We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast stochastic algorithms that provably converge to a stationary point for constant minibatches. Furthermore, using a variant of these algorithms, we show provably faster convergence than batch proximal gradient descent. Finally, we prove global linear convergence rate for an interesting subclass of nonsmooth nonconvex functions, that subsumes several recent works. This paper builds upon our recent series of papers on fast stochastic methods for smooth nonconvex optimization [22, 23], with a novel analysis for nonconvex and nonsmooth functions.

There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al. pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.

Supplying job schedulers with information on how long the jobs are expected to run enabled the development of the backfilling algorithms, which leverage this information to pack the jobs more efficiently and improve system utilization [1]. These algorithms, however, were designed for parallel systems, in which the jobs require many processors in order to execute, and processor fragmentation (idleness) is a big concern. In those environments the scheduler needs to know the actual runtimes of the jobs (use numeric predictions) to be able to optimize the schedule and improve performance [10]. Our work targets systems in which most jobs are serial, like server farms that are used for software testing. In those environments sophisticated scheduling algorithms are not required, and in order to improve performance it is enough to simply separate the short jobs from the long and assign them to different queues in the system [12]. This separation reduces the likelihood that short jobs will be delayed after long ones, improves the average turnaround times of the jobs and overall system throughput.

Spectral clustering has become a popular technique due to its high performance in many contexts. It comprises three main steps: create a similarity graph between N objects to cluster, compute the first k eigenvectors of its Laplacian matrix to define a feature vector for each object, and run k-means on these features to separate objects into k classes. Each of these three steps becomes computationally intensive for large N and/or k. We propose to speed up the last two steps based on recent results in the emerging field of graph signal processing: graph filtering of random signals, and random sampling of bandlimited graph signals. We prove that our method, with a gain in computation time that can reach several orders of magnitude, is in fact an approximation of spectral clustering, for which we are able to control the error. We test the performance of our method on artificial and real-world network data.

The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several Bayesian leave-one-out (LOO) cross-validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data. Our main objective is to assess the accuracy of the approximative LOO cross-validation estimators. That is, for each method (Laplace and EP) we compare the approximate fast computation with the exact brute force LOO computation. Secondarily, we evaluate the accuracy of the Laplace and EP approximations themselves against a ground truth established through extensive Markov chain Monte Carlo simulation. Our empirical results show that the approach based upon a Gaussian approximation to the LOO marginal distribution (the so-called cavity distribution) gives the most accurate and reliable results among the fast methods.

Subspace recovery from corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, existing robust Matrix Completion (MC) methods mostly concentrate on recovering a low-rank matrix from few corrupted coefficients w.r.t. standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few coefficients w.r.t. general basis, though $r$ and the number of corrupted samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is $\lambda=1/\sqrt{\log n}$. By our $\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust Low-Rank Representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. Experiments verify our theories.

ALISON STEWART, PBS NEWSHOUR WEEKEND ANCHOR: Joining me now via Skype to discuss the significance of the U.S. military strike on the Taliban's leader is Jennifer Glasse, a freelance reporter, now in Afghanistan's capital, Kabul. Jennifer, tell us a little bit more about Mansour, who he was within the hierarchy of the Taliban? JENNIFER GLASSE, FREELANCE REPORTER: He was the Taliban leader who took over in last summer. He was a bit of a controversy because when he took over after the announcement, that Mullah Omar had been dead for more than two years. There was a bit of a power struggle and division among the Taliban.

How would you perform accurate classification on a very large dataset, by just looking at a sample of it? One of his recent papers is about big data and similarity metrics. In this work Rocco proposes a deterministic method to obtain subsets from Big Data which are a good representative of the inherent structure in the data itself. This allows one to consider only a subset of the entire dataset, still performing at high accuracy if not better than traditional (eg. As you can see, there is always a solution in Big Data.

This controversial question is argued by many who hold very strong opinions.

Google wants to put the art back in artificial intelligence. During the last session at Moogfest, a four-day music and technology festival, in Durham, North Carolina, Douglas Eck, a researcher on Google Brain, the company's artificial-intelligence research project, outlined a new group that's going to focus on figuring out if computers can truly create. The group, called Magenta, will launch more publicly at the start of June, but attendees at Moogfest were given a taste of what it's going to be working on. Magenta will use TensorFlow, the machine-learning engine that Google built and opened up to the public at the end of 2015, to determine whether AI systems can be trained to create original pieces of music, art, or video. This is no simple task, given that even the most advanced artificially intelligent systems have enough trouble copying the styles of existing artists and musicians, let alone coming up with entirely new ideas themselves.