We present an optimizer which uses Bayesian optimization to tune the system parameters of distributed stochastic gradient descent (SGD). Given a specific context, our goal is to quickly find efficient configurations which appropriately balance the load between the available machines to minimize the average SGD iteration time. Our experiments consider setups with over thirty parameters. Traditional Bayesian optimization, which uses a Gaussian process as its model, is not well suited to such high dimensional domains. To reduce convergence time, we exploit the available structure. We design a probabilistic model which simulates the behavior of distributed SGD and use it within Bayesian optimization. Our model can exploit many runtime measurements for inference per evaluation of the objective function. Our experiments show that our resulting optimizer converges to efficient configurations within ten iterations, the optimized configurations outperform those found by generic optimizer in thirty iterations by up to 2X.

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polynomial-time algorithms they are often slow in practice. On the other hand, limited theoretical guarantees exist for non-convex methods. In this paper we provide a general framework that provides theoretical guarantees for learning high-dimensional tensor regression models under different low-rank structural assumptions using the projected gradient descent algorithm applied to a potentially non-convex constraint set $\Theta$ in terms of its \emph{localized Gaussian width}. We juxtapose our theoretical results for non-convex projected gradient descent algorithms with previous results on regularized convex approaches. The two main differences between the convex and non-convex approach are: (i) from a computational perspective whether the non-convex projection operator is computable and whether the projection has desirable contraction properties and (ii) from a statistical upper bound perspective, the non-convex approach has a superior rate for a number of examples. We provide three concrete examples of low-dimensional structure which address these issues and explain the pros and cons for the non-convex and convex approaches. We supplement our theoretical results with simulations which show that, under several common settings of generalized low rank tensor regression, the projected gradient descent approach is superior both in terms of statistical error and run-time provided the step-sizes of the projected descent algorithm are suitably chosen.

Many inference problems in structured prediction are naturally solved by augmenting a tractable dependency structure with complex, non-local auxiliary objectives. This includes the mean field family of variational inference algorithms, soft- or hard-constrained inference using Lagrangian relaxation or linear programming, collective graphical models, and forms of semi-supervised learning such as posterior regularization. We present a method to discriminatively learn broad families of inference objectives, capturing powerful non-local statistics of the latent variables, while maintaining tractable and provably fast inference using non-Euclidean projected gradient descent with a distance-generating function given by the Bethe entropy. We demonstrate the performance and flexibility of our method by (1) extracting structured citations from research papers by learning soft global constraints, (2) achieving state-of-the-art results on a widely-used handwriting recognition task using a novel learned non-convex inference procedure, and (3) providing a fast and highly scalable algorithm for the challenging problem of inference in a collective graphical model applied to bird migration.

In today's blog post, I demonstrated how to train a simple neural network using Python and Keras. We then applied our neural network to the Kaggle Dogs vs. Cats dataset and obtained 67.376% accuracy utilizing only the raw pixel intensities of the images. Starting next week, I'll begin discussing optimization methods such as gradient descent and Stochastic Gradient Descent (SGD). I'll also include a tutorial on backpropagation to help you understand the inner-workings of this important algorithm.

Developed back in the 50s by Rosenblatt and colleagues, this extremely simple algorithm can be viewed as the foundation for some of the most successful classifiers today, including suport vector machines and logistic regression, solved using stochastic gradient descent. The convergence proof for the Perceptron algorithm is one of the most elegant pieces of math I've seen in ML. Most useful: Boosting, especially boosted decision trees. This intuitive approach allows you to build highly accurate ML models, by combining many simple ones. Boosting is one of the most practical methods in ML, it's widely used in industry, can handle a wide variety of data types, and can be implemented at scale.

Recent work in model-agnostic explanations of black-box machine learning has demonstrated that interpretability of complex models does not have to come at the cost of accuracy or model flexibility. However, it is not clear what kind of explanations, such as linear models, decision trees, and rule lists, are the appropriate family to consider, and different tasks and models may benefit from different kinds of explanations. Instead of picking a single family of representations, in this work we propose to use "programs" as model-agnostic explanations. We show that small programs can be expressive yet intuitive as explanations, and generalize over a number of existing interpretable families. We propose a prototype program induction method based on simulated annealing that approximates the local behavior of black-box classifiers around a specific prediction using random perturbations. Finally, we present preliminary application on small datasets and show that the generated explanations are intuitive and accurate for a number of classifiers.

Word embeddings are a powerful approach for capturing semantic similarity among terms in a vocabulary. In this paper, we develop exponential family embeddings, a class of methods that extends the idea of word embeddings to other types of high-dimensional data. As examples, we studied neural data with real-valued observations, count data from a market basket analysis, and ratings data from a movie recommendation system. The main idea is to model each observation conditioned on a set of other observations. This set is called the context, and the way the context is defined is a modeling choice that depends on the problem. In language the context is the surrounding words; in neuroscience the context is close-by neurons; in market basket data the context is other items in the shopping cart. Each type of embedding model defines the context, the exponential family of conditional distributions, and how the latent embedding vectors are shared across data. We infer the embeddings with a scalable algorithm based on stochastic gradient descent. On all three applications - neural activity of zebrafish, users' shopping behavior, and movie ratings - we found exponential family embedding models to be more effective than other types of dimension reduction. They better reconstruct held-out data and find interesting qualitative structure.

A growing body of recent research is shedding new light on the role of nonconvex optimization for tackling large scale problems in machine learning, signal processing, and convex programming. This work is developing techniques that help to explain the surprising effectiveness of relatively simple first-order algorithms for certain nonconvex optimizations. When applied to problems that can be formulated as semidefinite programs, these techniques can often be viewed as part of a framework proposed by Burer and Monteiro [4]. The Burer-Monteiro technique is based on factoring the semidefinite variable, and applying classical optimization techniques to the resulting nonconvex objective over the factor. While worst-case complexity considerations imply that such an approach cannot succeed in general, a series of recent papers [11, 40, 35, 13, 1] has shown the strategy to be remarkably effective for a number of problems of practical interest, with analytical convergence guarantees and strong empirical performance. In this paper, we enlarge the collection of problems to which the Burer-Monteiro technique can be successfully applied, by analyzing the convergence properties of gradient descent applied to the problem of rectangular matrix completion from incomplete measurements.

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In this paper, we discuss the problem of minimizing the sum of two convex functions: a smooth function plus a non-smooth function. Further, the smooth part can be expressed by the average of a large number of smooth component functions, and the non-smooth part is equipped with a simple proximal mapping. We propose a proximal stochastic second-order method, which is efficient and scalable. It incorporates the Hessian in the smooth part of the function and exploits multistage scheme to reduce the variance of the stochastic gradient. We prove that our method can achieve linear rate of convergence.