This paper introduces a new probabilistic model for online learning which dynamically incorporates information from stochastic gradients of an arbitrary loss function. Similar to probabilistic filtering, the model maintains a Gaussian belief over the optimal weight parameters. Unlike traditional Bayesian updates, the model incorporates a small number of gradient evaluations at locations chosen using Thompson sampling, making it computationally tractable. The belief is then transformed via a linear flow field which optimally updates the belief distribution using rules derived from information theoretic principles. Several versions of the algorithm are shown using different constraints on the flow field and compared with conventional online learning algorithms. Results are given for several classification tasks including logistic regression and multilayer neural networks.

Modern classification problems frequently present mild to severe label imbalance as well as specific requirements on classification characteristics, and require optimizing performance measures that are non-decomposable over the dataset, such as F-measure. Such measures have spurred much interest and pose specific challenges to learning algorithms since their non-additive nature precludes a direct application of well-studied large scale optimization methods such as stochastic gradient descent. In this paper we reveal that for two large families of performance measures that can be expressed as functions of true positive/negative rates, it is indeed possible to implement point stochastic updates. The families we consider are concave and pseudo-linear functions of TPR, TNR which cover several popularly used performance measures such as F-measure, G-mean and H-mean. Our core contribution is an adaptive linearization scheme for these families, using which we develop optimization techniques that enable truly point-based stochastic updates. For concave performance measures we propose SPADE, a stochastic primal dual solver; for pseudo-linear measures we propose STAMP, a stochastic alternate maximization procedure. Both methods have crisp convergence guarantees, demonstrate significant speedups over existing methods - often by an order of magnitude or more, and give similar or more accurate predictions on test data.

Metagenomics characterizes the taxonomic diversity of microbial communities by sequencing DNA directly from an environmental sample. One of the main challenges in metagenomics data analysis is the binning step, where each sequenced read is assigned to a taxonomic clade. Due to the large volume of metagenomics datasets, binning methods need fast and accurate algorithms that can operate with reasonable computing requirements. While standard alignment-based methods provide state-of-the-art performance, compositional approaches that assign a taxonomic class to a DNA read based on the k-mers it contains have the potential to provide faster solutions. In this work, we investigate the potential of modern, large-scale machine learning implementations for taxonomic affectation of next-generation sequencing reads based on their k-mers profile. We show that machine learning-based compositional approaches benefit from increasing the number of fragments sampled from reference genome to tune their parameters, up to a coverage of about 10, and from increasing the k-mer size to about 12. Tuning these models involves training a machine learning model on about 10 8 samples in 10 7 dimensions, which is out of reach of standard soft-wares but can be done efficiently with modern implementations for large-scale machine learning. The resulting models are competitive in terms of accuracy with well-established alignment tools for problems involving a small to moderate number of candidate species, and for reasonable amounts of sequencing errors. We show, however, that compositional approaches are still limited in their ability to deal with problems involving a greater number of species, and more sensitive to sequencing errors. We finally confirm that compositional approach achieve faster prediction times, with a gain of 3 to 15 times with respect to the BWA-MEM short read mapper, depending on the number of candidate species and the level of sequencing noise.

The problem of maximizing precision at the top of a ranked list, often dubbed Precision@k (prec@k), finds relevance in myriad learning applications such as ranking, multi-label classification, and learning with severe label imbalance. However, despite its popularity, there exist significant gaps in our understanding of this problem and its associated performance measure. The most notable of these is the lack of a convex upper bounding surrogate for prec@k. We also lack scalable perceptron and stochastic gradient descent algorithms for optimizing this performance measure. In this paper we make key contributions in these directions. At the heart of our results is a family of truly upper bounding surrogates for prec@k. These surrogates are motivated in a principled manner and enjoy attractive properties such as consistency to prec@k under various natural margin/noise conditions. These surrogates are then used to design a class of novel perceptron algorithms for optimizing prec@k with provable mistake bounds. We also devise scalable stochastic gradient descent style methods for this problem with provable convergence bounds. Our proofs rely on novel uniform convergence bounds which require an in-depth analysis of the structural properties of prec@k and its surrogates. We conclude with experimental results comparing our algorithms with state-of-the-art cutting plane and stochastic gradient algorithms for maximizing prec@k.

