This paper proposes a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform confidence subgraph. Due to the presence of unknown marginal transformations, we propose a pseudo likelihood based inferential approach. In sharp contrast to the existing high dimensional score test method, our method is free of tuning parameters given an initial estimator, and extends the scope of the existing likelihood based inferential framework. Furthermore, we propose a U-statistic multiplier bootstrap method to construct the confidence subgraph. We show that the constructed subgraph is contained in the true graph with probability greater than a given nominal level. Compared with existing methods for constructing confidence subgraphs, our method does not rely on Gaussian or sub-Gaussian assumptions. The theoretical properties of the proposed inferential methods are verified by thorough numerical experiments and real data analysis.

We present a general regularization-based framework for Multi-task learning (MTL), in which the similarity between tasks can be learned or refined using $\ell_p$-norm Multiple Kernel learning (MKL). Based on this very general formulation (including a general loss function), we derive the corresponding dual formulation using Fenchel duality applied to Hermitian matrices. We show that numerous established MTL methods can be derived as special cases from both, the primal and dual of our formulation. Furthermore, we derive a modern dual-coordinate descend optimization strategy for the hinge-loss variant of our formulation and provide convergence bounds for our algorithm. As a special case, we implement in C++ a fast LibLinear-style solver for $\ell_p$-norm MKL. In the experimental section, we analyze various aspects of our algorithm such as predictive performance and ability to reconstruct task relationships on biologically inspired synthetic data, where we have full control over the underlying ground truth. We also experiment on a new dataset from the domain of computational biology that we collected for the purpose of this paper. It concerns the prediction of transcription start sites (TSS) over nine organisms, which is a crucial task in gene finding. Our solvers including all discussed special cases are made available as open-source software as part of the SHOGUN machine learning toolbox (available at \url{http://shogun.ml}).

Cross-validation (CV) is one of the main tools for performance estimation and parameter tuning in machine learning. The general recipe for computing CV estimate is to run a learning algorithm separately for each CV fold, a computationally expensive process. In this paper, we propose a new approach to reduce the computational burden of CV-based performance estimation. As opposed to all previous attempts, which are specific to a particular learning model or problem domain, we propose a general method applicable to a large class of incremental learning algorithms, which are uniquely fitted to big data problems. In particular, our method applies to a wide range of supervised and unsupervised learning tasks with different performance criteria, as long as the base learning algorithm is incremental. We show that the running time of the algorithm scales logarithmically, rather than linearly, in the number of CV folds. Furthermore, the algorithm has favorable properties for parallel and distributed implementation. Experiments with state-of-the-art incremental learning algorithms confirm the practicality of the proposed method.

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a collection of recursively solved sub-problems. The proposed method is applicable to a broad class of probabilistic graphical models, including models with loops. Unlike a standard SMC sampler, the proposed Divide-and-Conquer SMC employs multiple independent populations of weighted particles, which are resampled, merged, and propagated as the method progresses. We illustrate empirically that this approach can outperform standard methods in terms of the accuracy of the posterior expectation and marginal likelihood approximations. Divide-and-Conquer SMC also opens up novel parallel implementation options and the possibility of concentrating the computational effort on the most challenging sub-problems. We demonstrate its performance on a Markov random field and on a hierarchical logistic regression problem.

Decentralized POMDPs provide an expressive framework for multiagent sequential decision making. However, the complexity of these models---NEXP-Complete even for two agents---has limited their scalability. We present a promising new class of approximation algorithms by developing novel connections between multiagent planning and machine learning. We show how the multiagent planning problem can be reformulated as inference in a mixture of dynamic Bayesian networks (DBNs). This planning-as-inference approach paves the way for the application of efficient inference techniques in DBNs to multiagent decision making. To further improve scalability, we identify certain conditions that are sufficient to extend the approach to multiagent systems with dozens of agents. Specifically, we show that the necessary inference within the expectation-maximization framework can be decomposed into processes that often involve a small subset of agents, thereby facilitating scalability. We further show that a number of existing multiagent planning models satisfy these conditions. Experiments on large planning benchmarks confirm the benefits of our approach in terms of runtime and scalability with respect to existing techniques.

We study the $K$-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. We introduce a tight asymptotic regret lower bound that is based on the information divergence. An algorithm that is inspired by the Deterministic Minimum Empirical Divergence algorithm (Honda and Takemura, 2010) is proposed, and its regret is analyzed. The proposed algorithm is found to be the first one with a regret upper bound that matches the lower bound. Experimental comparisons of dueling bandit algorithms show that the proposed algorithm significantly outperforms existing ones.

Machine learning methods for solving the equations of dynamical mean-field theory are developed. The method is demonstrated on the three dimensional Hubbard model. The key technical issues are defining a mapping of an input function to an output function, and distinguishing metallic from insulating solutions. Both metallic and Mott insulator solutions can be predicted. The validity of the machine learning scheme is assessed by comparing predictions of full correlation functions, of quasi-particle weight and particle density to values directly computed. The results indicate that with modest further development, machine learning approach may be an attractive computational efficient option for real materials predictions for strongly correlated systems.

In this paper we implement a Local Linear Regression Ensemble Committee (LOLREC) to predict 1-day-ahead returns of 453 assets form the S&P500. The estimates and the historical returns of the committees are used to compute the weights of the portfolio from the 453 stock. The proposed method outperforms benchmark portfolio selection strategies that optimize the growth rate of the capital. We investigate the effect of algorithm parameter m: the number of selected stocks on achieved average annual yields. Results suggest the algorithm's practical usefulness in everyday trading.

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S2 for this task. At each step, S2 selects the vertex to be labeled based on the structure of the graph and all previously gathered labels. Specifically, S2 queries for the label of the vertex that bisects the *shortest shortest* path between any pair of oppositely labeled vertices. We present a theoretical estimate of the number of queries S2 needs in terms of a novel parametrization of the complexity of binary functions on graphs. We also present experimental results demonstrating the performance of S2 on both real and synthetic data. While other graph-based active learning algorithms have shown promise in practice, our algorithm is the first with both good performance and theoretical guarantees. Finally, we demonstrate the implications of the S2 algorithm to the theory of nonparametric active learning. In particular, we show that S2 achieves near minimax optimal excess risk for an important class of nonparametric classification problems.

Single Index Models (SIMs) are simple yet flexible semi-parametric models for classification and regression. Response variables are modeled as a nonlinear, monotonic function of a linear combination of features. Estimation in this context requires learning both the feature weights, and the nonlinear function. While methods have been described to learn SIMs in the low dimensional regime, a method that can efficiently learn SIMs in high dimensions has not been forthcoming. We propose three variants of a computationally and statistically efficient algorithm for SIM inference in high dimensions. We establish excess risk bounds for the proposed algorithms and experimentally validate the advantages that our SIM learning methods provide relative to Generalized Linear Model (GLM) and low dimensional SIM based learning methods.