We consider in this paper a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. We present a new class of first-order methods, namely the gradient sliding algorithms, which can skip the computation of the gradient for the smooth component from time to time. As a consequence, these algorithms require only ${\cal O}(1/\sqrt{\epsilon})$ gradient evaluations for the smooth component in order to find an $\epsilon$-solution for the composite problem, while still maintaining the optimal ${\cal O}(1/\epsilon^2)$ bound on the total number of subgradient evaluations for the nonsmooth component. We then present a stochastic counterpart for these algorithms and establish similar complexity bounds for solving an important class of stochastic composite optimization problems. Moreover, if the smooth component in the composite function is strongly convex, the developed gradient sliding algorithms can significantly reduce the number of graduate and subgradient evaluations for the smooth and nonsmooth component to ${\cal O} (\log (1/\epsilon))$ and ${\cal O}(1/\epsilon)$, respectively. Finally, we generalize these algorithms to the case when the smooth component is replaced by a nonsmooth one possessing a certain bi-linear saddle point structure.

The extraction of logically-independent fragments out of an ontology ABox can be useful for solving the tractability problem of querying ontologies with large ABoxes. In this paper, we propose a formal definition of an ABox module, such that it guarantees complete preservation of facts about a given set of individuals, and thus can be reasoned independently w.r.t. the ontology TBox. With ABox modules of this type, isolated or distributed (parallel) ABox reasoning becomes feasible, and more efficient data retrieval from ontology ABoxes can be attained. To compute such an ABox module, we present a theoretical approach and also an approximation for $\mathcal{SHIQ}$ ontologies. Evaluation of the module approximation on different types of ontologies shows that, on average, extracted ABox modules are significantly smaller than the entire ABox, and the time for ontology reasoning based on ABox modules can be improved significantly.

Gaussian graphical models (GGM) have been widely used in many high-dimensional applications ranging from biological and financial data to recommender systems. Sparsity in GGM plays a central role both statistically and computationally. Unfortunately, real-world data often does not fit well to sparse graphical models. In this paper, we focus on a family of latent variable Gaussian graphical models (LVGGM), where the model is conditionally sparse given latent variables, but marginally non-sparse. In LVGGM, the inverse covariance matrix has a low-rank plus sparse structure, and can be learned in a regularized maximum likelihood framework. We derive novel parameter estimation error bounds for LVGGM under mild conditions in the high-dimensional setting. These results complement the existing theory on the structural learning, and open up new possibilities of using LVGGM for statistical inference.

We propose a new framework for estimating generative models via an adversarial process, in which we simultaneously train two models: a generative model G that captures the data distribution, and a discriminative model D that estimates the probability that a sample came from the training data rather than G. The training procedure for G is to maximize the probability of D making a mistake. This framework corresponds to a minimax two-player game. In the space of arbitrary functions G and D, a unique solution exists, with G recovering the training data distribution and D equal to 1/2 everywhere. In the case where G and D are defined by multilayer perceptrons, the entire system can be trained with backpropagation. There is no need for any Markov chains or unrolled approximate inference networks during either training or generation of samples. Experiments demonstrate the potential of the framework through qualitative and quantitative evaluation of the generated samples.

We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernels lies in the fact that (a) only one side of the matrix scales with the number of data points, and (b) cross-kernels, as opposed to the usual kernel matrices, can be used to certify for the data manifold. Our theoretical framework, which is based on a duality involving the feature space and vanishing ideals, indicates that cross-kernels have the potential to be used for any kind of kernel learning. We present a novel algorithm, Ideal PCA (IPCA), which cross-kernelizes PCA. We demonstrate on real and synthetic data that IPCA allows to (a) obtain PCA-like features faster and (b) to extract novel and empirically validated features certifying for the data manifold.

The purpose of this paper is to introduce a concept of equivalence between machine learning algorithms. We define two notions of algorithmic equivalence, namely, weak and strong equivalence. These notions are of paramount importance for identifying when learning prop erties from one learning algorithm can be transferred to another. Using regularized kernel machines as a case study, we illustrate the importance of the introduced equivalence concept by analyzing the relation between kernel ridge regression (KRR) and m-power regularized least squares regression (M-RLSR) algorithms.

Shimon Ullman Weizmann Institute of Science shimon.ullman@weizmann.ac.il Abstract We consider the problem of learning from a similarity matrix (such as spectral clustering and low-dimensional embedding), when computing pairwise similarities are costly, and only a limited number of entries can be observed. We provide a theoretical analysis using standard notions of graph approximation, significantly generalizing previous results (which focused on spectral clustering with two clusters). We also propose a new algorithmic approach based on adaptive sampling, which experimentally matches or improves on previous methods, while being considerably more general and computationally cheaper. 1 Introduction Many unsupervised learning algorithms, such as spectral clustering [18], [2] and low-dimensional embedding via Laplacian eigenmaps and diffusion maps [3],[16], need as input a matrix of pairwise similaritiesW between the different objects in our data. In some cases, obtaining the full matrix can be a costly matter. For example, w ij may be based on some expensive-to-compute metric such as W2D [5]; based on some physical measurement (such as in some computational biology applications); or is given by a human annotator. In such cases, we would like to have a good approximation of the (initially unknown) matrix, while querying only a limited number of entries. An alternative but equivalent viewpoint is the problem of approximating an unknown weighted undirected graph, by querying a limited number of edges. This question has received previous attention in works such as [17] and [9], which focus on the task of spectral clustering into two clusters, and assuming two such distinct clusters indeed exist (i.e. that there is a big gap between the second and third eigenvalues of the Laplacian matrix). In this work we consider, both theoretically and algorithmically, the question of query-based graph approximation more generally, obtaining results relevant beyond two clusters and beyond spectral clustering. When considering graph approximations, the first question is what notion of approximation to consider. One important notion is cut approximation [14] where we wish for every cut in the approximated graph to have weight close to the weight of the cut in the original graph up to a multiplicative factor. Many machine learning algorithms (and many more general algorithms) such as cut based clustering [11], energy minimization [22], etc. [1] are based on cuts, so this notion of approximation is natural for these uses.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance. This work extends the results of Pascanu et al. (2014).

We propose a novel information-theoretic approach for Bayesian optimization called Predictive Entropy Search (PES). At each iteration, PES selects the next evaluation point that maximizes the expected information gained with respect to the global maximum. PES codifies this intractable acquisition function in terms of the expected reduction in the differential entropy of the predictive distribution. This reformulation allows PES to obtain approximations that are both more accurate and efficient than other alternatives such as Entropy Search (ES). Furthermore, PES can easily perform a fully Bayesian treatment of the model hyperparameters while ES cannot. We evaluate PES in both synthetic and real-world applications, including optimization problems in machine learning, finance, biotechnology, and robotics. We show that the increased accuracy of PES leads to significant gains in optimization performance.

We introduce a Bayesian solution for the problem in forensic speaker recognition, where there may be very little background material for estimating score calibration parameters. We work within the Bayesian paradigm of evidence reporting and develop a principled probabilistic treatment of the problem, which results in a Bayesian likelihood-ratio as the vehicle for reporting weight of evidence. We show in contrast, that reporting a likelihood-ratio distribution does not solve this problem. Our solution is experimentally exercised on a simulated forensic scenario, using NIST SRE'12 scores, which demonstrates a clear advantage for the proposed method compared to the traditional plugin calibration recipe.