ICLR 2017 Papers

Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons needed by a deep network for a given degree of function approximation. First, we consider univariate functions on a bounded interval and require a neural network to achieve an approximation error of $\varepsilon$ uniformly over the interval. We show that shallow networks (i.e., networks whose depth does not depend on $\varepsilon$) require $\Omega(\text{poly}(1/\varepsilon))$ neurons while deep networks (i.e., networks whose depth grows with $1/\varepsilon$) require $\mathcal{O}(\text{polylog}(1/\varepsilon))$ neurons. We then extend these results to certain classes of important multivariate functions. Our results are derived for neural networks which use a combination of rectifier linear units (ReLUs) and binary step units, two of the most popular type of activation functions. Our analysis builds on a simple observation: the multiplication of two bits can be represented by a ReLU.

The past year saw the introduction of new architectures such as Highway networks and Residual networks which, for the first time, enabled the training of feedforward networks with dozens to hundreds of layers using simple gradient descent. While depth of representation has been posited as a primary reason for their success, there are indications that these architectures defy a popular view of deep learning as a hierarchical computation of increasingly abstract features at each layer. In this report, we argue that this view is incomplete and does not adequately explain several recent findings. We propose an alternative viewpoint based on unrolled iterative estimation---a group of successive layers iteratively refine their estimates of the same features instead of computing an entirely new representation. We demonstrate that this viewpoint directly leads to the construction of highway and residual networks. Finally we provide preliminary experiments to discuss the similarities and differences between the two architectures.

As more computational resources become available, machine learning researchers train ever larger neural networks on millions of data points using stochastic gradient descent (SGD). Although SGD scales well in terms of both the size of dataset and the number of parameters of the model, it has rapidly diminishing returns as parallel computing resources increase. Second-order optimization methods have an affinity for well-estimated gradients and large mini-batches, and can therefore benefit much more from parallel computation in principle. Unfortunately, they often employ severe approximations to the curvature matrix in order to scale to large models with millions of parameters, limiting their effectiveness in practice versus well-tuned SGD with momentum. The recently proposed K-FAC method(Martens and Grosse, 2015) uses a stronger and more sophisticated curvature approximation, and has been shown to make much more per-iteration progress than SGD, while only introducing a modest overhead. In this paper, we develop a version of K-FAC that distributes the computation of gradients and additional quantities required by K-FAC across multiple machines, thereby taking advantage of method’s superior scaling to large mini-batches and mitigating its additional overheads. We provide a Tensorflow implementation of our approach which is easy to use and can be applied to many existing codebases without modification. Additionally, we develop several algorithmic enhancements to K-FAC which can improve its computational performance for very large models. Finally, we show that our distributed K-FAC method speeds up training of various state-of-the-art ImageNet classification models by a factor of two compared to Batch Normalization(Ioffe and Szegedy, 2015).

Most contemporary multi-task learning methods assume linear models. This setting is considered shallow in the era of deep learning. In this paper, we present a new deep multi-task representation learning framework that learns cross-task sharing structure at every layer in a deep network. Our approach is based on generalising the matrix factorisation techniques explicitly or implicitly used by many conventional MTL algorithms to tensor factorisation, to realise automatic learning of end-to-end knowledge sharing in deep networks. This is in contrast to existing deep learning approaches that need a user-defined multi-task sharing strategy. Our approach applies to both homogeneous and heterogeneous MTL. Experiments demonstrate the efficacy of our deep multi-task representation learning in terms of both higher accuracy and fewer design choices.

A framework is presented for unsupervised learning of representations based on infomax principle for large-scale neural populations. We use an asymptotic approximation to the Shannon's mutual information for a large neural population to demonstrate that a good initial approximation to the global information-theoretic optimum can be obtained by a hierarchical infomax method. Starting from the initial solution, an efficient algorithm based on gradient descent of the final objective function is proposed to learn representations from the input datasets, and the method works for complete, overcomplete, and undercomplete bases. As confirmed by numerical experiments, our method is robust and highly efficient for extracting salient features from input datasets. Compared with the main existing methods, our algorithm has a distinct advantage in both the training speed and the robustness of unsupervised representation learning. Furthermore, the proposed method is easily extended to the supervised or unsupervised model for training deep structure networks.

