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We show that every packing of congruent regular pentagons in the Euclidean plane has density at most $(5-\sqrt5)/3$, which is about 0.92. More specifically, this article proves the pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice, formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons. The strategy is based on estimates of the areas of Delaunay triangles. Our strategy reduces the pentagonal ice-ray conjecture to area minimization problems that involve at most four Delaunay triangles. These minimization problems are solved by computer. The computer-assisted portions of the proof use techniques such as interval arithmetic, automatic differentiation, and a meet-in-the-middle algorithm.

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Learning efficient representations for concepts has been proven to be an important basis for many applications such as machine translation or document classification. Proper representations of medical concepts such as diagnosis, medication, procedure codes and visits will have broad applications in healthcare analytics. However, in Electronic Health Records (EHR) the visit sequences of patients include multiple concepts (diagnosis, procedure, and medication codes) per visit. This structure provides two types of relational information, namely sequential order of visits and co-occurrence of the codes within each visit. In this work, we propose Med2Vec, which not only learns distributed representations for both medical codes and visits from a large EHR dataset with over 3 million visits, but also allows us to interpret the learned representations confirmed positively by clinical experts. In the experiments, Med2Vec displays significant improvement in key medical applications compared to popular baselines such as Skip-gram, GloVe and stacked autoencoder, while providing clinically meaningful interpretation.

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Despite widespread adoption, machine learning models remain mostly black boxes. Understanding the reasons behind predictions is, however, quite important in assessing trust, which is fundamental if one plans to take action based on a prediction, or when choosing whether to deploy a new model. Such understanding also provides insights into the model, which can be used to transform an untrustworthy model or prediction into a trustworthy one. In this work, we propose LIME, a novel explanation technique that explains the predictions of any classifier in an interpretable and faithful manner, by learning an interpretable model locally around the prediction. We also propose a method to explain models by presenting representative individual predictions and their explanations in a non-redundant way, framing the task as a submodular optimization problem. We demonstrate the flexibility of these methods by explaining different models for text (e.g. random forests) and image classification (e.g. neural networks). We show the utility of explanations via novel experiments, both simulated and with human subjects, on various scenarios that require trust: deciding if one should trust a prediction, choosing between models, improving an untrustworthy classifier, and identifying why a classifier should not be trusted.

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For any positive integer $k$, there exist neural networks with $\Theta(k^3)$ layers, $\Theta(1)$ nodes per layer, and $\Theta(1)$ distinct parameters which can not be approximated by networks with $\mathcal{O}(k)$ layers unless they are exponentially large --- they must possess $\Omega(2^k)$ nodes. This result is proved here for a class of nodes termed "semi-algebraic gates" which includes the common choices of ReLU, maximum, indicator, and piecewise polynomial functions, therefore establishing benefits of depth against not just standard networks with ReLU gates, but also convolutional networks with ReLU and maximization gates, sum-product networks, and boosted decision trees (in this last case with a stronger separation: $\Omega(2^{k^3})$ total tree nodes are required).

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Unsupervised methods for learning distributed representations of words are ubiquitous in today's NLP research, but far less is known about the best ways to learn distributed phrase or sentence representations from unlabelled data. This paper is a systematic comparison of models that learn such representations. We find that the optimal approach depends critically on the intended application. Deeper, more complex models are preferable for representations to be used in supervised systems, but shallow log-linear models work best for building representation spaces that can be decoded with simple spatial distance metrics. We also propose two new unsupervised representation-learning objectives designed to optimise the trade-off between training time, domain portability and performance.

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We investigate a new method to augment recurrent neural networks with extra memory without increasing the number of network parameters. The system has an associative memory based on complex-valued vectors and is closely related to Holographic Reduced Representations and Long Short-Term Memory networks. Holographic Reduced Representations have limited capacity: as they store more information, each retrieval becomes noisier due to interference. Our system in contrast creates redundant copies of stored information, which enables retrieval with reduced noise. Experiments demonstrate faster learning on multiple memorization tasks.

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Attention mechanisms in neural networks have proved useful for problems in which the input and output do not have fixed dimension. Often there exist features that are locally translation invariant and would be valuable for directing the model's attention, but previous attentional architectures are not constructed to learn such features specifically. We introduce an attentional neural network that employs convolution on the input tokens to detect local time-invariant and long-range topical attention features in a context-dependent way. We apply this architecture to the problem of extreme summarization of source code snippets into short, descriptive function name-like summaries. Using those features, the model sequentially generates a summary by marginalizing over two attention mechanisms: one that predicts the next summary token based on the attention weights of the input tokens and another that is able to copy a code token as-is directly into the summary. We demonstrate our convolutional attention neural network's performance on 10 popular Java projects showing that it achieves better performance compared to previous attentional mechanisms.

