AISTATS 2015 Papers

We present a Monte Carlo integration method, antithetic Markov chain sampling (AMCS), that incorporates local Markov transitions in an underlying importance sampler. Like sequential Monte Carlo sampling, the proposed method uses a sequence of Markov transitions to adapt the sampling to favour more influential regions of the integrand (modes). However, AMCS differs in the type of transitions that may be used, the number of Markov chains, and the method of chain termination. In particular, from each point sampled from an initial proposal, AMCS collects a sequence of points by simulating two independent, but antithetic, Markov chains, each terminated by a sample-dependent stopping rule. This approach provides greater flexibility for targeting influential areas while eliminating the need to fix the length of the Markov chain a priori. We show that the resulting estimator is unbiased and can reduce variance on peaked multi-modal integrands that challenge existing methods.

We prove the equivalence of first-order local consistency relaxations and the MAX SAT relaxation of Goemans and Williamson (1994) for a class of MRFs we refer to as logical MRFs. This allows us to combine the advantages of each into a single MAP inference technique: solving the local consistency relaxation with any of a number of highly scalable message-passing algorithms, and then obtaining a high-quality discrete solution via a guaranteed rounding procedure when the relaxation is not tight. Logical MRFs are a general class of models that can incorporate many common dependencies, such as logical implications and mixtures of supermodular and submodular potentials. They can be used for many structured prediction tasks, including natural language processing, computer vision, and computational social science. We show that our new inference technique can improve solution quality by as much as 20% without sacrificing speed on problems with over one million dependencies.

Linear discriminant analysis (LDA) represents a simple yet powerful technique for partitioning a p-dimensional feature vector into one of K classes based on a linear projection learned from N labeled observations. However, it is well-established that in the high-dimensional setting (p > N) the underlying projection estimator degenerates. Moreover, any linear discriminate function involving a large number of features may be difficult to interpret. To ameliorate these issues, two general categories of sparse LDA modifications have been proposed, both to reduce the number of active features and to stabilize the resulting projections. The first, based on optimal scoring, is more straightforward to implement and analyze but has been heavily criticized for its ambiguous connection with the original LDA formulation. In contrast, a second strategy applies sparse penalty functions directly to the original LDA objective but requires additional heuristic trade-off parameters, has unknown global and local minima properties, and requires a greedy sequential optimization procedure. In all cases the choice of sparse regularizer can be important, but no rigorous guidelines have been provided regarding which penalty might be preferable. Against this backdrop, we winnow down the broad space of candidate sparse LDA algorithms and promote a specific selection based on optimal scoring coupled with a particular, complementary sparse regularizer. This overall process ultimately progresses our understanding of sparse LDA in general, while leading to targeted modifications of existing algorithms that produce superior results in practice on three high-dimensional gene data sets.

We focus on the distribution regression problem: regressing to a real-valued response from a probability distribution. Although there exist a large number of similarity measures between distributions, very little is known about their generalization performance in specific learning tasks. Learning problems formulated on distributions have an inherent two-stage sampled difficulty: in practice only samples from sampled distributions are observable, and one has to build an estimate on similarities computed between sets of points. To the best of our knowledge, the only existing method with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step (which suffers from slow convergence issues in high dimensions), and the domain of the distributions to be compact Euclidean. In this paper, we provide theoretical guarantees for a remarkably simple algorithmic alternative to solve the distribution regression problem: embed the distributions to a reproducing kernel Hilbert space, and learn a ridge regressor from the embeddings to the outputs. Our main contribution is to prove the consistency of this technique in the two-stage sampled setting under mild conditions (on separable, topological domains endowed with kernels). As a special case, we answer a 15-year-old open question: we establish the consistency of the classical set kernel [Haussler, 1999; Gaertner et. al, 2002] in regression, and cover more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010].

We introduce a family of adaptive estimators on graphs, based on penalizing the \ell_1 norm of discrete graph differences. This generalizes the idea of trend filtering (Kim et al., 2009, Tibshirani, 2014) used for univariate nonparametric regression, to graphs. Analogous to the univariate case, graph trend filtering exhibits a level of local adaptivity unmatched by the usual \ell_2-based graph smoothers. It is also defined by a convex minimization problem that is readily solved (e.g., by fast ADMM or Newton algorithms). We demonstrate the merits of graph trend filtering through examples and theory.

