We study large-scale, nonsmooth, nonconconvex optimization problems. In particular, we focus on nonconvex problems with \emph{composite} objectives. This class of problems includes the extensively studied convex, composite objective problems as a special case. To tackle composite nonconvex problems, we introduce a powerful new framework based on asymptotically \emph{nonvanishing} errors, avoiding the common convenient assumption of eventually vanishing errors. Within our framework we derive both batch and incremental nonconvex proximal splitting algorithms. To our knowledge, our framework is first to develop and analyze incremental \emph{nonconvex} proximal-splitting algorithms, even if we disregard the ability to handle nonvanishing errors. We illustrate our theoretical framework by showing how it applies to difficult large-scale, nonsmooth, and nonconvex problems.

In this paper, we derive a novel algorithm to cluster hidden Markov models (HMMs) according to their probability distributions. We propose a variational hierarchical EM algorithm that i) clusters a given collection of HMMs into groups of HMMs that are similar, in terms of the distributions they represent, and ii) characterizes each group by a ``cluster center'', i.e., a novel HMM that is representative for the group. We illustrate the benefits of the proposed algorithm on hierarchical clustering of motion capture sequences as well as on automatic music tagging.

The Restricted Boltzmann Machine (RBM) is a popular density model that is also good for extracting features. A main source of tractability in RBM models is the model's assumption that given an input, hidden units activate independently from one another. Sparsity and competition in the hidden representation is believed to be beneficial, and while an RBM with competition among its hidden units would acquire some of the attractive properties of sparse coding, such constraints are not added due to the widespread belief that the resulting model would become intractable. In this work, we show how a dynamic programming algorithm developed in 1981 can be used to implement exact sparsity in the RBM's hidden units. We then expand on this and show how to pass derivatives through a layer of exact sparsity, which makes it possible to fine-tune a deep belief network (DBN) consisting of RBMs with sparse hidden layers. We show that sparsity in the RBM's hidden layer improves the performance of both the pre-trained representations and of the fine-tuned model.

In this paper we dicuss a novel framework for multiclass learning, defined by a suitable coding/decoding strategy, namely the simplex coding, that allows to generalize to multiple classes a relaxation approach commonly used in binary classification. In this framework a relaxation error analysis can be developed avoiding constraints on the considered hypotheses class. Moreover, we show that in this setting it is possible to derive the first provably consistent regularized methods with training/tuning complexity which is {\em independent} to the number of classes. Tools from convex analysis are introduced that can be used beyond the scope of this paper.

We present a method for approximate inference for a broad class of non-conjugate probabilistic models. In particular, for the family of generalized linear model target densities we describe a rich class of variational approximating densities which can be best fit to the target by minimizing the Kullback-Leibler divergence. Our approach is based on using the Fourier representation which we show results in efficient and scalable inference.

Topic modeling is a generalization of clustering that posits that observations (words in a document) are generated by \emph{multiple} latent factors (topics), as opposed to just one. This increased representational power comes at the cost of a more challenging unsupervised learning problem of estimating the topic-word distributions when only words are observed, and the topics are hidden. This work provides a simple and efficient learning procedure that is guaranteed to recover the parameters for a wide class of topic models, including Latent Dirichlet Allocation (LDA). For LDA, the procedure correctly recovers both the topic-word distributions and the parameters of the Dirichlet prior over the topic mixtures, using only trigram statistics (\emph{i.e.}, third order moments, which may be estimated with documents containing just three words). The method, called Excess Correlation Analysis, is based on a spectral decomposition of low-order moments via two singular value decompositions (SVDs). Moreover, the algorithm is scalable, since the SVDs are carried out only on $k \times k$ matrices, where $k$ is the number of latent factors (topics) and is typically much smaller than the dimension of the observation (word) space.

Bipartite matching problems characterize many situations, ranging from ranking in information retrieval to correspondence in vision. Exact inference in real-world applications of these problems is intractable, making efficient approximation methods essential for learning and inference. In this paper we propose a novel {\it sequential matching} sampler based on the generalization of the Plackett-Luce model, which can effectively make large moves in the space of matchings. This allows the sampler to match the difficult target distributions common in these problems: highly multimodal distributions with well separated modes. We present experimental results with bipartite matching problems - ranking and image correspondence - which show that the sequential matching sampler efficiently approximates the target distribution, significantly outperforming other sampling approaches.

This paper suggests a learning-theoretic perspective on how synaptic plasticity benefits global brain functioning. We introduce a model, the selectron, that (i) arises as the fast time constant limit of leaky integrate-and-fire neurons equipped with spiking timing dependent plasticity (STDP) and (ii) is amenable to theoretical analysis. We show that the selectron encodes reward estimates into spikes and that an error bound on spikes is controlled by a spiking margin and the sum of synaptic weights. Moreover, the efficacy of spikes (their usefulness to other reward maximizing selectrons) also depends on total synaptic strength. Finally, based on our analysis, we propose a regularized version of STDP, and show the regularization improves the robustness of neuronal learning when faced with multiple stimuli.

Principal components analysis (PCA) is a standard tool for identifying good low-dimensional approximations to data sets in high dimension. Many current data sets of interest contain private or sensitive information about individuals. Algorithms which operate on such data should be sensitive to the privacy risks in publishing their outputs. Differential privacy is a framework for developing tradeoffs between privacy and the utility of these outputs. In this paper we investigate the theory and empirical performance of differentially private approximations to PCA and propose a new method which explicitly optimizes the utility of the output. We demonstrate that on real data, there this a large performance gap between the existing methods and our method. We show that the sample complexity for the two procedures differs in the scaling with the data dimension, and that our method is nearly optimal in terms of this scaling.

Residue-residue contact prediction is a fundamental problem in protein structure prediction. Hower, despite considerable research efforts, contact prediction methods are still largely unreliable. Here we introduce a novel deep machine-learning architecture which consists of a multidimensional stack of learning modules. For contact prediction, the idea is implemented as a three-dimensional stack of Neural Networks NN^k_{ij}, where i and j index the spatial coordinates of the contact map and k indexes ''time''. The temporal dimension is introduced to capture the fact that protein folding is not an instantaneous process, but rather a progressive refinement. Networks at level k in the stack can be trained in supervised fashion to refine the predictions produced by the previous level, hence addressing the problem of vanishing gradients, typical of deep architectures. Increased accuracy and generalization capabilities of this approach are established by rigorous comparison with other classical machine learning approaches for contact prediction. The deep approach leads to an accuracy for difficult long-range contacts of about 30%, roughly 10% above the state-of-the-art. Many variations in the architectures and the training algorithms are possible, leaving room for further improvements. Furthermore, the approach is applicable to other problems with strong underlying spatial and temporal components.