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.

This thesis studies two problems in modern statistics. First, we study selective inference, or inference for hypothesis that are chosen after looking at the data. The motiving application is inference for regression coefficients selected by the lasso. We present the Condition-on-Selection method that allows for valid selective inference, and study its application to the lasso, and several other selection algorithms. In the second part, we consider the problem of learning the structure of a pairwise graphical model over continuous and discrete variables. We present a new pairwise model for graphical models with both continuous and discrete variables that is amenable to structure learning. In previous work, authors have considered structure learning of Gaussian graphical models and structure learning of discrete models. Our approach is a natural generalization of these two lines of work to the mixed case. The penalization scheme involves a novel symmetric use of the group-lasso norm and follows naturally from a particular parametrization of the model. We provide conditions under which our estimator is model selection consistent in the high-dimensional regime.

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}).

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 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.

In this paper we present a sequential investment strategy - a portfolio selection strategy or portfolio optimization technique - that could be used in financial markets. Sequential investment means that at the end of one trading period the investor is allowed to redistribute his current capital among a set of available assets. The investor's goal is to maximize his capital. The portfolio selection is based on historical data collected from the market. Local linear regression base models or experts are used in an ensemble called a committee to model the nextday return of an asset. The committees use different voting strategies to provide the estimate for each asset. The estimates along with historical performances will be used to generate portfolio weights for a given trading period. Numerical results will be presented to show the performance of the portfolio selection strategy.

The linking genotype to phenotype is the fundamental aim of modern genetics. We focus on study of links between gene expression data and phenotype data through integrative analysis. We propose three approaches. 1) The inherent complexity of phenotypes makes high-throughput phenotype profiling a very difficult and laborious process. We propose a method of automated multi-dimensional profiling which uses gene expression similarity. Large-scale analysis show that our method can provide robust profiling that reveals different phenotypic aspects of samples. This profiling technique is also capable of interpolation and extrapolation beyond the phenotype information given in training data. It can be used in many applications, including facilitating experimental design and detecting confounding factors. 2) Phenotype association analysis problems are complicated by small sample size and high dimensionality. Consequently, phenotype-associated gene subsets obtained from training data are very sensitive to selection of training samples, and the constructed sample phenotype classifiers tend to have poor generalization properties. To eliminate these obstacles, we propose a novel approach that generates sequences of increasingly discriminative gene cluster combinations. Our experiments on both simulated and real datasets show robust and accurate classification performance. 3) Many complex phenotypes, such as cancer, are the product of not only gene expression, but also gene interaction. We propose an integrative approach to find gene network modules that activate under different phenotype conditions. Using our method, we discovered cancer subtype-specific network modules, as well as the ways in which these modules coordinate. In particular, we detected a breast-cancer specific tumor suppressor network module with a hub gene, PDGFRL, which may play an important role in this module.

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.

Mixture of Experts (MoE) is a popular framework for modeling heterogeneity in data for regression, classification and clustering. For continuous data which we consider here in the context of regression and cluster analysis, MoE usually use normal experts, that is, expert components following the Gaussian distribution. However, for a set of data containing a group or groups of observations with asymmetric behavior, heavy tails or atypical observations, the use of normal experts may be unsuitable and can unduly affect the fit of the MoE model. In this paper, we introduce new non-normal mixture of experts (NNMoE) which can deal with these issues regarding possibly skewed, heavy-tailed data and with outliers. The proposed models are the skew-normal MoE and the robust $t$ MoE and skew $t$ MoE, respectively named SNMoE, TMoE and STMoE. We develop dedicated expectation-maximization (EM) and expectation conditional maximization (ECM) algorithms to estimate the parameters of the proposed models by monotonically maximizing the observed data log-likelihood. We describe how the presented models can be used in prediction and in model-based clustering of regression data. Numerical experiments carried out on simulated data show the effectiveness and the robustness of the proposed models in terms modeling non-linear regression functions as well as in model-based clustering. Then, to show their usefulness for practical applications, the proposed models are applied to the real-world data of tone perception for musical data analysis, and the one of temperature anomalies for the analysis of climate change data.

In this paper, a nonparametric maximum likelihood (ML) estimator for band-limited (BL) probability density functions (pdfs) is proposed. The BLML estimator is consistent and computationally efficient. To compute the BLML estimator, three approximate algorithms are presented: a binary quadratic programming (BQP) algorithm for medium scale problems, a Trivial algorithm for large-scale problems that yields a consistent estimate if the underlying pdf is strictly positive and BL, and a fast implementation of the Trivial algorithm that exploits the band-limited assumption and the Nyquist sampling theorem ("BLMLQuick"). All three BLML estimators outperform kernel density estimation (KDE) algorithms (adaptive and higher order KDEs) with respect to the mean integrated squared error for data generated from both BL and infinite-band pdfs. Further, the BLMLQuick estimate is remarkably faster than the KD algorithms. Finally, the BLML method is applied to estimate the conditional intensity function of a neuronal spike train (point process) recorded from a rat's entorhinal cortex grid cell, for which it outperforms state-of-the-art estimators used in neuroscience.