In a recent paper Joachims (2006) presented SVM-Perf, a cutting plane method (CPM) for training linear Support Vector Machines (SVMs) which converges to an $\epsilon$ accurate solution in $O(1/\epsilon^{2})$ iterations. By tightening the analysis, Teo et al. (2010) showed that $O(1/\epsilon)$ iterations suffice. Given the impressive convergence speed of CPM on a number of practical problems, it was conjectured that these rates could be further improved. In this paper we disprove this conjecture. We present counter examples which are not only applicable for training linear SVMs with hinge loss, but also hold for support vector methods which optimize a \emph{multivariate} performance score. However, surprisingly, these problems are not inherently hard. By exploiting the structure of the objective function we can devise an algorithm that converges in $O(1/\sqrt{\epsilon})$ iterations.

We study the problem of learning a sparse linear regression vector under additional conditions on the structure of its sparsity pattern. We present a family of convex penalty functions, which encode this prior knowledge by means of a set of constraints on the absolute values of the regression coefficients. This family subsumes the $\ell_1$ norm and is flexible enough to include different models of sparsity patterns, which are of practical and theoretical importance. We establish some important properties of these functions and discuss some examples where they can be computed explicitly. Moreover, we present a convergent optimization algorithm for solving regularized least squares with these penalty functions. Numerical simulations highlight the benefit of structured sparsity and the advantage offered by our approach over the Lasso and other related methods.

We consider the problem of retrieving the database points nearest to a given {\em hyperplane} query without exhaustively scanning the database. We propose two hashing-based solutions. Our first approach maps the data to two-bit binary keys that are locality-sensitive for the angle between the hyperplane normal and a database point. Our second approach embeds the data into a vector space where the Euclidean norm reflects the desired distance between the original points and hyperplane query. Both use hashing to retrieve near points in sub-linear time. Our first method's preprocessing stage is more efficient, while the second has stronger accuracy guarantees. We apply both to pool-based active learning: taking the current hyperplane classifier as a query, our algorithm identifies those points (approximately) satisfying the well-known minimal distance-to-hyperplane selection criterion. We empirically demonstrate our methods' tradeoffs, and show that they make it practical to perform active selection with millions of unlabeled points.

We address the problem of semi-supervised learning in an adversarial setting. Instead of assuming that labels are missing at random, we analyze a less favorable scenario where the label information can be missing partially and arbitrarily, which is motivated by several practical examples. We present nearly matching upper and lower generalization bounds for learning in this setting under reasonable assumptions about available label information. Motivated by the analysis, we formulate a convex optimization problem for parameter estimation, derive an efficient algorithm, and analyze its convergence. We provide experimental results on several standard data sets showing the robustness of our algorithm to the pattern of missing label information, outperforming several strong baselines.

Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted. We present an efficient convex optimization-based algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the *exact* optimal low-dimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct *column space* of the uncorrupted matrix, rather than the exact matrix itself.

We address the problem of estimating the F-measure of a given model as accurately as possible on a fixed labeling budget. This problem occurs whenever an estimate cannot be obtained from held-out training data; for instance, when data that have been used to train the model are held back for reasons of privacy or do not reflect the test distribution. In this case, new test instances have to be drawn and labeled at a cost. An active estimation procedure selects instances according to an instrumental sampling distribution. An analysis of the sources of estimation error leads to an optimal sampling distribution that minimizes estimator variance. We explore conditions under which active estimates of F-measures are more accurate than estimates based on instances sampled from the test distribution.

We introduce CST, an algorithm for constructing skill trees from demonstration trajectories in continuous reinforcement learning domains. CST uses a changepoint detection method to segment each trajectory into a skill chain by detecting a change of appropriate abstraction, or that a segment is too complex to model as a single skill. The skill chains from each trajectory are then merged to form a skill tree. We demonstrate that CST constructs an appropriate skill tree that can be further refined through learning in a challenging continuous domain, and that it can be used to segment demonstration trajectories on a mobile manipulator into chains of skills where each skill is assigned an appropriate abstraction.

We develop a theory of online learning by defining several complexity measures. Among them are analogues of Rademacher complexity, covering numbers and fat-shattering dimension from statistical learning theory. Relationship among these complexity measures, their connection to online learning, and tools for bounding them are provided. We apply these results to various learning problems. We provide a complete characterization of online learnability in the supervised setting.

In discriminative machine learning one is interested in training a system to optimize a certain desired measure of performance, or loss. In binary classification one typically tries to minimizes the error rate. But in structured prediction each task often has its own measure of performance such as the BLEU score in machine translation or the intersection-over-union score in PASCAL segmentation. The most common approaches to structured prediction, structural SVMs and CRFs, do not minimize the task loss: the former minimizes a surrogate loss with no guarantees for task loss and the latter minimizes log loss independent of task loss. The main contribution of this paper is a theorem stating that a certain perceptron-like learning rule, involving features vectors derived from loss-adjusted inference, directly corresponds to the gradient of task loss. We give empirical results on phonetic alignment of a standard test set from the TIMIT corpus, which surpasses all previously reported results on this problem.

We present a model that describes the structure in the responses of different brain areas to a set of stimuli in terms of stimulus categories" (clusters of stimuli) and "functional units" (clusters of voxels). We assume that voxels within a unit respond similarly to all stimuli from the same category, and design a nonparametric hierarchical model to capture inter-subject variability among the units. The model explicitly captures the relationship between brain activations and fMRI time courses. A variational inference algorithm derived based on the model can learn categories, units, and a set of unit-category activation probabilities from data. When applied to data from an fMRI study of object recognition, the method finds meaningful and consistent clusterings of stimuli into categories and voxels into units."