arXiv subjects

Aaron Courville

Publications and source records attributed to Aaron Courville.

230 records · Page 13Linked to original sources

arXiv2014

An empirical analysis of dropout in piecewise linear networks

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.

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arXiv2013

Maxout Networks

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.

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arXiv2013

Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation

Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of {\em conditional computation}, where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful.

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arXiv2013

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

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.

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arXiv2013

Joint Training Deep Boltzmann Machines for Classification

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.

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arXiv2013

Metric-Free Natural Gradient for Joint-Training of Boltzmann Machines

This paper introduces the Metric-Free Natural Gradient (MFNG) algorithm for training Boltzmann Machines. Similar in spirit to the Hessian-Free method of Martens [8], our algorithm belongs to the family of truncated Newton methods and exploits an efficient matrix-vector product to avoid explicitely storing the natural gradient metric $L$. This metric is shown to be the expected second derivative of the log-partition function (under the model distribution), or equivalently, the variance of the vector of partial derivatives of the energy function. We evaluate our method on the task of joint-training a 3-layer Deep Boltzmann Machine and show that MFNG does indeed have faster per-epoch convergence compared to Stochastic Maximum Likelihood with centering, though wall-clock performance is currently not competitive.

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arXiv2012

Joint Training of Deep Boltzmann Machines

We introduce a new method for training deep Boltzmann machines jointly. Prior methods require an initial learning pass that trains the deep Boltzmann machine greedily, one layer at a time, or do not perform well on classifi- cation tasks.

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arXiv2012

Texture Modeling with Convolutional Spike-and-Slab RBMs and Deep Extensions

We apply the spike-and-slab Restricted Boltzmann Machine (ssRBM) to texture modeling. The ssRBM with tiled-convolution weight sharing (TssRBM) achieves or surpasses the state-of-the-art on texture synthesis and inpainting by parametric models. We also develop a novel RBM model with a spike-and-slab visible layer and binary variables in the hidden layer. This model is designed to be stacked on top of the TssRBM. We show the resulting deep belief network (DBN) is a powerful generative model that improves on single-layer models and is capable of modeling not only single high-resolution and challenging textures but also multiple textures.

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arXiv2012

Disentangling Factors of Variation via Generative Entangling

Here we propose a novel model family with the objective of learning to disentangle the factors of variation in data. Our approach is based on the spike-and-slab restricted Boltzmann machine which we generalize to include higher-order interactions among multiple latent variables. Seen from a generative perspective, the multiplicative interactions emulates the entangling of factors of variation. Inference in the model can be seen as disentangling these generative factors. Unlike previous attempts at disentangling latent factors, the proposed model is trained using no supervised information regarding the latent factors. We apply our model to the task of facial expression classification.

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arXiv2012

Large-Scale Feature Learning With Spike-and-Slab Sparse Coding

We consider the problem of object recognition with a large number of classes. In order to overcome the low amount of labeled examples available in this setting, we introduce a new feature learning and extraction procedure based on a factor model we call spike-and-slab sparse coding (S3C). Prior work on S3C has not prioritized the ability to exploit parallel architectures and scale S3C to the enormous problem sizes needed for object recognition. We present a novel inference procedure for appropriate for use with GPUs which allows us to dramatically increase both the training set size and the amount of latent factors that S3C may be trained with. We demonstrate that this approach improves upon the supervised learning capabilities of both sparse coding and the spike-and-slab Restricted Boltzmann Machine (ssRBM) on the CIFAR-10 dataset. We use the CIFAR-100 dataset to demonstrate that our method scales to large numbers of classes better than previous methods. Finally, we use our method to win the NIPS 2011 Workshop on Challenges In Learning Hierarchical Models? Transfer Learning Challenge.

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arXiv2012

Spike-and-Slab Sparse Coding for Unsupervised Feature Discovery

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.

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arXiv2012

On Training Deep Boltzmann Machines

The deep Boltzmann machine (DBM) has been an important development in the quest for powerful "deep" probabilistic models. To date, simultaneous or joint training of all layers of the DBM has been largely unsuccessful with existing training methods. We introduce a simple regularization scheme that encourages the weight vectors associated with each hidden unit to have similar norms. We demonstrate that this regularization can be easily combined with standard stochastic maximum likelihood to yield an effective training strategy for the simultaneous training of all layers of the deep Boltzmann machine.

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arXiv2011

The Statistical Inefficiency of Sparse Coding for Images (or, One Gabor to Rule them All)

Sparse coding is a proven principle for learning compact representations of images. However, sparse coding by itself often leads to very redundant dictionaries. With images, this often takes the form of similar edge detectors which are replicated many times at various positions, scales and orientations. An immediate consequence of this observation is that the estimation of the dictionary components is not statistically efficient. We propose a factored model in which factors of variation (e.g. position, scale and orientation) are untangled from the underlying Gabor-like filters. There is so much redundancy in sparse codes for natural images that our model requires only a single dictionary element (a Gabor-like edge detector) to outperform standard sparse coding. Our model scales naturally to arbitrary-sized images while achieving much greater statistical efficiency during learning. We validate this claim with a number of experiments showing, in part, superior compression of out-of-sample data using a sparse coding dictionary learned with only a single image.

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arXiv2010

Adaptive Parallel Tempering for Stochastic Maximum Likelihood Learning of RBMs

Restricted Boltzmann Machines (RBM) have attracted a lot of attention of late, as one the principle building blocks of deep networks. Training RBMs remains problematic however, because of the intractibility of their partition function. The maximum likelihood gradient requires a very robust sampler which can accurately sample from the model despite the loss of ergodicity often incurred during learning. While using Parallel Tempering in the negative phase of Stochastic Maximum Likelihood (SML-PT) helps address the issue, it imposes a trade-off between computational complexity and high ergodicity, and requires careful hand-tuning of the temperatures. In this paper, we show that this trade-off is unnecessary. The choice of optimal temperatures can be automated by minimizing average return time (a concept first proposed by [Katzgraber et al., 2006]) while chains can be spawned dynamically, as needed, thus minimizing the computational overhead. We show on a synthetic dataset, that this results in better likelihood scores.

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