Training deep feature hierarchies to solve supervised learning tasks has achieving state of the art performance on many problems in computer vision. However, a principled way in which to train such hierarchies in the unsupervised setting has remained elusive. In this work we suggest a new architecture and loss for training deep feature hierarchies that linearize the transformations observed in unlabelednatural video sequences. This is done by training a generative model to predict video frames. We also address the problem of inherent uncertainty in prediction by introducing a latent variables that are non-deterministic functions of the input into the network architecture.

We consider the problem of efficiently computing the maximum likelihood estimator in Generalized Linear Models (GLMs)when the number of observations is much larger than the number of coefficients (n > > p > > 1). In this regime, optimization algorithms can immensely benefit fromapproximate second order information.We propose an alternative way of constructing the curvature information by formulatingit as an estimation problem and applying a Stein-type lemma, which allows further improvements through sub-sampling andeigenvalue thresholding.Our algorithm enjoys fast convergence rates, resembling that of second order methods, with modest per-iteration cost. We provide its convergence analysis for the case where the rows of the design matrix are i.i.d. samples with bounded support.We show that the convergence has two phases, aquadratic phase followed by a linear phase. Finally,we empirically demonstrate that our algorithm achieves the highest performancecompared to various algorithms on several datasets.

Practitioners of Bayesian statistics have long depended on Markov chain Monte Carlo (MCMC) to obtain samples from intractable posterior distributions. Unfortunately, MCMC algorithms are typically serial, and do not scale to the large datasets typical of modern machine learning. The recently proposed consensus Monte Carlo algorithm removes this limitation by partitioning the data and drawing samples conditional on each partition in parallel (Scott et al, 2013). A fixed aggregation function then combines these samples, yielding approximate posterior samples. We introduce variational consensus Monte Carlo (VCMC), a variational Bayes algorithm that optimizes over aggregation functions to obtain samples from a distribution that better approximates the target. The resulting objective contains an intractable entropy term; we therefore derive a relaxation of the objective and show that the relaxed problem is blockwise concave under mild conditions. We illustrate the advantages of our algorithm on three inference tasks from the literature, demonstrating both the superior quality of the posterior approximation and the moderate overhead of the optimization step. Our algorithm achieves a relative error reduction (measured against serial MCMC) of up to 39% compared to consensus Monte Carlo on the task of estimating 300-dimensional probit regression parameter expectations; similarly, it achieves an error reduction of 92% on the task of estimating cluster comembership probabilities in a Gaussian mixture model with 8 components in 8 dimensions. Furthermore, these gains come at moderate cost compared to the runtime of serial MCMC, achieving near-ideal speedup in some instances.

In regression problems involving vector-valued outputs (or equivalently, multiple responses), it is well known that the maximum likelihood estimator (MLE), which takes noise covariance structure into account, can be significantly more accurate than the ordinary least squares (OLS) estimator. However, existing literature compares OLS and MLE in terms of their asymptotic, not finite sample, guarantees. More crucially, computing the MLE in general requires solving a non-convex optimization problem and is not known to be efficiently solvable. We provide finite sample upper and lower bounds on the estimation error of OLS and MLE, in two popular models: a) Pooled model, b) Seemingly Unrelated Regression (SUR) model. We provide precise instances where the MLE is significantly more accurate than OLS. Furthermore, for both models, we show that the output of a computationally efficient alternating minimization procedure enjoys the same performance guarantee as MLE, up to universal constants. Finally, we show that for high-dimensional settings as well, the alternating minimization procedure leads to significantly more accurate solutions than the corresponding OLS solutions but with error bound that depends only logarithmically on the data dimensionality.

We consider the estimation of sparse graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. The penalized maximum likelihood estimation of this model involves minimizing a non-convex objective function. In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with the optimal statistical rate of convergence as well as consistent graph recovery. Notably, such an estimator achieves estimation consistency with only one tensor sample, which is unobserved in previous work. Our theoretical results are backed by thorough numerical studies.

