Structured prediction is a powerful framework for coping with joint prediction of interacting outputs. A central difficulty in using this framework is that often the correct label dependence structure is unknown. At the same time, we would like to avoid an overly complex structure that will lead to intractable prediction. In this work we address the challenge of learning tree structured predictive models that achieve high accuracy while at the same time facilitate efficient (linear time) inference. We start by proving that this task is in general NP-hard, and then suggest an approximate alternative. Briefly, our CRANK approach relies on a novel Circuit-RANK regularizer that penalizes non-tree structures and that can be optimized using a CCCP procedure. We demonstrate the effectiveness of our approach on several domains and show that, despite the relative simplicity of the structure, prediction accuracy is competitive with a fully connected model that is computationally costly at prediction time.

We consider the problem of maximum a posteriori (MAP) inference in discrete graphical models. We present a parallel MAP inference algorithm called Bethe-ADMM based on two ideas: tree-decomposition of the graph and the alternating direction method of multipliers (ADMM). However, unlike the standard ADMM, we use an inexact ADMM augmented with a Bethe-divergence based proximal function, which makes each subproblem in ADMM easy to solve in parallel using the sum-product algorithm. We rigorously prove global convergence of Bethe-ADMM. The proposed algorithm is extensively evaluated on both synthetic and real datasets to illustrate its effectiveness. Further, the parallel Bethe-ADMM is shown to scale almost linearly with increasing number of cores.

We consider the problem of providing nonparametric confidence guarantees for undirected graphs under weak assumptions. In particular, we do not assume sparsity, incoherence or Normality. We allow the dimension $D$ to increase with the sample size $n$. First, we prove lower bounds that show that if we want accurate inferences with low assumptions then there are limitations on the dimension as a function of sample size. When the dimension increases slowly with sample size, we show that methods based on Normal approximations and on the bootstrap lead to valid inferences and we provide Berry-Esseen bounds on the accuracy of the Normal approximation. When the dimension is large relative to sample size, accurate inferences for graphs under low assumptions are not possible. Instead we propose to estimate something less demanding than the entire partial correlation graph. In particular, we consider: cluster graphs, restricted partial correlation graphs and correlation graphs.

In this paper we propose a multi-armed bandit inspired, pool based active learning algorithm for the problem of binary classification. By carefully constructing an analogy between active learning and multi-armed bandits, we utilize ideas such as lower confidence bounds, and self-concordant regularization from the multi-armed bandit literature to design our proposed algorithm. Our algorithm is a sequential algorithm, which in each round assigns a sampling distribution on the pool, samples one point from this distribution, and queries the oracle for the label of this sampled point. The design of this sampling distribution is also inspired by the analogy between active learning and multi-armed bandits. We show how to derive lower confidence bounds required by our algorithm. Experimental comparisons to previously proposed active learning algorithms show superior performance on some standard UCI datasets.

Graphical models for structured domains are powerful tools, but the computational complexities of combinatorial prediction spaces can force restrictions on models, or require approximate inference in order to be tractable. Instead of working in a combinatorial space, we use hinge-loss Markov random fields (HL-MRFs), an expressive class of graphical models with log-concave density functions over continuous variables, which can represent confidences in discrete predictions. This paper demonstrates that HL-MRFs are general tools for fast and accurate structured prediction. We introduce the first inference algorithm that is both scalable and applicable to the full class of HL-MRFs, and show how to train HL-MRFs with several learning algorithms. Our experiments show that HL-MRFs match or surpass the predictive performance of state-of-the-art methods, including discrete models, in four application domains.

In this paper, we present the Bennett-type generalization bounds of the learning process for i.i.d. samples, and then show that the generalization bounds have a faster rate of convergence than the traditional results. In particular, we first develop two types of Bennett-type deviation inequality for the i.i.d. learning process: one provides the generalization bounds based on the uniform entropy number; the other leads to the bounds based on the Rademacher complexity. We then adopt a new method to obtain the alternative expressions of the Bennett-type generalization bounds, which imply that the bounds have a faster rate o(N^{-1/2}) of convergence than the traditional results O(N^{-1/2}). Additionally, we find that the rate of the bounds will become faster in the large-deviation case, which refers to a situation where the empirical risk is far away from (at least not close to) the expected risk. Finally, we analyze the asymptotical convergence of the learning process and compare our analysis with the existing results.

Matching pursuit (MP) methods are a promising class of feature construction algorithms for value function approximation. Yet existing MP methods require creating a pool of potential features, mandating expert knowledge or enumeration of a large feature pool, both of which hinder scalability. This paper introduces batch incremental feature dependency discovery (Batch-iFDD) as an MP method that inherits a provable convergence property. Additionally, Batch-iFDD does not require a large pool of features, leading to lower computational complexity. Empirical policy evaluation results across three domains with up to one million states highlight the scalability of Batch-iFDD over the previous state of the art MP algorithm.

We tackle the problem of online reward maximisation over a large finite set of actions described by their contexts. We focus on the case when the number of actions is too big to sample all of them even once. However we assume that we have access to the similarities between actions' contexts and that the expected reward is an arbitrary linear function of the contexts' images in the related reproducing kernel Hilbert space (RKHS). We propose KernelUCB, a kernelised UCB algorithm, and give a cumulative regret bound through a frequentist analysis. For contextual bandits, the related algorithm GP-UCB turns out to be a special case of our algorithm, and our finite-time analysis improves the regret bound of GP-UCB for the agnostic case, both in the terms of the kernel-dependent quantity and the RKHS norm of the reward function. Moreover, for the linear kernel, our regret bound matches the lower bound for contextual linear bandits.

We propose maximum likelihood estimation for learning Gaussian graphical models with a Gaussian (ell_2^2) prior on the parameters. This is in contrast to the commonly used Laplace (ell_1) prior for encouraging sparseness. We show that our optimization problem leads to a Riccati matrix equation, which has a closed form solution. We propose an efficient algorithm that performs a singular value decomposition of the training data. Our algorithm is O(NT^2)-time and O(NT)-space for N variables and T samples. Our method is tailored to high-dimensional problems (N gg T), in which sparseness promoting methods become intractable. Furthermore, instead of obtaining a single solution for a specific regularization parameter, our algorithm finds the whole solution path. We show that the method has logarithmic sample complexity under the spiked covariance model. We also propose sparsification of the dense solution with provable performance guarantees. We provide techniques for using our learnt models, such as removing unimportant variables, computing likelihoods and conditional distributions. Finally, we show promising results in several gene expressions datasets.

Predictive State Representations (PSRs) are an expressive class of models for controlled stochastic processes. PSRs represent state as a set of predictions of future observable events. Because PSRs are defined entirely in terms of observable data, statistically consistent estimates of PSR parameters can be learned efficiently by manipulating moments of observed training data. Most learning algorithms for PSRs have assumed that actions and observations are finite with low cardinality. In this paper, we generalize PSRs to infinite sets of observations and actions, using the recent concept of Hilbert space embeddings of distributions. The essence is to represent the state as a nonparametric conditional embedding operator in a Reproducing Kernel Hilbert Space (RKHS) and leverage recent work in kernel methods to estimate, predict, and update the representation. We show that these Hilbert space embeddings of PSRs are able to gracefully handle continuous actions and observations, and that our learned models outperform competing system identification algorithms on several prediction benchmarks.