The paper describes the refinement algorithm for the Calculus of (Co)Inductive Constructions (CIC) implemented in the interactive theorem prover Matita. The refinement algorithm is in charge of giving a meaning to the terms, types and proof terms directly written by the user or generated by using tactics, decision procedures or general automation. The terms are written in an "external syntax" meant to be user friendly that allows omission of information, untyped binders and a certain liberal use of user defined sub-typing. The refiner modifies the terms to obtain related well typed terms in the internal syntax understood by the kernel of the ITP. In particular, it acts as a type inference algorithm when all the binders are untyped. The proposed algorithm is bi-directional: given a term in external syntax and a type expected for the term, it propagates as much typing information as possible towards the leaves of the term. Traditional mono-directional algorithms, instead, proceed in a bottom-up way by inferring the type of a sub-term and comparing (unifying) it with the type expected by its context only at the end. We propose some novel bi-directional rules for CIC that are particularly effective. Among the benefits of bi-directionality we have better error message reporting and better inference of dependent types. Moreover, thanks to bi-directionality, the coercion system for sub-typing is more effective and type inference generates simpler unification problems that are more likely to be solved by the inherently incomplete higher order unification algorithms implemented. Finally we introduce in the external syntax the notion of vector of placeholders that enables to omit at once an arbitrary number of arguments. Vectors of placeholders allow a trivial implementation of implicit arguments and greatly simplify the implementation of primitive and simple tactics.

The use of Reinforcement Learning in real-world scenarios is strongly limited by issues of scale. Most RL learning algorithms are unable to deal with problems composed of hundreds or sometimes even dozens of possible actions, and therefore cannot be applied to many real-world problems. We consider the RL problem in the supervised classification framework where the optimal policy is obtained through a multiclass classifier, the set of classes being the set of actions of the problem. We introduce error-correcting output codes (ECOCs) in this setting and propose two new methods for reducing complexity when using rollouts-based approaches. The first method consists in using an ECOC-based classifier as the multiclass classifier, reducing the learning complexity from O(A2) to O(Alog(A)). We then propose a novel method that profits from the ECOC's coding dictionary to split the initial MDP into O(log(A)) seperate two-action MDPs. This second method reduces learning complexity even further, from O(A2) to O(log(A)), thus rendering problems with large action sets tractable. We finish by experimentally demonstrating the advantages of our approach on a set of benchmark problems, both in speed and performance.

The original contributions of this paper are twofold: a new understanding of the influence of noise on the eigenvectors of the graph Laplacian of a set of image patches, and an algorithm to estimate a denoised set of patches from a noisy image. The algorithm relies on the following two observations: (1) the low-index eigenvectors of the diffusion, or graph Laplacian, operators are very robust to random perturbations of the weights and random changes in the connections of the patch-graph; and (2) patches extracted from smooth regions of the image are organized along smooth low-dimensional structures in the patch-set, and therefore can be reconstructed with few eigenvectors. Experiments demonstrate that our denoising algorithm outperforms the denoising gold-standards.

Hidden Markov models (HMMs) are a fundamental tool for data analysis and exploration. Many variants of the basic HMM have been developed in response to shortcomings in the original HMM formulation [9]. In this paper we address inference in the explicit state duration HMM (EDHMM). By state duration we mean the amount of time an HMM dwells in a state. In the standard HMM specification, a state's duration is implicit and, a priori, distributed geometrically. The EDHMM (or, equivalently, the hidden semi-Markov model [12]) was developed to allow explicit parameterization and direct inference of state duration distributions. EDHMM estimation and inference can be performed using the forward-backward algorithm; though only if the sequence is short or a tight "allowable" duration interval for each state is hard-coded a priori [13]. If the sequence is short then forward-backward can be run on a state representation that allows for all possible durations up to the observed sequence length. If the sequence is long then forward-backward only remains computationally tractable if only transitions between durations that lie within pre-specified allowable intervals are considered.

Many combinatorial problems deal with preferences and violations, the goal of which is to find solutions with the minimum cost. Weighted constraint satisfaction is a framework for modeling such problems, which consists of a set of cost functions to measure the degree of violation or preferences of different combinations of variable assignments. Typical solution methods for weighted constraint satisfaction problems (WCSPs) are based on branch-and-bound search, which are made practical through the use of powerful consistency techniques such as AC*, FDAC*, EDAC* to deduce hidden cost information and value pruning during search. These techniques, however, are designed to be efficient only on binary and ternary cost functions which are represented in table form. In tackling many real-life problems, high arity (or global) cost functions are required. We investigate efficient representation scheme and algorithms to bring the benefits of the consistency techniques to also high arity cost functions, which are often derived from hard global constraints from classical constraint satisfaction. The literature suggests some global cost functions can be represented as flow networks, and the minimum cost flow algorithm can be used to compute the minimum costs of such networks in polynomial time. We show that naive adoption of this flow-based algorithmic method for global cost functions can result in a stronger form of null-inverse consistency. We further show how the method can be modified to handle cost projections and extensions to maintain generalized versions of AC* and FDAC* for cost functions with more than two variables. Similar generalization for the stronger EDAC* is less straightforward. We reveal the oscillation problem when enforcing EDAC* on cost functions sharing more than one variable. To avoid oscillation, we propose a weak version of EDAC* and generalize it to weak EDGAC* for non-binary cost functions. Using various benchmarks involving the soft variants of hard global constraints ALLDIFFERENT, GCC, SAME, and REGULAR, empirical results demonstrate that our proposal gives improvements of up to an order of magnitude when compared with the traditional constraint optimization approach, both in terms of time and pruning.