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We introduce the value iteration network (VIN): a fully differentiable neural network with a `planning module' embedded within. VINs can learn to plan, and are suitable for predicting outcomes that involve planning-based reasoning, such as policies for reinforcement learning. Key to our approach is a novel differentiable approximation of the value-iteration algorithm, which can be represented as a convolutional neural network, and trained end-to-end using standard backpropagation. We evaluate VIN based policies on discrete and continuous path-planning domains, and on a natural-language based search task. We show that by learning an explicit planning computation, VIN policies generalize better to new, unseen domains.

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Machine learning (ML) models, e.g., state-of-the-art deep neural networks (DNNs), are vulnerable to adversarial examples: malicious inputs modified to yield erroneous model outputs, while appearing unmodified to human observers. Potential attacks include having malicious content like malware identified as legitimate or controlling vehicle behavior. Yet, all existing adversarial example attacks require knowledge of either the model internals or its training data. We introduce the first practical demonstration of an attacker controlling a remotely hosted DNN with no such knowledge. Indeed, the only capability of our black-box adversary is to observe labels given by the DNN to chosen inputs. Our attack strategy consists in training a local model to substitute for the target DNN, using inputs synthetically generated by an adversary and labeled by the target DNN. We then use the local substitute to craft adversarial examples, and find that they are misclassified by the targeted DNN. To perform a real-world and properly-blinded evaluation, we attack a DNN hosted by MetaMind, an online deep learning API. After labeling 6,400 synthetic inputs to train our substitute, we find that their DNN misclassifies adversarial examples crafted with our substitute at a rate of 84.24%. We demonstrate that our strategy generalizes to many ML techniques like logistic regression or SVMs, regardless of the ML model chosen for the substitute. We instantiate the same attack against models hosted by Amazon and Google, using logistic regression substitutes trained with only 800 label queries. They yield adversarial examples misclassified by Amazon and Google at rates of 96.19% and 88.94%. We also find that this black-box attack strategy is capable of evading defense strategies previously found to make adversarial example crafting harder.

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Image-generating machine learning models are typically trained with loss functions based on distance in the image space. This often leads to over-smoothed results. We propose a class of loss functions, which we call deep perceptual similarity metrics (DeePSiM), that mitigate this problem. Instead of computing distances in the image space, we compute distances between image features extracted by deep neural networks. This metric better reflects perceptually similarity of images and thus leads to better results. We show three applications: autoencoder training, a modification of a variational autoencoder, and inversion of deep convolutional networks. In all cases, the generated images look sharp and resemble natural images.

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In this work we explore recent advances in Recurrent Neural Networks for large scale Language Modeling, a task central to language understanding. We extend current models to deal with two key challenges present in this task: corpora and vocabulary sizes, and complex, long term structure of language. We perform an exhaustive study on techniques such as character Convolutional Neural Networks or Long-Short Term Memory, on the One Billion Word Benchmark. Our best single model significantly improves state-of-the-art perplexity from 51.3 down to 30.0 (whilst reducing the number of parameters by a factor of 20), while an ensemble of models sets a new record by improving perplexity from 41.0 down to 23.7. We also release these models for the NLP and ML community to study and improve upon.

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Recurrent neural networks are increasing popular models for sequential learning. Unfortunately, although the most effective RNN architectures are perhaps excessively complicated, extensive searches have not found simpler alternatives. This paper imports ideas from physics and functional programming into RNN design to provide guiding principles. From physics, we introduce type constraints, analogous to the constraints that forbids adding meters to seconds. From functional programming, we require that strongly-typed architectures factorize into stateless learnware and state-dependent firmware, reducing the impact of side-effects. The features learned by strongly-typed nets have a simple semantic interpretation via dynamic average-pooling on one-dimensional convolutions. We also show that strongly-typed gradients are better behaved than in classical architectures, and characterize the representational power of strongly-typed nets. Finally, experiments show that, despite being more constrained, strongly-typed architectures achieve lower training and comparable generalization error to classical architectures.

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We present Submatrix-wise Vector Embedding Learner (Swivel), a method for generating low-dimensional feature embeddings from a feature co-occurrence matrix. Swivel performs approximate factorization of the point-wise mutual information matrix via stochastic gradient descent. It uses a piecewise loss with special handling for unobserved co-occurrences, and thus makes use of all the information in the matrix. While this requires computation proportional to the size of the entire matrix, we make use of vectorized multiplication to process thousands of rows and columns at once to compute millions of predicted values. Furthermore, we partition the matrix into shards in order to parallelize the computation across many nodes. This approach results in more accurate embeddings than can be achieved with methods that consider only observed co-occurrences, and can scale to much larger corpora than can be handled with sampling methods.

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We propose sparsemax, a new activation function similar to the traditional softmax, but able to output sparse probabilities. After deriving its properties, we show how its Jacobian can be efficiently computed, enabling its use in a network trained with backpropagation. Then, we propose a new smooth and convex loss function which is the sparsemax analogue of the logistic loss. We reveal an unexpected connection between this new loss and the Huber classification loss. We obtain promising empirical results in multi-label classification problems and in attention-based neural networks for natural language inference. For the latter, we achieve a similar performance as the traditional softmax, but with a selective, more compact, attention focus.

