Online optimization has emerged as powerful tool in large scale optimization. In this paper, we introduce efficient online algorithms based on the alternating directions method (ADM). We introduce a new proof technique for ADM in the batch setting, which yields the O(1/T) convergence rate of ADM and forms the basis of regret analysis in the online setting. We consider two scenarios in the online setting, based on whether the solution needs to lie in the feasible set or not. In both settings, we establish regret bounds for both the objective function as well as constraint violation for general and strongly convex functions. Preliminary results are presented to illustrate the performance of the proposed algorithms.

Functional neuroimaging can measure the brain?s response to an external stimulus. It is used to perform brain mapping: identifying from these observations the brain regions involved. This problem can be cast into a linear supervised learning task where the neuroimaging data are used as predictors for the stimulus. Brain mapping is then seen as a support recovery problem. On functional MRI (fMRI) data, this problem is particularly challenging as i) the number of samples is small due to limited acquisition time and ii) the variables are strongly correlated. We propose to overcome these difficulties using sparse regression models over new variables obtained by clustering of the original variables. The use of randomization techniques, e.g. bootstrap samples, and clustering of the variables improves the recovery properties of sparse methods. We demonstrate the benefit of our approach on an extensive simulation study as well as two fMRI datasets.

This manuscript develops the theory of agglomerative clustering with Bregman divergences. Geometric smoothing techniques are developed to deal with degenerate clusters. To allow for cluster models based on exponential families with overcomplete representations, Bregman divergences are developed for nondifferentiable convex functions.

Visual perception is a challenging problem in part due to illumination variations. A possible solution is to first estimate an illumination invariant representation before using it for recognition. The object albedo and surface normals are examples of such representations. In this paper, we introduce a multilayer generative model where the latent variables include the albedo, surface normals, and the light source. Combining Deep Belief Nets with the Lambertian reflectance assumption, our model can learn good priors over the albedo from 2D images. Illumination variations can be explained by changing only the lighting latent variable in our model. By transferring learned knowledge from similar objects, albedo and surface normals estimation from a single image is possible in our model. Experiments demonstrate that our model is able to generalize as well as improve over standard baselines in one-shot face recognition.

Motivated by value function estimation in reinforcement learning, we study statistical linear inverse problems, i.e., problems where the coefficients of a linear system to be solved are observed in noise. We consider penalized estimators, where performance is evaluated using a matrix-weighted two-norm of the defect of the estimator measured with respect to the true, unknown coefficients. Two objective functions are considered depending whether the error of the defect measured with respect to the noisy coefficients is squared or unsquared. We propose simple, yet novel and theoretically well-founded data-dependent choices for the regularization parameters for both cases that avoid data-splitting. A distinguishing feature of our analysis is that we derive deterministic error bounds in terms of the error of the coefficients, thus allowing the complete separation of the analysis of the stochastic properties of these errors. We show that our results lead to new insights and bounds for linear value function estimation in reinforcement learning.

We carefully study how well minimizing convex surrogate loss functions, corresponds to minimizing the misclassification error rate for the problem of binary classification with linear predictors. In particular, we show that amongst all convex surrogate losses, the hinge loss gives essentially the best possible bound, of all convex loss functions, for the misclassification error rate of the resulting linear predictor in terms of the best possible margin error rate. We also provide lower bounds for specific convex surrogates that show how different commonly used losses qualitatively differ from each other.

We examine the problem of learning a probabilistic model for melody directly from musical sequences belonging to the same genre. This is a challenging task as one needs to capture not only the rich temporal structure evident in music, but also the complex statistical dependencies among different music components. To address this problem we introduce the Variable-gram Topic Model, which couples the latent topic formalism with a systematic model for contextual information. We evaluate the model on next-step prediction. Additionally, we present a novel way of model evaluation, where we directly compare model samples with data sequences using the Maximum Mean Discrepancy of string kernels, to assess how close is the model distribution to the data distribution. We show that the model has the highest performance under both evaluation measures when compared to LDA, the Topic Bigram and related non-topic models.

Traditional approaches to ranking in web search follow the paradigm of rank-by-score: a learned function gives each query-URL combination an absolute score and URLs are ranked according to this score. This paradigm ensures that if the score of one URL is better than another then one will always be ranked higher than the other. Scoring contradicts prior work in behavioral economics that showed that users' preferences between two items depend not only on the items but also on the presented alternatives. Thus, for the same query, users' preference between items A and B depends on the presence/absence of item C. We propose a new model of ranking, the Random Shopper Model, that allows and explains such behavior. In this model, each feature is viewed as a Markov chain over the items to be ranked, and the goal is to find a weighting of the features that best reflects their importance. We show that our model can be learned under the empirical risk minimization framework, and give an efficient learning algorithm. Experiments on commerce search logs demonstrate that our algorithm outperforms scoring-based approaches including regression and listwise ranking.

We study the problem of unsupervised domain adaptation, which aims to adapt classifiers trained on a labeled source domain to an unlabeled target domain. Many existing approaches first learn domain-invariant features and then construct classifiers with them. We propose a novel approach that jointly learn the both. Specifically, while the method identifies a feature space where data in the source and the target domains are similarly distributed, it also learns the feature space discriminatively, optimizing an information-theoretic metric as an proxy to the expected misclassification error on the target domain. We show how this optimization can be effectively carried out with simple gradient-based methods and how hyperparameters can be cross-validated without demanding any labeled data from the target domain. Empirical studies on benchmark tasks of object recognition and sentiment analysis validated our modeling assumptions and demonstrated significant improvement of our method over competing ones in classification accuracies.

Natural image statistics exhibit hierarchical dependencies across multiple scales. Representing such prior knowledge in non-factorial latent tree models can boost performance of image denoising, inpainting, deconvolution or reconstruction substantially, beyond standard factorial "sparse" methodology. We derive a large scale approximate Bayesian inference algorithm for linear models with non-factorial (latent tree-structured) scale mixture priors. Experimental results on a range of denoising and inpainting problems demonstrate substantially improved performance compared to MAP estimation or to inference with factorial priors.