Clustering large data is a fundamental problem with a vast number of applications. Due to the increasing size of data, practitioners interested in clustering have turned to distributed computation methods. In this work, we consider the widely used k-center clustering problem and its variant used to handle noisy data, k-center with outliers. In the noise-free setting we demonstrate how a previously-proposed distributed method is actually an O(1)-approximation algorithm, which accurately explains its strong empirical performance. Additionally, in the noisy setting, we develop a novel distributed algorithm that is also an O(1)-approximation. These algorithms are highly parallel and lend themselves to virtually any distributed computing framework. We compare both empirically against the best known noisy sequential clustering methods and show that both distributed algorithms are consistently close to their sequential versions. The algorithms are all one can hope for in distributed settings: they are fast, memory efficient and they match their sequential counterparts.

We consider the problem of binary classification when the covariates conditioned on the each of the response values follow multivariate Gaussian distributions. We focus on the setting where the covariance matrices for the two conditional distributions are the same. The corresponding generative model classifier, derived via the Bayes rule, also called Linear Discriminant Analysis, has been shown to behave poorly in high-dimensional settings. We present a novel analysis of the classification error of any linear discriminant approach given conditional Gaussian models. This allows us to compare the generative model classifier, other recently proposed discriminative approaches that directly learn the discriminant function, and then finally logistic regression which is another classical discriminative model classifier. As we show, under a natural sparsity assumption, and letting $s$ denote the sparsity of the Bayes classifier, $p$ the number of covariates, and $n$ the number of samples, the simple ($\ell_1$-regularized) logistic regression classifier achieves the fast misclassification error rates of $O\left(\frac{s \log p}{n}\right)$, which is much better than the other approaches, which are either inconsistent under high-dimensional settings, or achieve a slower rate of $O\left(\sqrt{\frac{s \log p}{n}}\right)$.

In many learning problems, ranging from clustering to ranking through metric learning, empirical estimates of the risk functional consist of an average over tuples (e.g., pairs or triplets) of observations, rather than over individual observations. In this paper, we focus on how to best implement a stochastic approximation approach to solve such risk minimization problems. We argue that in the large-scale setting, gradient estimates should be obtained by sampling tuples of data points with replacement (incomplete U-statistics) instead of sampling data points without replacement (complete U-statistics based on subsamples). We develop a theoretical framework accounting for the substantial impact of this strategy on the generalization ability of the prediction model returned by the Stochastic Gradient Descent (SGD) algorithm. It reveals that the method we promote achieves a much better trade-off between statistical accuracy and computational cost. Beyond the rate bound analysis, experiments on AUC maximization and metric learning provide strong empirical evidence of the superiority of the proposed approach.

We introduce a unifying generalization of the Lovász theta function, and the associated geometric embedding, for graphs with weights on both nodes and edges. We show how it can be computed exactly by semidefinite programming, and how to approximate it using SVM computations. We show how the theta function can be interpreted as a measure of diversity in graphs and use this idea, and the graph embedding in algorithms for Max-Cut, correlation clustering and document summarization, all of which are well represented as problems on weighted graphs.

Modern prediction problems arising in multilabel learning and learning to rank pose unique challenges to the classical theory of supervised learning. These problems have large prediction and label spaces of a combinatorial nature and involve sophisticated loss functions. We offer a general framework to derive mistake driven online algorithms and associated loss bounds. The key ingredients in our framework are a general loss function, a general vector space representation of predictions, and a notion of margin with respect to a general norm. Our general algorithm, Predtron, yields the perceptron algorithm and its variants when instantiated on classic problems such as binary classification, multiclass classification, ordinal regression, and multilabel classification. For multilabel ranking and subset ranking, we derive novel algorithms, notions of margins, and loss bounds. A simulation study confirms the behavior predicted by our bounds and demonstrates the flexibility of the design choices in our framework.