Bayesian model averaging (BMA) is the state of the art approach for overcoming model uncertainty. Yet, especially on small data sets, the results yielded by BMA might be sensitive to the prior over the models. Credal Model Averaging (CMA) addresses this problem by substituting the single prior over the models by a set of priors (credal set). Such approach solves the problem of how to choose the prior over the models and automates sensitivity analysis. We discuss various CMA algorithms for building an ensemble of logistic regressors characterized by different sets of covariates. We show how CMA can be appropriately tuned to the case in which one is prior-ignorant and to the case in which instead domain knowledge is available. CMA detects prior-dependent instances, namely instances in which a different class is more probable depending on the prior over the models. On such instances CMA suspends the judgment, returning multiple classes. We thoroughly compare different BMA and CMA variants on a real case study, predicting presence of Alpine marmot burrows in an Alpine valley. We find that BMA is almost a random guesser on the instances recognized as prior-dependent by CMA.

Stochastic Gradient Descent (SGD) is a popular optimization method which has been applied to many important machine learning tasks such as Support Vector Machines and Deep Neural Networks. In order to parallelize SGD, minibatch training is often employed. The standard approach is to uniformly sample a minibatch at each step, which often leads to high variance. In this paper we propose a stratified sampling strategy, which divides the whole dataset into clusters with low within-cluster variance; we then take examples from these clusters using a stratified sampling technique. It is shown that the convergence rate can be significantly improved by the algorithm. Encouraging experimental results confirm the effectiveness of the proposed method.

In online advertising, display ads are increasingly being placed based on real-time auctions where the advertiser who wins gets to serve the ad. This is called real-time bidding (RTB). In RTB, auctions have very tight time constraints on the order of 100ms. Therefore mechanisms for bidding intelligently such as clickthrough rate prediction need to be sufficiently fast. In this work, we propose to use dimensionality reduction of the user-website interaction graph in order to produce simplified features of users and websites that can be used as predictors of clickthrough rate. We demonstrate that the Infinite Relational Model (IRM) as a dimensionality reduction offers comparable predictive performance to conventional dimensionality reduction schemes, while achieving the most economical usage of features and fastest computations at run-time. For applications such as real-time bidding, where fast database I/O and few computations are key to success, we thus recommend using IRM based features as predictors to exploit the recommender effects from bipartite graphs.

This paper provides new insight into maximizing F1 scores in the context of binary classification and also in the context of multilabel classification. The harmonic mean of precision and recall, F1 score is widely used to measure the success of a binary classifier when one class is rare. Micro average, macro average, and per instance average F1 scores are used in multilabel classification. For any classifier that produces a real-valued output, we derive the relationship between the best achievable F1 score and the decision-making threshold that achieves this optimum. As a special case, if the classifier outputs are well-calibrated conditional probabilities, then the optimal threshold is half the optimal F1 score. As another special case, if the classifier is completely uninformative, then the optimal behavior is to classify all examples as positive. Since the actual prevalence of positive examples typically is low, this behavior can be considered undesirable. As a case study, we discuss the results, which can be surprising, of applying this procedure when predicting 26,853 labels for Medline documents.

We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably \emph{improves} with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM.

Classical methods such as Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are ubiquitous in statistics. However, these techniques are only able to reveal linear relationships in data. Although nonlinear variants of PCA and CCA have been proposed, these are computationally prohibitive in the large scale. In a separate strand of recent research, randomized methods have been proposed to construct features that help reveal nonlinear patterns in data. For basic tasks such as regression or classification, random features exhibit little or no loss in performance, while achieving drastic savings in computational requirements. In this paper we leverage randomness to design scalable new variants of nonlinear PCA and CCA; our ideas extend to key multivariate analysis tools such as spectral clustering or LDA. We demonstrate our algorithms through experiments on real-world data, on which we compare against the state-of-the-art. A simple R implementation of the presented algorithms is provided.

Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system-such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization.

We consider nonparametric estimation of $L_2$, Renyi-$\alpha$ and Tsallis-$\alpha$ divergences between continuous distributions. Our approach is to construct estimators for particular integral functionals of two densities and translate them into divergence estimators. For the integral functionals, our estimators are based on corrections of a preliminary plug-in estimator. We show that these estimators achieve the parametric convergence rate of $n^{-1/2}$ when the densities' smoothness, $s$, are both at least $d/4$ where $d$ is the dimension. We also derive minimax lower bounds for this problem which confirm that $s > d/4$ is necessary to achieve the $n^{-1/2}$ rate of convergence. We validate our theoretical guarantees with a number of simulations.

A standard technique for understanding underlying dependency structures among a set of variables posits a shared conditional probability distribution for the variables measured on individuals within a group. This approach is often referred to as module networks, where individuals are represented by nodes in a network, groups are termed modules, and the focus is on estimating the network structure among modules. However, estimation solely from node-specific variables can lead to spurious dependencies, and unverifiable structural assumptions are often used for regularization. Here, we propose an extended model that leverages direct observations about the network in addition to node-specific variables. By integrating complementary data types, we avoid the need for structural assumptions. We illustrate theoretical and practical significance of the model and develop a reversible-jump MCMC learning procedure for learning modules and model parameters. We demonstrate the method accuracy in predicting modular structures from synthetic data and capability to learn influence structures in twitter data and regulatory modules in the Mycobacterium tuberculosis gene regulatory network.

Batch Reinforcement Learning (RL) algorithms attempt to choose a policy from a designer-provided class of policies given a fixed set of training data. Choosing the policy which maximizes an estimate of return often leads to over-fitting when only limited data is available, due to the size of the policy class in relation to the amount of data available. In this work, we focus on learning policy classes that are appropriately sized to the amount of data available. We accomplish this by using the principle of Structural Risk Minimization, from Statistical Learning Theory, which uses Rademacher complexity to identify a policy class that maximizes a bound on the return of the best policy in the chosen policy class, given the available data. Unlike similar batch RL approaches, our bound on return requires only extremely weak assumptions on the true system.