Faced with massive data, is it possible to trade off (statistical) risk, and (computational) space and time? This challenge lies at the heart of large-scale machine learning. Using k-means clustering as a prototypical unsupervised learning problem, we show how we can strategically summarize the data (control space) in order to trade off risk and time when data is generated by a probabilistic model. Our summarization is based on coreset constructions from computational geometry. We also develop an algorithm, TRAM, to navigate the space/time/data/risk tradeoff in practice. In particular, we show that for a fixed risk (or data size), as the data size increases (resp. risk increases) the running time of TRAM decreases. Our extensive experiments on real data sets demonstrate the existence and practical utility of such tradeoffs, not only for k-means but also for Gaussian Mixture Models.

This paper studies the off-policy evaluation problem, where one aims to estimate the value of a target policy based on a sample of observations collected by another policy. We first consider the multi-armed bandit case, establish a finite-time minimax risk lower bound, and analyze the risk of three standard estimators. It is shown that in a large class of settings the so-called regression estimator is minimax optimal up to a constant that depends on the number of actions, while the other two can be arbitrarily worse even in the limit of infinitely many data points, despite their empirical success and popularity. The performance of these estimators are studied in synthetic and real problems; illustrating the nontriviality of this simple task. Finally the results are extended to the problem of off-policy evaluation in contextual bandits and fixed-horizon Markov decision processes.

A stochastic combinatorial semi-bandit is an online learning problem where at each step a learning agent chooses a subset of ground items subject to constraints, and then observes stochastic weights of these items and receives their sum as a payoff. In this paper, we close the problem of computationally and sample efficient learning in stochastic combinatorial semi-bandits. In particular, we analyze a UCB-like algorithm for solving the problem, which is known to be computationally efficient; and prove O(K L (1 / ∆) \log n) and O(\sqrtK L n \log n) upper bounds on its n-step regret, where L is the number of ground items, K is the maximum number of chosen items, and ∆is the gap between the expected returns of the optimal and best suboptimal solutions. The gap-dependent bound is tight up to a constant factor and the gap-free bound is tight up to a polylogarithmic factor.

Latent Dirichlet allocation (LDA) is an increasingly popular tool for data analysis in many domains. If LDA output affects decision making (especially when money is involved), there is an incentive for attackers to compromise it. We ask the question: how can an attacker minimally poison the corpus so that LDA produces topics that the attacker wants the LDA user to see? Answering this question is important to characterize such attacks, and to develop defenses in the future. We give a novel bilevel optimization formulation to identify the optimal poisoning attack. We present an efficient solution (up to local optima) using descent method and implicit functions. We demonstrate poisoning attacks on LDA with extensive experiments, and discuss possible defenses.

We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between large- and small-size networks where for the latter poor quality local minima have non-zero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting.

Sparse Gaussian graphical models characterize sparse dependence relationships between random variables in a network. To estimate multiple related Gaussian graphical models on the same set of variables, we formulate a hierarchical model, which leads to an optimization problem with a nonconvex log-shift penalty function. We show that under mild conditions the optimization problem is convex despite the inclusion of a nonconvex penalty, and derive an efficient optimization algorithm. Experiments on both synthetic and real data show that the proposed method is able to achieve good selection and estimation performance simultaneously, because the nonconvexity of the log-shift penalty allows for weak signals to be thresholded to zero without excessive shrinkage on the strong signals.

We present the Bayesian Echo Chamber, a new Bayesian generative model for social interaction data. By modeling the evolution of people’s language usage over time, this model discovers latent influence relationships between them. Unlike previous work on inferring influence, which has primarily focused on simple temporal dynamics evidenced via turn-taking behavior, our model captures more nuanced influence relationships, evidenced via linguistic accommodation patterns in interaction content. The model, which is based on a discrete analog of the multivariate Hawkes process, permits a fully Bayesian inference algorithm. We validate our model’s ability to discover latent influence patterns using transcripts of arguments heard by the US Supreme Court and the movie "12 Angry Men." We showcase our model’s capabilities by using it to infer latent influence patterns from Federal Open Market Committee meeting transcripts, demonstrating state-of-the-art performance at uncovering social dynamics in group discussions.

