We analyse a plug-in estimator for a large class of integral functionals of one or more continuous probability densities. This class includes important families of entropy, divergence, mutual information, and their conditional versions. For densities on the d-dimensional unit cube [0,1]^d that lie in a beta-Holder smoothness class, we prove our estimator converges at the rate O(n^(1/(beta+d))). Furthermore, we prove that the estimator obeys an exponential concentration inequality about its mean, whereas most previous related results have bounded only expected error of estimators. Finally, we demonstrate our bounds to the case of conditional Renyi mutual information.

The idea that animals might use information-driven planning to explore an unknown environment and build an internal model of it has been proposed for quite some time. Recent work has demonstrated that agents using this principle can efficiently learn models of probabilistic environments with discrete, bounded state spaces. However, animals and robots are commonly confronted with unbounded environments. To address this more challenging situation, we study information-based learning strategies of agents in unbounded state spaces using non-parametric Bayesian models. Specifically, we demonstrate that the Chinese Restaurant Process (CRP) model is able to solve this problem and that an Empirical Bayes version is able to efficiently explore bounded and unbounded worlds by relying on little prior information.

Importance sampling is an essential component of off-policy model-free reinforcement learning algorithms. However, its most effective variant, \emph{weighted} importance sampling, does not carry over easily to function approximation and, because of this, it is not utilized in existing off-policy learning algorithms. In this paper, we take two steps toward bridging this gap. First, we show that weighted importance sampling can be viewed as a special case of weighting the error of individual training samples, and that this weighting has theoretical and empirical benefits similar to those of weighted importance sampling. Second, we show that these benefits extend to a new weighted-importance-sampling version of off-policy LSTD(lambda). We show empirically that our new WIS-LSTD(lambda) algorithm can result in much more rapid and reliable convergence than conventional off-policy LSTD(lambda) (Yu 2010, Bertsekas & Yu 2009).

We consider the problem of finding clusters in an unweighted graph, when the graph is partially observed. We analyze two programs, one which works for dense graphs and one which works for both sparse and dense graphs, but requires some a priori knowledge of the total cluster size, that are based on the convex optimization approach for low-rank matrix recovery using nuclear norm minimization. For the commonly used Stochastic Block Model, we obtain \emph{explicit} bounds on the parameters of the problem (size and sparsity of clusters, the amount of observed data) and the regularization parameter characterize the success and failure of the programs. We corroborate our theoretical findings through extensive simulations. We also run our algorithm on a real data set obtained from crowdsourcing an image classification task on the Amazon Mechanical Turk, and observe significant performance improvement over traditional methods such as k-means.

Markov Logic Networks (MLNs) are weighted first-order logic templates for generating large (ground) Markov networks. Lifted inference algorithms for them bring the power of logical inference to probabilistic inference. These algorithms operate as much as possible at the compact first-order level, grounding or propositionalizing the MLN only as necessary. As a result, lifted inference algorithms can be much more scalable than propositional algorithms that operate directly on the much larger ground network. Unfortunately, existing lifted inference algorithms suffer from two interrelated problems, which severely affects their scalability in practice. First, for most real-world MLNs having complex structure, they are unable to exploit symmetries and end up grounding most atoms (the grounding problem). Second, they suffer from the evidence problem, which arises because evidence breaks symmetries, severely diminishing the power of lifted inference. In this paper, we address both problems by presenting a scalable, lifted importance sampling-based approach that never grounds the full MLN. Specifically, we show how to scale up the two main steps in importance sampling: sampling from the proposal distribution and weight computation. Scalable sampling is achieved by using an informed, easy-to-sample proposal distribution derived from a compressed MLN-representation. Fast weight computation is achieved by only visiting a small subset of the sampled groundings of each formula instead of all of its possible groundings. We show that our new algorithm yields an asymptotically unbiased estimate. Our experiments on several MLNs clearly demonstrate the promise of our approach.

We consider the challenging practical problem of optimizing the power production of a complex of hydroelectric power plants, which involves control over three continuous action variables, uncertainty in the amount of water inflows and a variety of constraints that need to be satisfied. We propose a policy-search-based approach coupled with predictive modelling to address this problem. This approach has some key advantages compared to other alternatives, such as dynamic programming: the policy representation and search algorithm can conveniently incorporate domain knowledge; the resulting policies are easy to interpret, and the algorithm is naturally parallelizable. Our algorithm obtains a policy which outperforms the solution found by dynamic programming both quantitatively and qualitatively.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new approach to second-order optimization, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep or recurrent neural network training, and provide numerical evidence for its superior optimization performance.

Neuronal encoding models range from the detailed biophysically-based Hodgkin Huxley model, to the statistical linear time invariant model specifying firing rates in terms of the extrinsic signal. Decoding the former becomes intractable, while the latter does not adequately capture the nonlinearities present in the neuronal encoding system. For use in practical applications, we wish to record the output of neurons, namely spikes, and decode this signal fast in order to drive a machine, for example a prosthetic device. Here, we introduce a causal, real-time decoder of the biophysically-based Integrate and Fire encoding neuron model. We show that the upper bound of the real-time reconstruction error decreases polynomially in time, and that the L2 norm of the error is bounded by a constant that depends on the density of the spikes, as well as the bandwidth and the decay of the input signal. We numerically validate the effect of these parameters on the reconstruction error.

We consider the Principal Component Analysis problem for large tensors of arbitrary order k under a single-spike (or rank-one plus noise) model. On the one hand, we use information theory, and recent results in probability theory to establish necessary and sufficient conditions under which the principal component can be estimated using unbounded computational resources. It turns out that this is possible as soon as the signal-to-noise ratio beta becomes larger than C\sqrt{k log k} (and in particular beta can remain bounded has the problem dimensions increase). On the other hand, we analyze several polynomial-time estimation algorithms, based on tensor unfolding, power iteration and message passing ideas from graphical models. We show that, unless the signal-to-noise ratio diverges in the system dimensions, none of these approaches succeeds. This is possibly related to a fundamental limitation of computationally tractable estimators for this problem. For moderate dimensions, we propose an hybrid approach that uses unfolding together with power iteration, and show that it outperforms significantly baseline methods. Finally, we consider the case in which additional side information is available about the unknown signal. We characterize the amount of side information that allow the iterative algorithms to converge to a good estimate.

Recursive Neural Networks have recently obtained state of the art performance on several natural language processing tasks. However, because of their feedforward architecture they cannot correctly predict phrase or word labels that are determined by context. This is a problem in tasks such as aspect-specific sentiment classification which tries to, for instance, predict that the word Android is positive in the sentence Android beats iOS. We introduce global belief recursive neural networks (GB-RNNs) which are based on the idea of extending purely feedforward neural networks to include one feedbackward step during inference. This allows phrase level predictions and representations to give feedback to words. We show the effectiveness of this model on the task of contextual sentiment analysis. We also show that dropout can improve RNN training and that a combination of unsupervised and supervised word vector representations performs better than either alone. The feedbackward step improves F1 performance by 3% over the standard RNN on this task, obtains state-of-the-art performance on the SemEval 2013 challenge and can accurately predict the sentiment of specific entities.