The learning of domain-invariant representations in the context of domain adaptation with neural networks is considered. We propose a new regularization method that minimizes the domain-specific latent feature representations directly in the hidden activation space. Although some standard distribution matching approaches exist that can be interpreted as the matching of weighted sums of moments, e.g. Maximum Mean Discrepancy (MMD), an explicit order-wise matching of higher order moments has not been considered before. We propose to match the higher order central moments of probability distributions by means of order-wise moment differences. Our model does not require computationally expensive distance and kernel matrix computations. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, called Central Moment Discrepancy (CMD). We prove that CMD is a metric on the set of probability distributions on a compact interval. We further prove that convergence of probability distributions on compact intervals w.r.t. the new metric implies convergence in distribution of the respective random variables. We test our approach on two different benchmark data sets for object recognition (Office) and sentiment analysis of product reviews (Amazon reviews). CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with MMD, Variational Fair Autoencoders and Domain Adversarial Neural Networks on Amazon reviews. In addition, a post-hoc parameter sensitivity analysis shows that the new approach is stable w. r. t. parameter changes in a certain interval. The source code of the experiments is publicly available.

We study the problem of transferring a sample in one domain to an analog sample in another domain. Given two related domains, S and T, we would like to learn a generative function G that maps an input sample from S to the domain T, such that the output of a given representation function f, which accepts inputs in either domains, would remain unchanged. Other than f, the training data is unsupervised and consist of a set of samples from each domain, without any mapping between them. The Domain Transfer Network (DTN) we present employs a compound loss function that includes a multiclass GAN loss, an f preserving component, and a regularizing component that encourages G to map samples from T to themselves. We apply our method to visual domains including digits and face images and demonstrate its ability to generate convincing novel images of previously unseen entities, while preserving their identity.

Learning tasks such as those involving genomic data often poses a serious challenge: the number of input features can be orders of magnitude larger than the number of training examples, making it difficult to avoid overfitting, even when using the known regularization techniques. We focus here on tasks in which the input is a description of the genetic variation specific to a patient, the single nucleotide polymorphisms (SNPs), yielding millions of ternary inputs. Improving the ability of deep learning to handle such datasets could have an important impact in medical research, more specifically in precision medicine, where high-dimensional data regarding a particular patient is used to make predictions of interest. Even though the amount of data for such tasks is increasing, this mismatch between the number of examples and the number of inputs remains a concern. Naive implementations of classifier neural networks involve a huge number of free parameters in their first layer (number of input features times number of hidden units): each input feature is associated with as many parameters as there are hidden units. We propose a novel neural network parametrization which considerably reduces the number of free parameters. It is based on the idea that we can first learn or provide a distributed representation for each input feature (e.g. for each position in the genome where variations are observed in data), and then learn (with another neural network called the parameter prediction network) how to map a feature's distributed representation (based on the feature's identity not its value) to the vector of parameters specific to that feature in the classifier neural network (the weights which link the value of the feature to each of the hidden units). This approach views the problem of producing the parameters associated with each feature as a multi-task learning problem. We show experimentally on a population stratification task of interest to medical studies that the proposed approach can significantly reduce both the number of parameters and the error rate of the classifier.

Machine learning and deep learning in particular has advanced tremendously on perceptual tasks in recent years. However, it remains vulnerable against adversarial perturbations of the input that have been crafted specifically to fool the system while being quasi-imperceptible to a human. In this work, we propose to augment deep neural networks with a small ``detector'' subnetwork which is trained on the binary classification task of distinguishing genuine data from data containing adversarial perturbations. Our method is orthogonal to prior work on addressing adversarial perturbations, which has mostly focused on making the classification network itself more robust. We show empirically that adversarial perturbations can be detected surprisingly well even though they are quasi-imperceptible to humans. Moreover, while the detectors have been trained to detect only a specific adversary, they generalize to similar and weaker adversaries. In addition, we propose an adversarial attack that fools both the classifier and the detector and a novel training procedure for the detector that counteracts this attack.

