Two recently introduced criteria for estimation of generative models are both based on a reduction to binary classification. Noise-contrastive estimation (NCE) is an estimation procedure in which a generative model is trained to be able to distinguish data samples from noise samples. Generative adversarial networks (GANs) are pairs of generator and discriminator networks, with the generator network learning to generate samples by attempting to fool the discriminator network into believing its samples are real data. Both estimation procedures use the same function to drive learning, which naturally raises questions about how they are related to each other, as well as whether this function is related to maximum likelihood estimation (MLE). NCE corresponds to training an internal data model belonging to the {\em discriminator} network but using a fixed generator network. We show that a variant of NCE, with a dynamic generator network, is equivalent to maximum likelihood estimation. Since pairing a learned discriminator with an appropriate dynamically selected generator recovers MLE, one might expect the reverse to hold for pairing a learned generator with a certain discriminator. However, we show that recovering MLE for a learned generator requires departing from the distinguishability game. Specifically: (i) The expected gradient of the NCE discriminator can be made to match the expected gradient of MLE, if one is allowed to use a non-stationary noise distribution for NCE, (ii) No choice of discriminator network can make the expected gradient for the GAN generator match that of MLE, and (iii) The existing theory does not guarantee that GANs will converge in the non-convex case. This suggests that the key next step in GAN research is to determine whether GANs converge, and if not, to modify their training algorithm to force convergence.

* This version adds a figure that appeared on the poster at ICLR, changes the template to say that the paper was accepted as a workshop contribution (previously it was under a review as a conference submission), and fixes some typos

* This version adds a figure that appeared on the poster at ICLR, changes the template to say that the paper was accepted as a workshop contribution (previously it was under a review as a conference submission), and fixes some typos

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We propose a new type of hidden layer for a multilayer perceptron, and demonstrate that it obtains the best reported performance for an MLP on the MNIST dataset.

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Explaining and Harnessing Adversarial Examples

Mar 20, 2015

Ian J. Goodfellow, Jonathon Shlens, Christian Szegedy

Several machine learning models, including neural networks, consistently misclassify adversarial examples---inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset, such that the perturbed input results in the model outputting an incorrect answer with high confidence. Early attempts at explaining this phenomenon focused on nonlinearity and overfitting. We argue instead that the primary cause of neural networks' vulnerability to adversarial perturbation is their linear nature. This explanation is supported by new quantitative results while giving the first explanation of the most intriguing fact about them: their generalization across architectures and training sets. Moreover, this view yields a simple and fast method of generating adversarial examples. Using this approach to provide examples for adversarial training, we reduce the test set error of a maxout network on the MNIST dataset.
Mar 20, 2015

Ian J. Goodfellow, Jonathon Shlens, Christian Szegedy

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Joint Training Deep Boltzmann Machines for Classification

May 01, 2013

Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

We introduce a new method for training deep Boltzmann machines jointly. Prior methods of training DBMs require an initial learning pass that trains the model greedily, one layer at a time, or do not perform well on classification tasks. In our approach, we train all layers of the DBM simultaneously, using a novel training procedure called multi-prediction training. The resulting model can either be interpreted as a single generative model trained to maximize a variational approximation to the generalized pseudolikelihood, or as a family of recurrent networks that share parameters and may be approximately averaged together using a novel technique we call the multi-inference trick. We show that our approach performs competitively for classification and outperforms previous methods in terms of accuracy of approximate inference and classification with missing inputs.
May 01, 2013

Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

* Major revision with new techniques and experiments. This version includes new material put on the poster for the ICLR workshop

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Spike-and-Slab Sparse Coding for Unsupervised Feature Discovery

Apr 03, 2012

Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

We consider the problem of using a factor model we call {\em spike-and-slab sparse coding} (S3C) to learn features for a classification task. The S3C model resembles both the spike-and-slab RBM and sparse coding. Since exact inference in this model is intractable, we derive a structured variational inference procedure and employ a variational EM training algorithm. Prior work on approximate inference for this model has not prioritized the ability to exploit parallel architectures and scale to enormous problem sizes. We present an inference procedure appropriate for use with GPUs which allows us to dramatically increase both the training set size and the amount of latent factors. We demonstrate that this approach improves upon the supervised learning capabilities of both sparse coding and the ssRBM on the CIFAR-10 dataset. We evaluate our approach's potential for semi-supervised learning on subsets of CIFAR-10. We demonstrate state-of-the art self-taught learning performance on the STL-10 dataset and use our method to win the NIPS 2011 Workshop on Challenges In Learning Hierarchical Models' Transfer Learning Challenge.
Apr 03, 2012

Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

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Qualitatively characterizing neural network optimization problems

May 21, 2015

Ian J. Goodfellow, Oriol Vinyals, Andrew M. Saxe

Training neural networks involves solving large-scale non-convex optimization problems. This task has long been believed to be extremely difficult, with fear of local minima and other obstacles motivating a variety of schemes to improve optimization, such as unsupervised pretraining. However, modern neural networks are able to achieve negligible training error on complex tasks, using only direct training with stochastic gradient descent. We introduce a simple analysis technique to look for evidence that such networks are overcoming local optima. We find that, in fact, on a straight path from initialization to solution, a variety of state of the art neural networks never encounter any significant obstacles.
May 21, 2015

Ian J. Goodfellow, Oriol Vinyals, Andrew M. Saxe

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On the Challenges of Physical Implementations of RBMs

Oct 24, 2014

Vincent Dumoulin, Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

Restricted Boltzmann machines (RBMs) are powerful machine learning models, but learning and some kinds of inference in the model require sampling-based approximations, which, in classical digital computers, are implemented using expensive MCMC. Physical computation offers the opportunity to reduce the cost of sampling by building physical systems whose natural dynamics correspond to drawing samples from the desired RBM distribution. Such a system avoids the burn-in and mixing cost of a Markov chain. However, hardware implementations of this variety usually entail limitations such as low-precision and limited range of the parameters and restrictions on the size and topology of the RBM. We conduct software simulations to determine how harmful each of these restrictions is. Our simulations are designed to reproduce aspects of the D-Wave quantum computer, but the issues we investigate arise in most forms of physical computation.
Oct 24, 2014

Vincent Dumoulin, Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

* Proc. AAAI 2014, pp. 1199-1205

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An empirical analysis of dropout in piecewise linear networks

Jan 02, 2014

David Warde-Farley, Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

The recently introduced dropout training criterion for neural networks has been the subject of much attention due to its simplicity and remarkable effectiveness as a regularizer, as well as its interpretation as a training procedure for an exponentially large ensemble of networks that share parameters. In this work we empirically investigate several questions related to the efficacy of dropout, specifically as it concerns networks employing the popular rectified linear activation function. We investigate the quality of the test time weight-scaling inference procedure by evaluating the geometric average exactly in small models, as well as compare the performance of the geometric mean to the arithmetic mean more commonly employed by ensemble techniques. We explore the effect of tied weights on the ensemble interpretation by training ensembles of masked networks without tied weights. Finally, we investigate an alternative criterion based on a biased estimator of the maximum likelihood ensemble gradient.
Jan 02, 2014

David Warde-Farley, Ian J. Goodfellow, Aaron Courville, Yoshua Bengio

* Extensive updates; 8 pages plus acknowledgements/references

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An Empirical Investigation of Catastrophic Forgetting in Gradient-Based Neural Networks

Mar 04, 2015

Ian J. Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, Yoshua Bengio

Catastrophic forgetting is a problem faced by many machine learning models and algorithms. When trained on one task, then trained on a second task, many machine learning models "forget" how to perform the first task. This is widely believed to be a serious problem for neural networks. Here, we investigate the extent to which the catastrophic forgetting problem occurs for modern neural networks, comparing both established and recent gradient-based training algorithms and activation functions. We also examine the effect of the relationship between the first task and the second task on catastrophic forgetting. We find that it is always best to train using the dropout algorithm--the dropout algorithm is consistently best at adapting to the new task, remembering the old task, and has the best tradeoff curve between these two extremes. We find that different tasks and relationships between tasks result in very different rankings of activation function performance. This suggests the choice of activation function should always be cross-validated.
Mar 04, 2015

Ian J. Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, Yoshua Bengio

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Multi-digit Number Recognition from Street View Imagery using Deep Convolutional Neural Networks

