We study the problem of learning a tensor from a set of linear measurements. A prominent methodology for this problem is based on the extension of trace norm regularization, which has been used extensively for learning low rank matrices, to the tensor setting. In this paper, we highlight some limitations of this approach and propose an alternative convex relaxation on the Euclidean unit ball. We then describe a technique to solve the associated regularization problem, which builds upon the alternating direction method of multipliers. Experiments on one synthetic dataset and two real datasets indicate that the proposed method improves significantly over tensor trace norm regularization in terms of estimation error, while remaining computationally tractable.

Humans and animals readily utilize active sensing, or the use of self-motion, to focus sensory and cognitive resources on the behaviorally most relevant stimuli and events in the environment. Understanding the computational basis of natural active sensing is important both for advancing brain sciences and for developing more powerful artificial systems. Recently, a goal-directed, context-sensitive, Bayesian control strategy for active sensing, termed C-DAC (Context-Dependent Active Controller), was proposed (Ahmad & Yu, 2013). In contrast to previously proposed algorithms for human active vision, which tend to optimize abstract statistical objectives and therefore cannot adapt to changing behavioral context or task goals, C-DAC directly minimizes behavioral costs and thus, automatically adapts itself to different task conditions. However, C-DAC is limited as a model of human active sensing, given its computational/representational requirements, especially for more complex, real-world situations. Here, we propose a myopic approximation to C-DAC, which also takes behavioral costs into account, but achieves a significant reduction in complexity by looking only one step ahead. We also present data from a human active visual search experiment, and compare the performance of the various models against human behavior. We find that C-DAC and its myopic variant both achieve better fit to human data than Infomax (Butko & Movellan, 2010), which maximizes expected cumulative future information gain. In summary, this work provides novel experimental results that differentiate theoretical models for human active sensing, as well as a novel active sensing algorithm that retains the context-sensitivity of the optimal controller while achieving significant computational savings.

We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.

We investigate the relationship between three fundamental problems in machine learning: binary classification, bipartite ranking, and binary class probability estimation (CPE). It is known that a good binary CPE model can be used to obtain a good binary classification model (by thresholding at 0.5), and also to obtain a good bipartite ranking model (by using the CPE model directly as a ranking model); it is also known that a binary classification model does not necessarily yield a CPE model. However, not much is known about other directions. Formally, these relationships involve regret transfer bounds. In this paper, we introduce the notion of weak regret transfer bounds, where the mapping needed to transform a model from one problem to another depends on the underlying probability distribution (and in practice, must be estimated from data). We then show that, in this weaker sense, a good bipartite ranking model can be used to construct a good classification model (by thresholding at a suitable point), and more surprisingly, also to construct a good binary CPE model (by calibrating the scores of the ranking model).

Many attempts to understand the success of simple decision heuristics have examined heuristics as an approximation to a linear decision rule. This research has identified three environmental structures that aid heuristics: dominance, cumulative dominance, and noncompensatoriness. Here, we further develop these ideas and examine their empirical relevance in 51 natural environments. We find that all three structures are prevalent, making it possible for some simple rules to reach the accuracy levels of the linear decision rule using less information.

Many problems in machine learning can be solved by rounding the solution of an appropriate linear program (LP). This paper shows that we can recover solutions of comparable quality by rounding an approximate LP solution instead of the exact one.These approximate LP solutions can be computed efficiently by applying a parallel stochastic-coordinate-descent method to a quadratic-penalty formulation ofthe LP. We derive worst-case runtime and solution quality guarantees of this scheme using novel perturbation and convergence analysis. Our experiments demonstrate that on such combinatorial problems as vertex cover, independent set and multiway-cut, our approximate rounding scheme is up to an order of magnitude fasterthan Cplex (a commercial LP solver) while producing solutions of similar quality.

Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when computing conditional probabilities. The reason is that conditioning on evidence breaks many of the model's symmetries, which preempts standard lifting techniques. Recent theoretical results show, for example, that conditioning on evidence which corresponds to binary relations is #P-hard, suggesting that no lifting is to be expected in the worst case. In this paper, we balance this grim result by identifying the Boolean rank of the evidence as a key parameter for characterizing the complexity of conditioning in lifted inference. In particular, we show that conditioning on binary evidence with bounded Boolean rank is efficient. This opens up the possibility of approximating evidence by a low-rank Boolean matrix factorization, which we investigate both theoretically and empirically.

We propose a Learning from Demonstration (LfD) algorithm which leverages expert data,even if they are very few or inaccurate. We achieve this by using both expert data, as well as reinforcement signals gathered through trial-and-error interactions withthe environment. The key idea of our approach, Approximate Policy Iteration with Demonstration (APID), is that expert's suggestions are used to define linearconstraints which guide the optimization performed by Approximate Policy Iteration. We prove an upper bound on the Bellman error of the estimate computed by APID at each iteration. Moreover, we show empirically that APID outperforms pure Approximate Policy Iteration, a state-of-the-art LfD algorithm, and supervised learning in a variety of scenarios, including when very few and/or suboptimal demonstrations are available. Our experiments include simulations as well as a real robot path-finding task.

We develop a probabilistic approach for accurate network modeling using node popularities within the framework of the mixed-membership stochastic blockmodel (MMSB). Our model integrates some of the basic properties of nodes in social networks: homophily and preferential connection to popular nodes. We develop a scalable algorithm for posterior inference, based on a novel nonconjugate variant of stochastic variational inference. We evaluate the link prediction accuracy of our algorithm on eight real-world networks with up to 60,000 nodes, and 24 benchmark networks. We demonstrate that our algorithm predicts better than the MMSB. Further, using benchmark networks we show that node popularities are essential to achieving high accuracy in the presence of skewed degree distribution and noisy links---both characteristics of real networks.

We study stochastic optimization problems when the \emph{data} is sparse, which is in a sense dual to the current understanding of high-dimensional statistical learning and optimization. We highlight both the difficulties---in terms of increased sample complexity that sparse data necessitates---and the potential benefits, in terms of allowing parallelism and asynchrony in the design of algorithms. Concretely, we derive matching upper and lower bounds on the minimax rate for optimization and learning with sparse data, and we exhibit algorithms achieving these rates. Our algorithms are adaptive: they achieve the best possible rate for the data observed. We also show how leveraging sparsity leads to (still minimax optimal) parallel and asynchronous algorithms, providing experimental evidence complementing our theoretical results on medium to large-scale learning tasks.