Although deep learning models have proven effective at solving problems in natural language processing, the mechanism by which they come to their conclusions is often unclear. As a result, these models are generally treated as black boxes, yielding no insight of the underlying learned patterns. In this paper we consider Long Short Term Memory networks (LSTMs) and demonstrate a new approach for tracking the importance of a given input to the LSTM for a given output. By identifying consistently important patterns of words, we are able to distill state of the art LSTMs on sentiment analysis and question answering into a set of representative phrases. This representation is then quantitatively validated by using the extracted phrases to construct a simple, rule-based classifier which approximates the output of the LSTM.

The goal of two-sample tests is to assess whether two samples, $S_P \sim P^n$ and $S_Q \sim Q^m$, are drawn from the same distribution. Perhaps intriguingly, one relatively unexplored method to build two-sample tests is the use of binary classifiers. In particular, construct a dataset by pairing the $n$ examples in $S_P$ with a positive label, and by pairing the $m$ examples in $S_Q$ with a negative label. If the null hypothesis ``$P = Q$'' is true, then the classification accuracy of a binary classifier on a held-out subset of this dataset should remain near chance-level. As we will show, such \emph{Classifier Two-Sample Tests} (C2ST) learn a suitable representation of the data on the fly, return test statistics in interpretable units, have a simple null distribution, and their predictive uncertainty allow to interpret where $P$ and $Q$ differ. The goal of this paper is to establish the properties, performance, and uses of C2ST. First, we analyze their main theoretical properties. Second, we compare their performance against a variety of state-of-the-art alternatives. Third, we propose their use to evaluate the sample quality of generative models with intractable likelihoods, such as Generative Adversarial Networks (GANs). Fourth, we showcase the novel application of GANs together with C2ST for causal discovery.

We present a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs. We motivate the choice of our convolutional architecture via a localized first-order approximation of spectral graph convolutions. Our model scales linearly in the number of graph edges and learns hidden layer representations that encode both local graph structure and features of nodes. In a number of experiments on citation networks and on a knowledge graph dataset we demonstrate that our approach outperforms related methods by a significant margin.

Despite recent advances, memory-augmented deep neural networks are still limited when it comes to life-long and one-shot learning, especially in remembering rare events. We present a large-scale life-long memory module for use in deep learning. The module exploits fast nearest-neighbor algorithms for efficiency and thus scales to large memory sizes. Except for the nearest-neighbor query, the module is fully differentiable and trained end-to-end with no extra supervision. It operates in a life-long manner, i.e., without the need to reset it during training. Our memory module can be easily added to any part of a supervised neural network. To show its versatility we add it to a number of networks, from simple convolutional ones tested on image classification to deep sequence-to-sequence and recurrent-convolutional models. In all cases, the enhanced network gains the ability to remember and do life-long one-shot learning. Our module remembers training examples shown many thousands of steps in the past and it can successfully generalize from them. We set new state-of-the-art for one-shot learning on the Omniglot dataset and demonstrate, for the first time, life-long one-shot learning in recurrent neural networks on a large-scale machine translation task.

In many computer vision tasks, the relevant information to solve the problem at hand is mixed to irrelevant, distracting information. This has motivated researchers to design attentional models that can dynamically focus on parts of images or videos that are salient, e.g., by down-weighting irrelevant pixels. In this work, we propose a spatiotemporal attentional model that learns where to look in a video directly from human fixation data. We model visual attention with a mixture of Gaussians at each frame. This distribution is used to express the probability of saliency for each pixel. Time consistency in videos is modeled hierarchically by: 1) deep 3D convolutional features to represent spatial and short-term time relations and 2) a long short-term memory network on top that aggregates the clip-level representation of sequential clips and therefore expands the temporal domain from few frames to seconds. The parameters of the proposed model are optimized via maximum likelihood estimation using human fixations as training data, without knowledge of the action in each video. Our experiments on Hollywood2 show state-of-the-art performance on saliency prediction for video. We also show that our attentional model trained on Hollywood2 generalizes well to UCF101 and it can be leveraged to improve action classification accuracy on both datasets.

