013, NIPS.

[19]  Dustin Tran,et al.  Hierarchical Implicit Models and Likelihood-Free Variational Inference , 2017, NIPS.

[20]  Aaron C. Courville,et al.  Adversarially Learned Inference , 2016, ICLR.

[21]  Gilles Louppe,et al.  Approximating Likelihood Ratios with Calibrated Discriminative Classifiers , 2015, 1506.02169.

[22]  Trevor Darrell,et al.  Adversarial Feature Learning , 2016, ICLR.

[23]  Aki Vehtari,et al.  ELFI: Engine for Likelihood Free Inference , 2016, J. Mach. Learn. Res..

[24]  Jukka Corander,et al.  Likelihood-Free Inference by Ratio Estimation , 2016, Bayesian Analysis.

[25]  Michael U. Gutmann,et al.  Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models , 2015, J. Mach. Learn. Res..

[26]  A. Mincholé,et al.  Variational Inference over Non-differentiable Cardiac Simulators using Bayesian Optimization , 2017, 1712.03353.

[27]  A. Dell'Acqua,et al.  Geant4 - A simulation toolkit , 2003 .

[28]  Mark M. Tanaka,et al.  Sequential Monte Carlo without likelihoods , 2007, Proceedings of the National Academy of Sciences.

[29]  Robert Leenders,et al.  Hamiltonian ABC , 2015, UAI.

[30]  T. Weber,et al.  Stochastic Gradient Estimation With Finite Differences , 2016 .

[31]  David Barber,et al.  Variational Optimization , 2012, ArXiv.

[32]  Aapo Hyvärinen,et al.  Noise-Contrastive Estimation of Unnormalized Statistical Models, with Applications to Natural Image Statistics , 2012, J. Mach. Learn. Res..

[33]  D. Rubin Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician , 1984 .

[34]  Ritabrata Dutta,et al.  Likelihood-free inference via classification , 2014, Stat. Comput..

[35]  David J. Nott,et al.  Variational Bayes With Intractable Likelihood , 2015, 1503.08621.

[36]  Sebastian Nowozin,et al.  The Numerics of GANs , 2017, NIPS.

[37]  Ronald J. Williams,et al.  Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning , 2004, Machine Learning.

[38]  Wojciech Zaremba,et al.  Improved Techniques for Training GANs , 2016, NIPS.

[39]  Sebastian Nowozin,et al.  Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks , 2017, ICML.

[40]  D. Balding,et al.  Approximate Bayesian computation in population genetics. , 2002, Genetics.

[41]  Yanan Fan,et al.  Likelihood-Free MCMC , 2011 .

[42]  David Barber,et al.  Optimization by Variational Bounding , 2013, ESANN.

[43]  J. Zico Kolter,et al.  Gradient descent GAN optimization is locally stable , 2017, NIPS.

[44]  Shakir Mohamed,et al.  Learning in Implicit Generative Models , 2016, ArXiv.

[45]  Jean-Michel Marin,et al.  Approximate Bayesian computational methods , 2011, Statistics and Computing.

[46]  Gabriel Peyré,et al.  Stochastic Optimization for Large-scale Optimal Transport , 2016, NIPS.

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

[48]  Jan Hasenauer,et al.  pyABC: distributed, likelihood-free inference , 2017, bioRxiv.

[49]  Sebastian Nowozin,et al.  Which Training Methods for GANs do actually Converge? , 2018, ICML.

[50]  Sebastian Nowozin,et al.  Stabilizing Training of Generative Adversarial Networks through Regularization , 2017, NIPS.

[51]  Tom Schaul,et al.  Natural Evolution Strategies , 2008, 2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence).

[52]  Gilles Louppe,et al.  Experiments using machine learning to approximate likelihood ratios for mixture models , 2016 .

[53]  Oriol Vinyals,et al.  Synthesizing Programs for Images using Reinforced Adversarial Learning , 2018, ICML.