This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free. Our results can be directly applied to many machine learning applications, including deep learning. As a particular concrete example of such an application, we show that our results can be used directly to establish sharp global convergence rates for matrix factorization. Our results rely on a novel characterization of the geometry around saddle points, which may be of independent interest to the non-convex optimization community.

Characterization of lung nodules as benign or malignant is one of the most important tasks in lung cancer diagnosis, staging and treatment planning. While the variation in the appearance of the nodules remains large, there is a need for a fast and robust computer aided system. In this work, we propose an end-to-end trainable multi-view deep Convolutional Neural Network (CNN) for nodule characterization. First, we use median intensity projection to obtain a 2D patch corresponding to each dimension. The three images are then concatenated to form a tensor, where the images serve as different channels of the input image. In order to increase the number of training samples, we perform data augmentation by scaling, rotating and adding noise to the input image. The trained network is used to extract features from the input image followed by a Gaussian Process (GP) regression to obtain the malignancy score. We also empirically establish the significance of different high level nodule attributes such as calcification, sphericity and others for malignancy determination. These attributes are found to be complementary to the deep multi-view CNN features and a significant improvement over other methods is obtained.

We draw a formal connection between using synthetic training data to optimize neural network parameters and approximate, Bayesian, model-based reasoning. In particular, training a neural network using synthetic data can be viewed as learning a proposal distribution generator for approximate inference in the synthetic-data generative model. We demonstrate this connection in a recognition task where we develop a novel Captcha-breaking architecture and train it using synthetic data, demonstrating both state-of-the-art performance and a way of computing task-specific posterior uncertainty. Using a neural network trained this way, we also demonstrate successful breaking of real-world Captchas currently used by Facebook and Wikipedia. Reasoning from these empirical results and drawing connections with Bayesian modeling, we discuss the robustness of synthetic data results and suggest important considerations for ensuring good neural network generalization when training with synthetic data.

When using reinforcement learning (RL) algorithms to evaluate a policy it is common, given a large state space, to introduce some form of approximation architecture for the value function (VF). The exact form of this architecture can have a significant effect on the accuracy of the VF estimate, however, and determining a suitable approximation architecture can often be a highly complex task. Consequently there is a large amount of interest in the potential for allowing RL algorithms to adaptively generate approximation architectures. We investigate a method of adapting approximation architectures which uses feedback regarding the frequency with which an agent has visited certain states to guide which areas of the state space to approximate with greater detail. This method is "unsupervised" in the sense that it makes no direct reference to reward or the VF estimate. We introduce an algorithm based upon this idea which adapts a state aggregation approximation architecture on-line. A common method of scoring a VF estimate is to weight the squared Bellman error of each state-action by the probability of that state-action occurring. Adopting this scoring method, and assuming $S$ states, we demonstrate theoretically that - provided (1) the number of cells $X$ in the state aggregation architecture is of order $\sqrt{S}\log_2{S}\ln{S}$ or greater, (2) the policy and transition function are close to deterministic, and (3) the prior for the transition function is uniformly distributed - our algorithm, used in conjunction with a suitable RL algorithm, can guarantee a score which is arbitrarily close to zero as $S$ becomes large. It is able to do this despite having only $O(X \log_2S)$ space complexity and negligible time complexity. The results take advantage of certain properties of the stationary distributions of Markov chains.

This paper presents a new multitask learning framework that learns a shared representation among the tasks, incorporating both task and feature clusters. The jointly-induced clusters yield a shared latent subspace where task relationships are learned more effectively and more generally than in state-of-the-art multitask learning methods. The proposed general framework enables the derivation of more specific or restricted state-of-the-art multitask methods. The paper also proposes a highly-scalable multitask learning algorithm, based on the new framework, using conjugate gradient descent and generalized \textit{Sylvester equations}. Experimental results on synthetic and benchmark datasets show that the proposed method systematically outperforms several state-of-the-art multitask learning methods.

In this paper we introduce a new classification algorithm called Optimization of Distributions Differences (ODD). The algorithm aims to find a transformation from the feature space to a new space where the instances in the same class are as close as possible to one another while the gravity centers of these classes are as far as possible from one another. This aim is formulated as a multiobjective optimization problem that is solved by a hybrid of an evolutionary strategy and the Quasi-Newton method. The choice of the transformation function is flexible and could be any continuous space function. We experiment with a linear and a non-linear transformation in this paper. We show that the algorithm can outperform 6 other state-of-the-art classification methods, namely naive Bayes, support vector machines, linear discriminant analysis, multi-layer perceptrons, decision trees, and k-nearest neighbors, in 12 standard classification datasets. Our results show that the method is less sensitive to the imbalanced number of instances comparing to these methods. We also show that ODD maintains its performance better than other classification methods in these datasets, hence, offers a better generalization ability.

Information that is stored in an encrypted format is, by definition, usually not amenable to statistical analysis or machine learning methods. In this paper we present detailed analysis of coordinate and accelerated gradient descent algorithms which are capable of fitting least squares and penalised ridge regression models, using data encrypted under a fully homomorphic encryption scheme. Gradient descent is shown to dominate in terms of encrypted computational speed, and theoretical results are proven to give parameter bounds which ensure correctness of decryption. The characteristics of encrypted computation are empirically shown to favour a non-standard acceleration technique. This demonstrates the possibility of approximating conventional statistical regression methods using encrypted data without compromising privacy.

In this paper, we introduce and provide a short overview of nonnegative matrix factorization (NMF). Several aspects of NMF are discussed, namely, the application in hyperspectral imaging, geometry and uniqueness of NMF solutions, complexity, algorithms, and its link with extended formulations of polyhedra. In order to put NMF into perspective, the more general problem class of constrained low-rank matrix approximation problems is first briefly introduced.

As machine learning systems become ubiquitous, there has been a surge of interest in interpretable machine learning: systems that provide explanation for their outputs. These explanations are often used to qualitatively assess other criteria such as safety or non-discrimination. However, despite the interest in interpretability, there is very little consensus on what interpretable machine learning is and how it should be measured. In this position paper, we first define interpretability and describe when interpretability is needed (and when it is not). Next, we suggest a taxonomy for rigorous evaluation and expose open questions towards a more rigorous science of interpretable machine learning.

Learning a natural language interface for database tables is a challenging task that involves deep language understanding and multi-step reasoning. The task is often approached by mapping natural language queries to logical forms or programs that provide the desired response when executed on the database. To our knowledge, this paper presents the first weakly supervised, end-to-end neural network model to induce such programs on a real-world dataset. We enhance the objective function of Neural Programmer, a neural network with built-in discrete operations, and apply it on WikiTableQuestions, a natural language question-answering dataset. The model is trained end-to-end with weak supervision of question-answer pairs, and does not require domain-specific grammars, rules, or annotations that are key elements in previous approaches to program induction. The main experimental result in this paper is that a single Neural Programmer model achieves 34.2% accuracy using only 10,000 examples with weak supervision. An ensemble of 15 models, with a trivial combination technique, achieves 37.7% accuracy, which is competitive to the current state-of-the-art accuracy of 37.1% obtained by a traditional natural language semantic parser.