Learning to navigate in complex environments with dynamic elements is an important milestone in developing AI agents. In this work we formulate the navigation question as a reinforcement learning problem and show that data efficiency and task performance can be dramatically improved by relying on additional auxiliary tasks to bootstrap learning. In particular we consider jointly learning the goal-driven reinforcement learning problem with an unsupervised depth prediction task and a self-supervised loop closure classification task. Using this approach we can learn to navigate from raw sensory input in complicated 3D mazes, approaching human-level performance even under conditions where the goal location changes frequently. We provide detailed analysis of the agent behaviour, its ability to localise, and its network activity dynamics, that show that the agent implicitly learns key navigation abilities, with only sparse rewards and without direct supervision.

We describe a neural attention model with a learnable retinal sampling lattice. The model is trained on a visual search task requiring the classification of an object embedded in a visual scene amidst background distractors using the smallest number of fixations. We explore the tiling properties that emerge in the model's retinal sampling lattice after training. Specifically, we show that this lattice resembles the eccentricity dependent sampling lattice of the primate retina, with a high resolution region in the fovea surrounded by a low resolution periphery. Furthermore, we find conditions where these emergent properties are amplified or eliminated providing clues to their function.

Sparse coding is a core building block in many data analysis and machine learning pipelines. Typically it is solved by relying on generic optimization techniques, such as the Iterative Soft Thresholding Algorithm and its accelerated version (ISTA, FISTA). These methods are optimal in the class of first-order methods for non-smooth, convex functions. However, they do not exploit the particular structure of the problem at hand nor the input data distribution. An acceleration using neural networks, coined LISTA, was proposed in \cite{Gregor10}, which showed empirically that one could achieve high quality estimates with few iterations by modifying the parameters of the proximal splitting appropriately. In this paper we study the reasons for such acceleration. Our mathematical analysis reveals that it is related to a specific matrix factorization of the Gram kernel of the dictionary, which attempts to nearly diagonalise the kernel with a basis that produces a small perturbation of the $\ell_1$ ball. When this factorization succeeds, we prove that the resulting splitting algorithm enjoys an improved convergence bound with respect to the non-adaptive version. Moreover, our analysis also shows that conditions for acceleration occur mostly at the beginning of the iterative process, consistent with numerical experiments. We further validate our analysis by showing that on dictionaries where this factorization does not exist, adaptive acceleration fails.

We propose a new formulation for pruning convolutional kernels in neural networks to enable efficient inference. We interleave greedy criteria-based pruning with fine-tuning by backpropagation-a computationally efficient procedure that maintains good generalization in the pruned network. We propose a new criterion based on Taylor expansion that approximates the change in the cost function induced by pruning network parameters. We focus on transfer learning, where large pretrained networks are adapted to specialized tasks. The proposed criterion demonstrates superior performance compared to other criteria, e.g. the norm of kernel weights or feature map activation, for pruning large CNNs after adaptation to fine-grained classification tasks (Birds-200 and Flowers-102) relaying only on the first order gradient information. We also show that pruning can lead to more than 10x theoretical reduction in adapted 3D-convolutional filters with a small drop in accuracy in a recurrent gesture classifier. Finally, we show results for the large-scale ImageNet dataset to emphasize the flexibility of our approach.

We present an approach to training neural networks to generate sequences using actor-critic methods from reinforcement learning (RL). Current log-likelihood training methods are limited by the discrepancy between their training and testing modes, as models must generate tokens conditioned on their previous guesses rather than the ground-truth tokens. We address this problem by introducing a textit{critic} network that is trained to predict the value of an output token, given the policy of an textit{actor} network. This results in a training procedure that is much closer to the test phase, and allows us to directly optimize for a task-specific score such as BLEU. Crucially, since we leverage these techniques in the supervised learning setting rather than the traditional RL setting, we condition the critic network on the ground-truth output. We show that our method leads to improved performance on both a synthetic task, and for German-English machine translation. Our analysis paves the way for such methods to be applied in natural language generation tasks, such as machine translation, caption generation, and dialogue modelling.