Much of the data being created on the web contains interactions between users and items. Stochastic blockmodels, and other methods for community detection and clustering of bipartite graphs, can infer latent user communities and latent item clusters from this interaction data. These methods, however, typically ignore the items' contents and the information they provide about item clusters, despite the tendency of items in the same latent cluster to share commonalities in content. We introduce content-augmented stochastic blockmodels (CASB), which use item content together with user-item interaction data to enhance the user communities and item clusters learned. Comparisons to several state-of-the-art benchmark methods, on datasets arising from scientists interacting with scientific articles, show that content-augmented stochastic blockmodels provide highly accurate clusters with respect to metrics representative of the underlying community structure.

We empirically evaluate a stochastic annealing strategy for Bayesian posterior optimization with variational inference. Variational inference is a deterministic approach to approximate posterior inference in Bayesian models in which a typically non-convex objective function is locally optimized over the parameters of the approximating distribution. We investigate an annealing method for optimizing this objective with the aim of finding a better local optimal solution and compare with deterministic annealing methods and no annealing. We show that stochastic annealing can provide clear improvement on the GMM and HMM, while performance on LDA tends to favor deterministic annealing methods.

An important form of prior information in clustering comes in form of cannot-link and must-link constraints. We present a generalization of the popular spectral clustering technique which integrates such constraints. Motivated by the recently proposed $1$-spectral clustering for the unconstrained problem, our method is based on a tight relaxation of the constrained normalized cut into a continuous optimization problem. Opposite to all other methods which have been suggested for constrained spectral clustering, we can always guarantee to satisfy all constraints. Moreover, our soft formulation allows to optimize a trade-off between normalized cut and the number of violated constraints. An efficient implementation is provided which scales to large datasets. We outperform consistently all other proposed methods in the experiments.

Lasso is a seminal contribution to high-dimensional statistics, but it hinges on a tuning parameter that is difficult to calibrate in practice. A partial remedy for this problem is Square-Root Lasso, because it inherently calibrates to the noise variance. However, Square-Root Lasso still requires the calibration of a tuning parameter to all other aspects of the model. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning parameters. We show that TREX can outperform cross-validated Lasso in terms of variable selection and computational efficiency. We also introduce a bootstrapped version of TREX that can further improve variable selection. We illustrate the promising performance of TREX both on synthetic data and on a recent high-dimensional biological data set that considers riboflavin production in B. subtilis.

Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced $k$-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced $k$-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimization of our tight continuous relaxation we propose a new algorithm for the difficult sum-of-ratios minimization problem which achieves monotonic descent. Extensive comparisons show that our method outperforms all existing approaches for ratio cut and other balanced $k$-cut criteria.

Covariance and histogram image descriptors provide an effective way to capture information about images. Both excel when used in combination with special purpose distance metrics. For covariance descriptors these metrics measure the distance along the non-Euclidean Riemannian manifold of symmetric positive definite matrices. For histogram descriptors the Earth Mover's distance measures the optimal transport between two histograms. Although more precise, these distance metrics are very expensive to compute, making them impractical in many applications, even for data sets of only a few thousand examples. In this paper we present two methods to compress the size of covariance and histogram datasets with only marginal increases in test error for k-nearest neighbor classification. Specifically, we show that we can reduce data sets to 16% and in some cases as little as 2% of their original size, while approximately matching the test error of kNN classification on the full training set. In fact, because the compressed set is learned in a supervised fashion, it sometimes even outperforms the full data set, while requiring only a fraction of the space and drastically reducing test-time computation.