Here we study the tensor prediction problem, where the goal is to accurately predict the entries of a low rank, third-order tensor (with noise) given as few observations as possible. We give algorithms based on the sixth level of the sum-of-squares hierarchy that work with roughly m=n3/2 observations, and we complement our result by showing that any attempt to solve tensor prediction with fewer observations through the sum-of-squares hierarchy would run in moderately exponential time. In contrast, information theoretically roughly…

Undirected graphical model has proven to be a useful abstraction in statistics and machine learning. The starting point is the graph of a distribution. While often the graph is assumed given, we are interested in estimating the graph from data. In this talk we present new nonparametric and semiparametric methods for graph estimation. The performance of these methods is illustrated and compared on several real and simulated examples.

To carry out posterior inference on datasets of unprecedented sizes, practitioners are turning to biased MCMC procedures that trade off asymptotic exactness for computational efficiency. The reasoning is sound: a reduction in variance due to more rapid sampling can outweigh the bias introduced. However, the inexactness creates new challenges for sampler and parameter selection, since standard measures of sample quality like effective sample size do not account for asymptotic bias. To address these challenges, we introduce a new computable quality…

We present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. We present computational guarantees for the method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply our method to the low-rank matrix completion problem, where low-dimensional faces correspond to low-rank solutions. We present computational results for large-scale low-rank matrix completion problems that demonstrate significant speed-ups in…

We consider the problem of influence maximization in fixed networks, for both stochastic and adversarial contagion models. In the stochastic setting, nodes are infected in waves according to linear threshold or independent cascade models. We establish upper and lower bounds for the influence of a subset of nodes in the network, where the influence is defined as the expected number of infected nodes at the conclusion of the epidemic. We quantify the gap between our upper and lower bounds in…

A independent set in a graph is a set of vertices that have no edges between them. How large can a independent set be in a random d-regular graph? How large can it be if we are to construct it using a (possibly randomized) algorithm that is local in nature? We will discuss a notion of local algorithms for combinatorial optimization problems on large, random d-regular graphs. We will then explain why, for asymptotically large d, local algorithms can only…

Classical approaches to causal inference largely rely on the assumption of “lack of interference”, according to which the outcome of each individual does not depend on the treatment assigned to others. In many applications, however, including healthcare interventions in schools, online education, and design of online auctions and political campaigns on social media, assuming lack of interference is untenable. In this talk, Prof. Airoldi will introduce some fundamental ideas to deal with interference in causal analyses, focusing on situations where…

ixture models represent the superposition of statistical processes, and are natural in machine learning and statistics. In mixed regression, the relationship between input and output is given by one of possibly several different (noisy) linear functions. Thus the solution encodes a combinatorial selection problem, and hence computing it is difficult in the worst case. Even in the average case, little is known in the realm of efficient algorithms with strong statistical guarantees. We give general conditions for linear convergence of…

Most big real-world networks (social, technological, biological) are sparse. Most of networks have noticeable structure, which can be formed by clusters (communities) and hubs. When and how can a hidden structure be recovered from a sparse network? Known approaches to this problem come from a variety of disciplines – probability, combinatorics, physics, statistics, optmization, information theory, etc. We will focus on the recently developed probabilistic approaches motivated by sparse recovery, where a network is regarded as a random measurement of…