Tensor factorization arises in many machine learning applications, such as knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity of matrix factorization methods. In this paper, we propose a new algorithm for CP tensor factorization that uses random projections to reduce the problem to simultaneous matrix diagonalization. Our method is conceptually simple and also applies to non-orthogonal and asymmetric tensors of arbitrary order. We prove that a small number random projections essentially preserves the spectral information in the tensor, allowing us to remove the dependence on the eigengap that plagued earlier tensor-to-matrix reductions. Experimentally, our method outperforms existing tensor factorization methods on both simulated data and two real datasets.

Iterative Proportional Fitting (IPF) generates from an input matrix W a sequence of matrices that converges, under certain conditions, to a specific limit matrix W*. This limit is the relative-entropy nearest solution to W among all matrices of prescribed row marginals r and column marginals c. We prove this known fact by a novel strategy that contributes a pure algorithmic intuition. Then we focus on the symmetric setting: W=W’ and r=c. Since IPF inherently generates non-symmetric matrices, we introduce two symmetrized variants of IPF. We prove convergence for both of them. Further, we give a novel characterization for the existence of W* in terms of expansion properties of the undirected weighted graph represented by W. Finally, we show how our results contribute to recent work in machine learning.

We introduce a class of discrete point processes that we call the \emphSubmodular Point Processes (SPPs). These processes are characterized via a submodular (or supermodular) function, and naturally model notions of \emphinformation, coverage and \emphdiversity, as well as \emphcooperation. Unlike Log-submodular and Log-supermodular distributions (Log-SPPs) such as determinantal point processes (DPPs), SPPs are themselves submodular (or supermodular). In this paper, we analyze the computational complexity of probabilistic inference in SPPs. We show that computing the partition function for SPPs (and Log-SPPs), requires exponential complexity in the worst case, and also provide algorithms which approximate SPPs up to polynomial factors. Moreover, for several subclasses of interesting submodular functions that occur in applications, we show how we can provide efficient closed form expressions for the partition functions, and thereby marginals and conditional distributions. We also show how SPPs are closed under mixtures, thus enabling maximum likelihood based strategies for learning mixtures of submodular functions. Finally, we argue how SPPs complement existing Log-SPP distributions, and are a natural model for several applications.

In theory, Bayesian nonparametric (BNP) models are well suited to streaming data scenarios due to their ability to adapt model complexity based on the amount of data observed. Unfortunately, such benefits have not been fully realized in practice; existing inference algorithms either are not applicable to streaming applications or are not extensible to nonparametric models. For the special case of Dirichlet processes, streaming inference has been considered. However, there is growing interest in more flexible BNP models, in particular building on the class of normalized random measures (NRMs). We work within this general framework and present a streaming variational inference algorithm for NRM mixture models based on assumed density filtering. Extensions to expectation propagation algorithms are possible in the batch data setting. We demonstrate the efficacy of the algorithm on clustering documents in large, streaming text corpora.

Stochastic variational inference makes it possible to approximate posterior distributions induced by large datasets quickly using stochastic optimization. The algorithm relies on the use of fully factorized variational distributions. However, this “mean-field” independence approximation limits the fidelity of the posterior approximation, and introduces local optima. We show how to relax the mean-field approximation to allow arbitrary dependencies between global parameters and local hidden variables, producing better parameter estimates by reducing bias, sensitivity to local optima, and sensitivity to hyperparameters.

Restricted Boltzmann Machines (RBMs) are widely used as building blocks for deep learning models. Learning typically proceeds by using stochastic gradient descent, and the gradients are estimated with sampling methods. However, the gradient estimation is a computational bottleneck, so better use of the gradients will speed up the descent algorithm. To this end, we first derive upper bounds on the RBM cost function, then show that descent methods can have natural ad- vantages by operating in the L∞and Shatten-∞norm. We introduce a new method called “Stochastic Spectral Descent” that updates parameters in the normed space. Empirical results show dramatic improvements over stochastic gradient descent, and have only have a fractional increase on the per-iteration cost.

There has been great interest in recent years on statistical models for dynamic networks. In this paper, I propose a stochastic block transition model (SBTM) for dynamic networks that is inspired by the well-known stochastic block model (SBM) for static networks and previous dynamic extensions of the SBM. Unlike most existing dynamic network models, it does not make a hidden Markov assumption on the edge-level dynamics, allowing the presence or absence of edges to directly influence future edge probabilities while retaining the interpretability of the SBM. I derive an approximate inference procedure for the SBTM and demonstrate that it is significantly better at reproducing durations of edges in real social network data.