ARTINGALES are one of the fundamental tools in probability theory and statistics for modeling and studying sequences of random variables. Some of the most well-known and widely used concentration inequalities for individual martingales are Hoeffding-Azuma's and Bernstein's inequalities [1], [2], [3]. We present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0, 1] interval by the expectation of the same function of independent Bernoulli variables. We apply this inequality in order to derive a tighter analog of Hoeffding-Azuma's inequality for martingales. More importantly, we present a set of inequalities that make it possible to control weighted averages of multiple simultaneously evolving and interdependent martingales (see Figure 1 for an illustration).

Distributions over rankings are used to model data in a multitude of real world settings such as preference analysis and political elections. Modeling such distributions presents several computational challenges, however, due to the factorial size of the set of rankings over an item set. Some of these challenges are quite familiar to the artificial intelligence community, such as how to compactly represent a distribution over a combinatorially large space, and how to efficiently perform probabilistic inference with these representations. With respect to ranking, however, there is the additional challenge of what we refer to as human task complexity -- users are rarely willing to provide a full ranking over a long list of candidates, instead often preferring to provide partial ranking information. Simultaneously addressing all of these challenges -- i.e., designing a compactly representable model which is amenable to efficient inference and can be learned using partial ranking data -- is a difficult task, but is necessary if we would like to scale to problems with nontrivial size. In this paper, we show that the recently proposed riffled independence assumptions cleanly and efficiently address each of the above challenges. In particular, we establish a tight mathematical connection between the concepts of riffled independence and of partial rankings. This correspondence not only allows us to then develop efficient and exact algorithms for performing inference tasks using riffled independence based representations with partial rankings, but somewhat surprisingly, also shows that efficient inference is not possible for riffle independent models (in a certain sense) with observations which do not take the form of partial rankings. Finally, using our inference algorithm, we introduce the first method for learning riffled independence based models from partially ranked data.

The minimisation problem of a sum of unary and pairwise functions of discrete variables is a general NP-hard problem with wide applications such as computing MAP configurations in Markov Random Fields (MRF), minimising Gibbs energy, or solving binary Valued Constraint Satisfaction Problems (VCSPs). We study the computational complexity of classes of discrete optimisation problems given by allowing only certain types of costs in every triangle of variable-value assignments to three distinct variables. We show that for several computational problems, the only non- trivial tractable classes are the well known maximum matching problem and the recently discovered joint-winner property. Our results, apart from giving complete classifications in the studied cases, provide guidance in the search for hybrid tractable classes; that is, classes of problems that are not captured by restrictions on the functions (such as submodularity) or the structure of the problem graph (such as bounded treewidth). Furthermore, we introduce a class of problems with convex cardinality functions on cross-free sets of assignments. We prove that while imposing only one of the two conditions renders the problem NP-hard, the conjunction of the two gives rise to a novel tractable class satisfying the cross-free convexity property, which generalises the joint-winner property to problems of unbounded arity.

In a minimal binary constraint network, every tuple of a constraint relation can be extended to a solution. The tractability or intractability of computing a solution to such a minimal network was a long standing open question. Dechter conjectured this computation problem to be NP-hard. We prove this conjecture. We also prove a conjecture by Dechter and Pearl stating that for k\geq2 it is NP-hard to decide whether a single constraint can be decomposed into an equivalent k-ary constraint network. We show that this holds even in case of bi-valued constraints where k\geq3, which proves another conjecture of Dechter and Pearl. Finally, we establish the tractability frontier for this problem with respect to the domain cardinality and the parameter k.

The rules of Sudoku are often specified using twenty seven \texttt{all\_different} constraints, referred to as the {\em big} \mrules. Using graphical proofs and exploratory logic programming, the following main and new result is obtained: many subsets of six of these big \mrules are redundant (i.e., they are entailed by the remaining twenty one \mrules), and six is maximal (i.e., removing more than six \mrules is not possible while maintaining equivalence). The corresponding result for binary inequality constraints, referred to as the {\em small} \mrules, is stated as a conjecture.

We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justification for their use. Furthermore, the equivalence allows the application of vector-valued regression methods and results to the problem of learning conditional distributions. Using this link we derive a sparse version of the embedding by considering alternative formulations. Further, by applying convergence results for vector-valued regression to the embedding problem we derive minimax convergence rates which are O(\log(n)/n) -- compared to current state of the art rates of O(n^{-1/4}) -- and are valid under milder and more intuitive assumptions. These minimax upper rates coincide with lower rates up to a logarithmic factor, showing that the embedding method achieves nearly optimal rates. We study our sparse embedding algorithm in a reinforcement learning task where the algorithm shows significant improvement in sparsity over an incomplete Cholesky decomposition.

We design iterative receiver schemes for a generic wireless communication system by treating channel estimation and information decoding as an inference problem in graphical models. We introduce a recently proposed inference framework that combines belief propagation (BP) and the mean field (MF) approximation and includes these algorithms as special cases. We also show that the expectation propagation and expectation maximization algorithms can be embedded in the BP-MF framework with slight modifications. By applying the considered inference algorithms to our probabilistic model, we derive four different message-passing receiver schemes. Our numerical evaluation demonstrates that the receiver based on the BP-MF framework and its variant based on BP-EM yield the best compromise between performance, computational complexity and numerical stability among all candidate algorithms.

This paper presents an unsupervised algorithm for nonlinear unmixing of hyperspectral images. The proposed model assumes that the pixel reflectances result from a nonlinear function of the abundance vectors associated with the pure spectral components. We assume that the spectral signatures of the pure components and the nonlinear function are unknown. The first step of the proposed method consists of the Bayesian estimation of the abundance vectors for all the image pixels and the nonlinear function relating the abundance vectors to the observations. The endmembers are subsequently estimated using Gaussian process regression. The performance of the unmixing strategy is evaluated with simulations conducted on synthetic and real data.

The exploration/exploitation (E/E) dilemma arises naturally in many subfields of Science. Multi-armed bandit problems formalize this dilemma in its canonical form. Most current research in this field focuses on generic solutions that can be applied to a wide range of problems. However, in practice, it is often the case that a form of prior information is available about the specific class of target problems. Prior knowledge is rarely used in current solutions due to the lack of a systematic approach to incorporate it into the E/E strategy. To address a specific class of E/E problems, we propose to proceed in three steps: (i) model prior knowledge in the form of a probability distribution over the target class of E/E problems; (ii) choose a large hypothesis space of candidate E/E strategies; and (iii), solve an optimization problem to find a candidate E/E strategy of maximal average performance over a sample of problems drawn from the prior distribution. We illustrate this meta-learning approach with two different hypothesis spaces: one where E/E strategies are numerically parameterized and another where E/E strategies are represented as small symbolic formulas. We propose appropriate optimization algorithms for both cases. Our experiments, with two-armed Bernoulli bandit problems and various playing budgets, show that the meta-learnt E/E strategies outperform generic strategies of the literature (UCB1, UCB1-Tuned, UCB-v, KL-UCB and epsilon greedy); they also evaluate the robustness of the learnt E/E strategies, by tests carried out on arms whose rewards follow a truncated Gaussian distribution.

In this paper we present DrillEdge — a commercial and award winning software system that monitors oil-well drilling operations in order to reduce non-productive time (NPT). DrillEdge utilizes case-based reasoning with temporal representations on streaming real-time data, pattern matching and agent systems to predict problems and give advice on how to mitigate the problems. The methods utilized, the architecture, the GUI and development cost in addition to two case studies are documented.