In this paper, we address the problem of global transmit power minimization in a self-congiguring network where radio devices are subject to operate at a minimum signal to interference plus noise ratio (SINR) level. We model the network as a parallel Gaussian interference channel and we introduce a fully decentralized algorithm (based on trial and error) able to statistically achieve a congiguration where the performance demands are met. Contrary to existing solutions, our algorithm requires only local information and can learn stable and efficient working points by using only one bit feedback. We model the network under two different game theoretical frameworks: normal form and satisfaction form. We show that the converging points correspond to equilibrium points, namely Nash and satisfaction equilibrium. Similarly, we provide sufficient conditions for the algorithm to converge in both formulations. Moreover, we provide analytical results to estimate the algorithm's performance, as a function of the network parameters. Finally, numerical results are provided to validate our theoretical conclusions. Keywords: Learning, power control, trial and error, Nash equilibrium, spectrum sharing.

The first decade of this century has seen the nascency of the first mathematical theory of general artificial intelligence. This theory of Universal Artificial Intelligence (UAI) has made significant contributions to many theoretical, philosophical, and practical AI questions. In a series of papers culminating in book (Hutter, 2005), an exciting sound and complete mathematical model for a super intelligent agent (AIXI) has been developed and rigorously analyzed. While nowadays most AI researchers avoid discussing intelligence, the award-winning PhD thesis (Legg, 2008) provided the philosophical embedding and investigated the UAI-based universal measure of rational intelligence, which is formal, objective and non-anthropocentric. Recently, effective approximations of AIXI have been derived and experimentally investigated in JAIR paper (Veness et al. 2011). This practical breakthrough has resulted in some impressive applications, finally muting earlier critique that UAI is only a theory. For the first time, without providing any domain knowledge, the same agent is able to self-adapt to a diverse range of interactive environments. For instance, AIXI is able to learn from scratch to play TicTacToe, Pacman, Kuhn Poker, and other games by trial and error, without even providing the rules of the games. These achievements give new hope that the grand goal of Artificial General Intelligence is not elusive. This article provides an informal overview of UAI in context. It attempts to gently introduce a very theoretical, formal, and mathematical subject, and discusses philosophical and technical ingredients, traits of intelligence, some social questions, and the past and future of UAI.

We focus on the problem of sequential decision making in partially observable environments shared with other agents of uncertain types having similar or conflicting objectives. This problem has been previously formalized by multiple frameworks one of which is the interactive dynamic influence diagram (I-DID), which generalizes the well-known influence diagram to the multiagent setting. I-DIDs are graphical models and may be used to compute the policy of an agent given its belief over the physical state and others' models, which changes as the agent acts and observes in the multiagent setting. As we may expect, solving I-DIDs is computationally hard. This is predominantly due to the large space of candidate models ascribed to the other agents and its exponential growth over time. We present two methods for reducing the size of the model space and stemming its exponential growth. Both these methods involve aggregating individual models into equivalence classes. Our first method groups together behaviorally equivalent models and selects only those models for updating which will result in predictive behaviors that are distinct from others in the updated model space. The second method further compacts the model space by focusing on portions of the behavioral predictions. Specifically, we cluster actionally equivalent models that prescribe identical actions at a single time step. Exactly identifying the equivalences would require us to solve all models in the initial set. We avoid this by selectively solving some of the models, thereby introducing an approximation. We discuss the error introduced by the approximation, and empirically demonstrate the improved efficiency in solving I-DIDs due to the equivalences.

We consider the problem of learning classifiers for labeled data that has been distributed across several nodes. Our goal is to find a single classifier, with small approximation error, across all datasets while minimizing the communication between nodes. This setting models real-world communication bottlenecks in the processing of massive distributed datasets. We present several very general sampling-based solutions as well as some two-way protocols which have a provable exponential speed-up over any one-way protocol. We focus on core problems for noiseless data distributed across two or more nodes. The techniques we introduce are reminiscent of active learning, but rather than actively probing labels, nodes actively communicate with each other, each node simultaneously learning the important data from another node.

We introduce a novel and efficient sampling algorithm for the Multiplicative Attribute Graph Model (MAGM - Kim and Leskovec (2010)}). Our algorithm is \emph{strictly} more efficient than the algorithm proposed by Yun and Vishwanathan (2012), in the sense that our method extends the \emph{best} time complexity guarantee of their algorithm to a larger fraction of parameter space. Both in theory and in empirical evaluation on sparse graphs, our new algorithm outperforms the previous one. To design our algorithm, we first define a stochastic \emph{ball-dropping process} (BDP). Although a special case of this process was introduced as an efficient approximate sampling algorithm for the Kronecker Product Graph Model (KPGM - Leskovec et al. (2010)}), neither \emph{why} such an approximation works nor \emph{what} is the actual distribution this process is sampling from has been addressed so far to the best of our knowledge. Our rigorous treatment of the BDP enables us to clarify the rational behind a BDP approximation of KPGM, and design an efficient sampling algorithm for the MAGM.