The development of statistical models for continuous-time longitudinal network data is of increasing interest in machine learning and social science. Leveraging ideas from survival and event history analysis, we introduce a continuous-time regression modeling framework for network event data that can incorporate both time-dependent network statistics and time-varying regression coefficients. We also develop an efficient inference scheme that allows our approach to scale to large networks. On synthetic and real-world data, empirical results demonstrate that the proposed inference approach can accurately estimate the coefficients of the regression model, which is useful for interpreting the evolution of the network; furthermore, the learned model has systematically better predictive performance compared to standard baseline methods.

The performance of Markov chain Monte Carlo methods is often sensitive to the scaling and correlations between the random variables of interest. An important source of information about the local correlation and scale is given by the Hessian matrix of the target distribution, but this is often either computationally expensive or infeasible. In this paper we propose MCMC samplers that make use of quasi-Newton approximations from the optimization literature, that approximate the Hessian of the target distribution from previous samples and gradients generated by the sampler. A key issue is that MCMC samplers that depend on the history of previous states are in general not valid. We address this problem by using limited memory quasi-Newton methods, which depend only on a fixed window of previous samples. On several real world datasets, we show that the quasi-Newton sampler is a more effective sampler than standard Hamiltonian Monte Carlo at a fraction of the cost of MCMC methods that require higher-order derivatives.

Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are non-convex and non-smooth, we provide an efficient first-order scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately.

The L_1 regularized Gaussian maximum likelihood estimator has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to other state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and also present experimental results using synthetic and real application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

We combine three important ideas present in previous work for building classifiers: thesemi-supervised hypothesis (the input distribution contains information about the classifier), the unsupervised manifold hypothesis (data density concentrates nearlow-dimensional manifolds), and the manifold hypothesis for classification (differentclasses correspond to disjoint manifolds separated by low density). We exploit a novel algorithm for capturing manifold structure (high-order contractive auto-encoders) and we show how it builds a topological atlas of charts, each chart being characterized by the principal singular vectors of the Jacobian of a representation mapping. This representation learning algorithm can be stacked to yield a deep architecture, and we combine it with a domain knowledge-free version of the TangentProp algorithm to encourage the classifier to be insensitive to local directions changes along the manifold. Record-breaking classification results are obtained.

Principal Components Analysis~(PCA) is often used as a feature extraction procedure. Given a matrix $X \in \mathbb{R}^{n \times d}$, whose rows represent $n$ data points with respect to $d$ features, the top $k$ right singular vectors of $X$ (the so-called \textit{eigenfeatures}), are arbitrary linear combinations of all available features. The eigenfeatures are very useful in data analysis, including the regularization of linear regression. Enforcing sparsity on the eigenfeatures, i.e., forcing them to be linear combinations of only a \textit{small} number of actual features (as opposed to all available features), can promote better generalization error and improve the interpretability of the eigenfeatures. We present deterministic and randomized algorithms that construct such sparse eigenfeatures while \emph{provably} achieving in-sample performance comparable to regularized linear regression. Our algorithms are relatively simple and practically efficient, and we demonstrate their performance on several data sets.

Thompson sampling is one of oldest heuristic to address the exploration / exploitation trade-off, but it is surprisingly not very popular in the literature. We present here some empirical results using Thompson sampling on simulated and real data, and show that it is highly competitive. And since this heuristic is very easy to implement, we argue that it should be part of the standard baselines to compare against.

We consider latent structural versions of probit loss and ramp loss. We show that these surrogate loss functions are consistent in the strong sense that for any feature map (finite or infinite dimensional) they yield predictors approaching the infimum task loss achievable by any linear predictor over the given features. We also give finite sample generalization bounds (convergence rates) for these loss functions. These bounds suggest that probit loss converges more rapidly. However, ramp loss is more easily optimized and may ultimately be more practical.

We consider the problem of assigning class labels to an unlabeled test data set, given several labeled training data sets drawn from similar distributions. This problem arises in several applications where data distributions fluctuate because of biological, technical, or other sources of variation. We develop a distribution-free, kernel-based approach to the problem. This approach involves identifying an appropriate reproducing kernel Hilbert space and optimizing a regularized empirical risk over the space. We present generalization error analysis, describe universal kernels, and establish universal consistency of the proposed methodology. Experimental results on flow cytometry data are presented.

Increasingly, optimization problems in machine learning, especially those arising from high-dimensional statistical estimation, have a large number of variables. Modern statistical estimators developed over the past decade have statistical or sample complexity that depends only weakly on the number of parameters when there is some structure to the problem, such as sparsity. A central question is whether similar advances can be made in their computational complexity as well. In this paper, we propose strategies that indicate that such advances can indeed be made. In particular, we investigate the greedy coordinate descent algorithm, and note that performing the greedy step efficiently weakens the costly dependence on the problem size provided the solution is sparse. We then propose a suite of methods that perform these greedy steps efficiently by a reduction to nearest neighbor search. We also devise a more amenable form of greedy descent for composite non-smooth objectives; as well as several approximate variants of such greedy descent. We develop a practical implementation of our algorithm that combines greedy coordinate descent with locality sensitive hashing. Without tuning the latter data structure, we are not only able to significantly speed up the vanilla greedy method, but also outperform cyclic descent when the problem size becomes large. Our results indicate the effectiveness of our nearest neighbor strategies, and also point to many open questions regarding the development of computational geometric techniques tailored towards first-order optimization methods.