A nonmonotonic logic of thresholded generalizations is presented. Given propositions A and B from a language L and a positive integer k, the thresholded generalization A=>B{k} means that the conditional probability P(B|A) falls short of one by no more than c*d^k. A two-level probability structure is defined. At the lower level, a model is defined to be a probability function on L. At the upper level, there is a probability distribution over models. A definition is given of what it means for a collection of thresholded generalizations to entail another thresholded generalization. This nonmonotonic entailment relation, called "entailment in probability", has the feature that its conclusions are "probabilistically trustworthy" meaning that, given true premises, it is improbable that an entailed conclusion would be false. A procedure is presented for ascertaining whether any given collection of premises entails any given conclusion. It is shown that entailment in probability is closely related to Goldszmidt and Pearl's System-Z^+, thereby demonstrating that the conclusions of System-Z^+ are probabilistically trustworthy.

The criterion commonly used in directed acyclic graphs (dags) for testing graphical independence is the well-known d-separation criterion. It allows us to build graphical representations of dependency models (usually probabilistic dependency models) in the form of belief networks, which make easy interpretation and management of independence relationships possible, without reference to numerical parameters (conditional probabilities). In this paper, we study the following combinatorial problem: finding the minimum d-separating set for two nodes in a dag. This set would represent the minimum information (in the sense of minimum number of variables) necessary to prevent these two nodes from influencing each other. The solution to this basic problem and some of its extensions can be useful in several ways, as we shall see later. Our solution is based on a two-step process: first, we reduce the original problem to the simpler one of finding a minimum separating set in an undirected graph, and second, we develop an algorithm for solving it.

Cooperative pathfinding is a problem of finding a set of non-conflicting trajectories for a number of mobile agents. Its applications include planning for teams of mobile robots, such as autonomous aircrafts, cars, or underwater vehicles. The state-of-the-art algorithms for cooperative pathfinding typically rely on some heuristic forward-search pathfinding technique, where A* is often the algorithm of choice. Here, we propose MA-RRT*, a novel algorithm for multi-agent path planning that builds upon a recently proposed asymptotically-optimal sampling-based algorithm for finding single-agent shortest path called RRT*. We experimentally evaluate the performance of the algorithm and show that the sampling-based approach offers better scalability than the classical forward-search approach in relatively large, but sparse environments, which are typical in real-world applications such as multi-aircraft collision avoidance.

Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms offer inexpensive iterations by sampling a subset of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full-gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate-of-convergence analysis shows that by controlling the sample size in an incremental gradient algorithm, it is possible to maintain the steady convergence rates of full-gradient methods. We detail a practical quasi-Newton implementation based on this approach. Numerical experiments illustrate its potential benefits.

Y ang, Member, IEEE, Roy Dong, Michel V erhaegen, S. Shankar Sastry, Fellow, IEEE Abstract--In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear . Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately characterize nonlinear models has the potential to improve the results in both existing compressive sensing applications and those where a linear approximation does not suffice, e.g., phase retrieval. In this paper, we extend the classical compressive sensing framework to a second-order T aylor expansion of the nonlinearity. Using a lifting technique and a method we call quadratic basis pursuit, we show that the sparse signal can be recovered exactly when the sampling rate is sufficiently high. We further present efficient numerical algorithms to recover sparse signals in second-order nonlinear systems, which are considerably more difficult to solve than their linear counterparts in sparse optimization. I NTRODUCTION Consider the problem of finding the sparsest signalx satisfying a system of linear equations: min x R n ‖ x ‖ 0 subj. One of the most well known approaches is to relax the zero norm and replace it with the 1-norm: min x R n ‖ x ‖ 1 subj. The ability to recover the optimal solution to (1) is essential in the theory of compressive sensing (CS) [4], [5] and a tremendous amount of work has been dedicated to solving and analyzing the solution of (1) and (2) in the last decade. Today CS is regarded as a powerful tool in signal processing and widely used in many applications. For a detailed review of the literature, the reader is referred to several recent publications such as [6], [7].

We analyse the complexity of environments according to the policies that need to be used to achieve high performance. The performance results for a population of policies leads to a distribution that is examined in terms of policy complexity and analysed through several diagrams and indicators. The notion of environment response curve is also introduced, by inverting the performance results into an ability scale. We apply all these concepts, diagrams and indicators to a minimalistic environment class, agent-populated elementary cellular automata, showing how the difficulty, discriminating power and ranges (previous to normalisation) may vary for several environments.

Social networks are increasingly being used to conduct polls. We introduce a simple model of such social polling. We suppose agents vote sequentially, but the order in which agents choose to vote is not necessarily fixed. We also suppose that an agent's vote is influenced by the votes of their friends who have already voted. Despite its simplicity, this model provides useful insights into a number of areas including social polling, sequential voting, and manipulation. We prove that the number of candidates and the network structure affect the computational complexity of computing which candidate necessarily or possibly can win in such a social poll. For social networks with bounded treewidth and a bounded number of candidates, we provide polynomial algorithms for both problems. In other cases, we prove that computing which candidates necessarily or possibly win are computationally intractable.

Deep Max-Pooling Convolutional Neural Networks are Deep Neural Networks (DNN) with convolutional and max-pooling layers. Convolutional Neural Networks (CNN) can be traced back to the Neocognitron [1] in 1980. They were first successfully applied to relatively small tasks such as digit recognition [2], image interpretation [3] and object recognition [4]. Back then their size was greatly limited by the low computational power of available hardware. Since 2010, however, DNN have greatly profited from Graphics Processing Units (GPU). Simple GPU-based multilayer perceptrons (MLP) establised new state of the art results [5] on the MNIST handwritten digit dataset [4] when made both deep and large (augmenting the training set by artificial samples helped to avoid overfitting).

The idea of fully accepting statements when the evidence has rendered them probable enough faces a number of difficulties. We leave the interpretation of probability largely open, but attempt to suggest a contextual approach to full belief. We show that the difficulties of probabilistic acceptance are not as severe as they are sometimes painted, and that though there are oddities associated with probabilistic acceptance they are in some instances less awkward than the difficulties associated with other nonmonotonic formalisms. We show that the structure at which we arrive provides a natural home for statistical inference.

Recursive graphical models usually underlie the statistical modelling concerning probabilistic expert systems based on Bayesian networks. This paper defines a version of these models, denoted as recursive exponential models, which have evolved by the desire to impose sophisticated domain knowledge onto local fragments of a model. Besides the structural knowledge, as specified by a given model, the statistical modelling may also include expert opinion about the values of parameters in the model. It is shown how to translate imprecise expert knowledge into approximately conjugate prior distributions. Based on possibly incomplete data, the score and the observed information are derived for these models. This accounts for both the traditional score and observed information, derived as derivatives of the log-likelihood, and the posterior score and observed information, derived as derivatives of the log-posterior distribution. Throughout the paper the specialization into recursive graphical models is accounted for by a simple example.