Apr 14, 2014

Ian J. Goodfellow, Yaroslav Bulatov, Julian Ibarz, Sacha Arnoud, Vinay Shet

Recognizing arbitrary multi-character text in unconstrained natural photographs is a hard problem. In this paper, we address an equally hard sub-problem in this domain viz. recognizing arbitrary multi-digit numbers from Street View imagery. Traditional approaches to solve this problem typically separate out the localization, segmentation, and recognition steps. In this paper we propose a unified approach that integrates these three steps via the use of a deep convolutional neural network that operates directly on the image pixels. We employ the DistBelief implementation of deep neural networks in order to train large, distributed neural networks on high quality images. We find that the performance of this approach increases with the depth of the convolutional network, with the best performance occurring in the deepest architecture we trained, with eleven hidden layers. We evaluate this approach on the publicly available SVHN dataset and achieve over $96\%$ accuracy in recognizing complete street numbers. We show that on a per-digit recognition task, we improve upon the state-of-the-art, achieving $97.84\%$ accuracy. We also evaluate this approach on an even more challenging dataset generated from Street View imagery containing several tens of millions of street number annotations and achieve over $90\%$ accuracy. To further explore the applicability of the proposed system to broader text recognition tasks, we apply it to synthetic distorted text from reCAPTCHA. reCAPTCHA is one of the most secure reverse turing tests that uses distorted text to distinguish humans from bots. We report a $99.8\%$ accuracy on the hardest category of reCAPTCHA. Our evaluations on both tasks indicate that at specific operating thresholds, the performance of the proposed system is comparable to, and in some cases exceeds, that of human operators.
Apr 14, 2014

Ian J. Goodfellow, Yaroslav Bulatov, Julian Ibarz, Sacha Arnoud, Vinay Shet

**Click to Read Paper and Get Code**

Realistic Evaluation of Deep Semi-Supervised Learning Algorithms

Oct 26, 2018

Avital Oliver, Augustus Odena, Colin Raffel, Ekin D. Cubuk, Ian J. Goodfellow

Semi-supervised learning (SSL) provides a powerful framework for leveraging unlabeled data when labels are limited or expensive to obtain. SSL algorithms based on deep neural networks have recently proven successful on standard benchmark tasks. However, we argue that these benchmarks fail to address many issues that these algorithms would face in real-world applications. After creating a unified reimplementation of various widely-used SSL techniques, we test them in a suite of experiments designed to address these issues. We find that the performance of simple baselines which do not use unlabeled data is often underreported, that SSL methods differ in sensitivity to the amount of labeled and unlabeled data, and that performance can degrade substantially when the unlabeled dataset contains out-of-class examples. To help guide SSL research towards real-world applicability, we make our unified reimplemention and evaluation platform publicly available.
Oct 26, 2018

Avital Oliver, Augustus Odena, Colin Raffel, Ekin D. Cubuk, Ian J. Goodfellow

* NIPS 2018 Proceedings

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Maxout Networks

Sep 20, 2013

Ian J. Goodfellow, David Warde-Farley, Mehdi Mirza, Aaron Courville, Yoshua Bengio

We consider the problem of designing models to leverage a recently introduced approximate model averaging technique called dropout. We define a simple new model called maxout (so named because its output is the max of a set of inputs, and because it is a natural companion to dropout) designed to both facilitate optimization by dropout and improve the accuracy of dropout's fast approximate model averaging technique. We empirically verify that the model successfully accomplishes both of these tasks. We use maxout and dropout to demonstrate state of the art classification performance on four benchmark datasets: MNIST, CIFAR-10, CIFAR-100, and SVHN.
Sep 20, 2013

Ian J. Goodfellow, David Warde-Farley, Mehdi Mirza, Aaron Courville, Yoshua Bengio

* JMLR WCP 28 (3): 1319-1327, 2013

* This is the version of the paper that appears in ICML 2013

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Generative Adversarial Networks

Jun 10, 2014

Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio

We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a discriminative model D that estimates the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This framework corresponds to a minimax two-player game. In the space of arbitrary functions G and D, a unique solution exists, with G recovering the training data distribution and D equal to 1/2 everywhere. In the case where G and D are defined by multilayer perceptrons, the entire system can be trained with backpropagation. There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate the potential of the framework through qualitative and quantitative evaluation of the generated samples.
Jun 10, 2014

Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio

**Click to Read Paper and Get Code**

Pylearn2: a machine learning research library

Aug 20, 2013

Ian J. Goodfellow, David Warde-Farley, Pascal Lamblin, Vincent Dumoulin, Mehdi Mirza, Razvan Pascanu, James Bergstra, Frédéric Bastien, Yoshua Bengio

Pylearn2 is a machine learning research library. This does not just mean that it is a collection of machine learning algorithms that share a common API; it means that it has been designed for flexibility and extensibility in order to facilitate research projects that involve new or unusual use cases. In this paper we give a brief history of the library, an overview of its basic philosophy, a summary of the library's architecture, and a description of how the Pylearn2 community functions socially.
Aug 20, 2013

Ian J. Goodfellow, David Warde-Farley, Pascal Lamblin, Vincent Dumoulin, Mehdi Mirza, Razvan Pascanu, James Bergstra, Frédéric Bastien, Yoshua Bengio

* 9 pages

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Challenges in Representation Learning: A report on three machine learning contests

Jul 01, 2013

Ian J. Goodfellow, Dumitru Erhan, Pierre Luc Carrier, Aaron Courville, Mehdi Mirza, Ben Hamner, Will Cukierski, Yichuan Tang, David Thaler, Dong-Hyun Lee, Yingbo Zhou, Chetan Ramaiah, Fangxiang Feng, Ruifan Li, Xiaojie Wang, Dimitris Athanasakis, John Shawe-Taylor, Maxim Milakov, John Park, Radu Ionescu, Marius Popescu, Cristian Grozea, James Bergstra, Jingjing Xie, Lukasz Romaszko, Bing Xu, Zhang Chuang, Yoshua Bengio

The ICML 2013 Workshop on Challenges in Representation Learning focused on three challenges: the black box learning challenge, the facial expression recognition challenge, and the multimodal learning challenge. We describe the datasets created for these challenges and summarize the results of the competitions. We provide suggestions for organizers of future challenges and some comments on what kind of knowledge can be gained from machine learning competitions.
Jul 01, 2013

Ian J. Goodfellow, Dumitru Erhan, Pierre Luc Carrier, Aaron Courville, Mehdi Mirza, Ben Hamner, Will Cukierski, Yichuan Tang, David Thaler, Dong-Hyun Lee, Yingbo Zhou, Chetan Ramaiah, Fangxiang Feng, Ruifan Li, Xiaojie Wang, Dimitris Athanasakis, John Shawe-Taylor, Maxim Milakov, John Park, Radu Ionescu, Marius Popescu, Cristian Grozea, James Bergstra, Jingjing Xie, Lukasz Romaszko, Bing Xu, Zhang Chuang, Yoshua Bengio

* 8 pages, 2 figures

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Theano: A Python framework for fast computation of mathematical expressions

