We study the problem of Robust Least Squares Regression (RLSR) where several response variables can be adversarially corrupted. More specifically, for a data matrix X \in R^{p x n} and an underlying model w*, the response vector is generated as y = X'w* + b where b \in R^n is the corruption vector supported over at most C.n coordinates. Existing exact recovery results for RLSR focus solely on L1-penalty based convex formulations and impose relatively strict model assumptions such as requiring the corruptions b to be selected independently of X. In this work, we study a simple hard-thresholding algorithm called TORRENT which, under mild conditions on X, can recover w* exactly even if b corrupts the response variables in an adversarial manner, i.e. both the support and entries of b are selected adversarially after observing X and w*. Our results hold under deterministic assumptions which are satisfied if X is sampled from any sub-Gaussian distribution. Finally unlike existing results that apply only to a fixed w*, generated independently of X, our results are universal and hold for any w* \in R^p. Next, we propose gradient descent-based extensions of TORRENT that can scale efficiently to large scale problems, such as high dimensional sparse recovery and prove similar recovery guarantees for these extensions. Empirically we find TORRENT, and more so its extensions, offering significantly faster recovery than the state-of-the-art L1 solvers. For instance, even on moderate-sized datasets (with p = 50K) with around 40% corrupted responses, a variant of our proposed method called TORRENT-HYB is more than 20x faster than the best L1 solver.

We propose Loco, an algorithm for large-scale ridge regression which distributes the features across workers on a cluster. Important dependencies between variables are preserved using structured random projections which are cheap to compute and must only be communicated once. We show that Loco obtains a solution which is close to the exact ridge regression solution in the fixed design setting. We verify this experimentally in a simulation study as well as an application to climate prediction. Furthermore, we show that Loco achieves significant speedups compared with a state-of-the-art distributed algorithm on a large-scale regression problem.

The analysis of complex, high-dimensional data is one of the major research challenges in contemporary computer science and statistics. In recent years, geometric and topological approaches to data analysis have begun to yield important insights into the structure of complex data sets (see, for instance, [1] for an example of spectral geometry applied to dimension reduction, and [6], [2] for surveys on homological methods of data analysis and visualization). The common point of departure of these methods is the assumption that data in highdimensional spaces is often concentrated around a low-dimensional manifold or other topological space. In this note, we begin from the assumption that the data comes from a uniform distribution supported on a topologically disconnected space, and that clusters in the data reflect this lack of topological connectivity. Geometric techniques for data analysis have concentrated on approximating the geometry of the data as a step toward nonlinear dimension reduction. Once the dimension is reduced, standard statistical techniques are then used to analyze the data in the lower-dimensional space.

We revisit the choice of SGD for training deep neural networks by reconsidering the appropriate geometry in which to optimize the weights. We argue for a geometry invariant to rescaling of weights that does not affect the output of the network, and suggest Path-SGD, which is an approximate steepest descent method with respect to a path-wise regularizer related to max-norm regularization. Path-SGD is easy and efficient to implement and leads to empirical gains over SGD and AdaGrad.

In this paper we generalize the framework of the feasible descent method (FDM) to a randomized (R-FDM) and a coordinate-wise random feasible descent method (RC-FDM) framework. We show that the famous SDCA algorithm for optimizing the SVM dual problem, or the stochastic coordinate descent method for the LASSO problem, fits into the framework of RC-FDM. We prove linear convergence for both R-FDM and RC-FDM under the weak strong convexity assumption. Moreover, we show that the duality gap converges linearly for RC-FDM, which implies that the duality gap also converges linearly for SDCA applied to the SVM dual problem.

In active learning, we are given a sample space X, a label space Y, a class of models that map X to Y, and a large set U of unlabelled samples. The goal of the learner is to learn a model in the class with small target error while interactively querying the labels of as few of the unlabelled samples as possible. Most theoretical work on active learning has focussed on the PAC or the agnostic PAC model, where the goal is to learn binary classifiers that belong to a particular hypothesis class [2, 13, 9, 6, 3, 4,22], andtherehasbeenonlyahandful ofexceptions[19, 8,20]. Inthispaper, weshift ourattention to a more general setting - maximum likelihood estimation (MLE), where Pr(Y X) is described by a model θ belonging to a model class Θ. We show that when data is generated by a model in this class, we can do active learning provided the model class Θ has the following simple property: the Fisher information matrix for any model θ Θ at any (x,y) depends only on x and θ. This condition is satisfied in a number of widely applicable model classes, such as Linear Regression and Generalized Linear Models (GLMs), which in turn includes models for Multiclass Classification and Conditional 1 Random Fields. Consequently, we can provide active learning algorithms for maximum likelihood estimation in all these model classes. The standard solution to active MLE estimation in the statistics literature is to select samples for label query by optimizing a class of summary statistics of the asymptotic covariance matrix of the estimator [5]. The literature, however, does not provide any guidance towards which summary statistic should be used, or any analysis of the solution quality when a finite number of labels or samples are available.

Several authors have recently developed risk-sensitive policy gradient methods that augment the standard expected cost minimization problem with a measure of variability in cost. These studies have focused on specific risk-measures, such as the variance or conditional value at risk (CVaR). In this work, we extend the policy gradient method to the whole class of coherent risk measures, which is widely accepted in finance and operations research, among other fields. We consider both static and time-consistent dynamic risk measures. For static risk measures, our approach is in the spirit of policy gradient algorithms and combines a standard sampling approach with convex programming. For dynamic risk measures, our approach is actor-critic style and involves explicit approximation of value function. Most importantly, our contribution presents a unified approach to risk-sensitive reinforcement learning that generalizes and extends previous results.

We consider the problem of designing locality sensitive hashes (LSH) for inner product similarity, and of the power of asymmetric hashes in this context. Shrivastava and Li (2014a) argue that there is no symmetric LSH for the problem and propose an asymmetric LSH based on different mappings for query and database points. However, we show there does exist a simple symmetric LSH that enjoys stronger guarantees and better empirical performance than the asymmetric LSH they suggest. We also show a variant of the settings where asymmetry is in-fact needed, but there a different asymmetric LSH is required.

In this paper, we analyze a generic algorithm scheme for sequential global optimization using Gaussian processes. The upper bounds we derive on the cumulative regret for this generic algorithm improve by an exponential factor the previously known bounds for algorithms like GP-UCB. We also introduce the novel Gaussian Process Mutual Information algorithm (GP-MI), which significantly improves further these upper bounds for the cumulative regret. We confirm the efficiency of this algorithm on synthetic and real tasks against the natural competitor, GP-UCB, and also the Expected Improvement heuristic. After the publication of our article, we found an error in the proof of Lemma 1 which invalidates the main theorem.

Bayesian predictive inference analyzes a dataset to make predictions about new observations. When a model does not match the data, predictive accuracy suffers. We develop population empirical Bayes (POP-EB), a hierarchical framework that explicitly models the empirical population distribution as part of Bayesian analysis. We introduce a new concept, the latent dataset, as a hierarchical variable and set the empirical population as its prior. This leads to a new predictive density that mitigates model mismatch. We efficiently apply this method to complex models by proposing a stochastic variational inference algorithm, called bumping variational inference (BUMP-VI). We demonstrate improved predictive accuracy over classical Bayesian inference in three models: a linear regression model of health data, a Bayesian mixture model of natural images, and a latent Dirichlet allocation topic model of scientific documents.