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We propose a conceptually simple and lightweight framework for deep reinforcement learning that uses asynchronous gradient descent for optimization of deep neural network controllers. We present asynchronous variants of four standard reinforcement learning algorithms and show that parallel actor-learners have a stabilizing effect on training allowing all four methods to successfully train neural network controllers. The best performing method, an asynchronous variant of actor-critic, surpasses the current state-of-the-art on the Atari domain while training for half the time on a single multi-core CPU instead of a GPU. Furthermore, we show that asynchronous actor-critic succeeds on a wide variety of continuous motor control problems as well as on a new task of navigating random 3D mazes using a visual input.

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State-of-the-art deep neural networks (DNNs) have hundreds of millions of connections and are both computationally and memory intensive, making them difficult to deploy on embedded systems with limited hardware resources and power budgets. While custom hardware helps the computation, fetching weights from DRAM is two orders of magnitude more expensive than ALU operations, and dominates the required power. Previously proposed 'Deep Compression' makes it possible to fit large DNNs (AlexNet and VGGNet) fully in on-chip SRAM. This compression is achieved by pruning the redundant connections and having multiple connections share the same weight. We propose an energy efficient inference engine (EIE) that performs inference on this compressed network model and accelerates the resulting sparse matrix-vector multiplication with weight sharing. Going from DRAM to SRAM gives EIE 120x energy saving; Exploiting sparsity saves 10x; Weight sharing gives 8x; Skipping zero activations from ReLU saves another 3x. Evaluated on nine DNN benchmarks, EIE is 189x and 13x faster when compared to CPU and GPU implementations of the same DNN without compression. EIE has a processing power of 102GOPS/s working directly on a compressed network, corresponding to 3TOPS/s on an uncompressed network, and processes FC layers of AlexNet at 1.88x10^4 frames/sec with a power dissipation of only 600mW. It is 24,000x and 3,400x more energy efficient than a CPU and GPU respectively. Compared with DaDianNao, EIE has 2.9x, 19x and 3x better throughput, energy efficiency and area efficiency.

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Modeling the distribution of natural images is a landmark problem in unsupervised learning. This task requires an image model that is at once expressive, tractable and scalable. We present a deep neural network that sequentially predicts the pixels in an image along the two spatial dimensions. Our method models the discrete probability of the raw pixel values and encodes the complete set of dependencies in the image. Architectural novelties include fast two-dimensional recurrent layers and an effective use of residual connections in deep recurrent networks. We achieve log-likelihood scores on natural images that are considerably better than the previous state of the art. Our main results also provide benchmarks on the diverse ImageNet dataset. Samples generated from the model appear crisp, varied and globally coherent.

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Based on the assumption that there exists a neural network that efficiently represents a set of Boolean functions between all binary inputs and outputs, we propose a process for developing and deploying neural networks whose weight parameters, bias terms, input, and intermediate hidden layer output signals, are all binary-valued, and require only basic bit logic for the feedforward pass. The proposed Bitwise Neural Network (BNN) is especially suitable for resource-constrained environments, since it replaces either floating or fixed-point arithmetic with significantly more efficient bitwise operations. Hence, the BNN requires for less spatial complexity, less memory bandwidth, and less power consumption in hardware. In order to design such networks, we propose to add a few training schemes, such as weight compression and noisy backpropagation, which result in a bitwise network that performs almost as well as its corresponding real-valued network. We test the proposed network on the MNIST dataset, represented using binary features, and show that BNNs result in competitive performance while offering dramatic computational savings.

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This paper studies a combination of generative Markov random field (MRF) models and discriminatively trained deep convolutional neural networks (dCNNs) for synthesizing 2D images. The generative MRF acts on higher-levels of a dCNN feature pyramid, controling the image layout at an abstract level. We apply the method to both photographic and non-photo-realistic (artwork) synthesis tasks. The MRF regularizer prevents over-excitation artifacts and reduces implausible feature mixtures common to previous dCNN inversion approaches, permitting synthezing photographic content with increased visual plausibility. Unlike standard MRF-based texture synthesis, the combined system can both match and adapt local features with considerable variability, yielding results far out of reach of classic generative MRF methods.

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In this paper we present a connection between two dynamical systems arising in entirely different contexts: one in signal processing and the other in biology. The first is the famous Iteratively Reweighted Least Squares (IRLS) algorithm used in compressed sensing and sparse recovery while the second is the dynamics of a slime mold (Physarum polycephalum). Both of these dynamics are geared towards finding a minimum l1-norm solution in an affine subspace. Despite its simplicity the convergence of the IRLS method has been shown only for a certain regularization of it and remains an important open problem. Our first result shows that the two dynamics are projections of the same dynamical system in higher dimensions. As a consequence, and building on the recent work on Physarum dynamics, we are able to prove convergence and obtain complexity bounds for a damped version of the IRLS algorithm.

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