May 09, 2016

The Theano Development Team, Rami Al-Rfou, Guillaume Alain, Amjad Almahairi, Christof Angermueller, Dzmitry Bahdanau, Nicolas Ballas, Frédéric Bastien, Justin Bayer, Anatoly Belikov, Alexander Belopolsky, Yoshua Bengio, Arnaud Bergeron, James Bergstra, Valentin Bisson, Josh Bleecher Snyder, Nicolas Bouchard, Nicolas Boulanger-Lewandowski, Xavier Bouthillier, Alexandre de Brébisson, Olivier Breuleux, Pierre-Luc Carrier, Kyunghyun Cho, Jan Chorowski, Paul Christiano, Tim Cooijmans, Marc-Alexandre Côté, Myriam Côté, Aaron Courville, Yann N. Dauphin, Olivier Delalleau, Julien Demouth, Guillaume Desjardins, Sander Dieleman, Laurent Dinh, Mélanie Ducoffe, Vincent Dumoulin, Samira Ebrahimi Kahou, Dumitru Erhan, Ziye Fan, Orhan Firat, Mathieu Germain, Xavier Glorot, Ian Goodfellow, Matt Graham, Caglar Gulcehre, Philippe Hamel, Iban Harlouchet, Jean-Philippe Heng, Balázs Hidasi, Sina Honari, Arjun Jain, Sébastien Jean, Kai Jia, Mikhail Korobov, Vivek Kulkarni, Alex Lamb, Pascal Lamblin, Eric Larsen, César Laurent, Sean Lee, Simon Lefrancois, Simon Lemieux, Nicholas Léonard, Zhouhan Lin, Jesse A. Livezey, Cory Lorenz, Jeremiah Lowin, Qianli Ma, Pierre-Antoine Manzagol, Olivier Mastropietro, Robert T. McGibbon, Roland Memisevic, Bart van Merriënboer, Vincent Michalski, Mehdi Mirza, Alberto Orlandi, Christopher Pal, Razvan Pascanu, Mohammad Pezeshki, Colin Raffel, Daniel Renshaw, Matthew Rocklin, Adriana Romero, Markus Roth, Peter Sadowski, John Salvatier, François Savard, Jan Schlüter, John Schulman, Gabriel Schwartz, Iulian Vlad Serban, Dmitriy Serdyuk, Samira Shabanian, Étienne Simon, Sigurd Spieckermann, S. Ramana Subramanyam, Jakub Sygnowski, Jérémie Tanguay, Gijs van Tulder, Joseph Turian, Sebastian Urban, Pascal Vincent, Francesco Visin, Harm de Vries, David Warde-Farley, Dustin J. Webb, Matthew Willson, Kelvin Xu, Lijun Xue, Li Yao, Saizheng Zhang, Ying Zhang

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficiently. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.
May 09, 2016

The Theano Development Team, Rami Al-Rfou, Guillaume Alain, Amjad Almahairi, Christof Angermueller, Dzmitry Bahdanau, Nicolas Ballas, Frédéric Bastien, Justin Bayer, Anatoly Belikov, Alexander Belopolsky, Yoshua Bengio, Arnaud Bergeron, James Bergstra, Valentin Bisson, Josh Bleecher Snyder, Nicolas Bouchard, Nicolas Boulanger-Lewandowski, Xavier Bouthillier, Alexandre de Brébisson, Olivier Breuleux, Pierre-Luc Carrier, Kyunghyun Cho, Jan Chorowski, Paul Christiano, Tim Cooijmans, Marc-Alexandre Côté, Myriam Côté, Aaron Courville, Yann N. Dauphin, Olivier Delalleau, Julien Demouth, Guillaume Desjardins, Sander Dieleman, Laurent Dinh, Mélanie Ducoffe, Vincent Dumoulin, Samira Ebrahimi Kahou, Dumitru Erhan, Ziye Fan, Orhan Firat, Mathieu Germain, Xavier Glorot, Ian Goodfellow, Matt Graham, Caglar Gulcehre, Philippe Hamel, Iban Harlouchet, Jean-Philippe Heng, Balázs Hidasi, Sina Honari, Arjun Jain, Sébastien Jean, Kai Jia, Mikhail Korobov, Vivek Kulkarni, Alex Lamb, Pascal Lamblin, Eric Larsen, César Laurent, Sean Lee, Simon Lefrancois, Simon Lemieux, Nicholas Léonard, Zhouhan Lin, Jesse A. Livezey, Cory Lorenz, Jeremiah Lowin, Qianli Ma, Pierre-Antoine Manzagol, Olivier Mastropietro, Robert T. McGibbon, Roland Memisevic, Bart van Merriënboer, Vincent Michalski, Mehdi Mirza, Alberto Orlandi, Christopher Pal, Razvan Pascanu, Mohammad Pezeshki, Colin Raffel, Daniel Renshaw, Matthew Rocklin, Adriana Romero, Markus Roth, Peter Sadowski, John Salvatier, François Savard, Jan Schlüter, John Schulman, Gabriel Schwartz, Iulian Vlad Serban, Dmitriy Serdyuk, Samira Shabanian, Étienne Simon, Sigurd Spieckermann, S. Ramana Subramanyam, Jakub Sygnowski, Jérémie Tanguay, Gijs van Tulder, Joseph Turian, Sebastian Urban, Pascal Vincent, Francesco Visin, Harm de Vries, David Warde-Farley, Dustin J. Webb, Matthew Willson, Kelvin Xu, Lijun Xue, Li Yao, Saizheng Zhang, Ying Zhang

* 19 pages, 5 figures

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