Variational studies and replica symmetry breaking in the generalization problem of the binary perceptron

We analyze the average performance of a general class of learning algorithms for the nondeterministic polynomial time complete problem of rule extraction by a binary perceptron. The examples are generated by a rule implemented by a teacher network of similar architecture. A variational approach is used in trying to identify the potential energy that leads to the largest generalization in the thermodynamic limit. We restrict our search to algorithms that always satisfy the binary constraints. A replica symmetric ansatz leads to a learning algorithm which presents a phase transition in violation of an information theoretical bound. Stability analysis shows that this is due to a failure of the replica symmetric ansatz and the first step of replica symmetry breaking (RSB) is studied. The variational method does not determine a unique potential but it allows construction of a class with a unique minimum within each first order valley. Members of this class improve on the performance of Gibbs algorithm but fail to reach the Bayesian limit in the low generalization phase. They even fail to reach the performance of the best binary, an optimal clipping of the barycenter of version space. We find a trade-off between a good low performance and early onset of perfect generalization. Although the RSB may be locally stable we discuss the possibility that it fails to be the correct saddle point globally. ©2000 The American Physical Society.

The Development of the Generalization Algorithm Based on the Rough Set Theory

This paper considers the problem of concept generalization in decision-making systems where such features of real-world databases as large size, incompleteness and inconsistence of the stored information are taken into account. The methods of the rough set theory (like lower and upper approximations, positive regions and reducts) are used for the solving of this problem. The new discretization algorithm of the continuous attributes is proposed. It essentially increases an overall performance of generalization algorithms and can be applied to processing of real value attributes in large data tables. Also the search algorithm of the significant attributes combined with a stage of discretization is developed. It allows avoiding splitting of continuous domains of insignificant attributes into intervals.

Semi-online neural-Q_leaming for real-time robot learning

Reinforcement learning (RL) is a very suitable technique for robot learning, as it can learn in unknown environments and in real-time computation. The main difficulties in adapting classic RL algorithms to robotic systems are the generalization problem and the correct observation of the Markovian state. This paper attempts to solve the generalization problem by proposing the semi-online neural-Q_learning algorithm (SONQL). The algorithm uses the classic Q_learning technique with two modifications. First, a neural network (NN) approximates the Q_function allowing the use of continuous states and actions. Second, a database of the most representative learning samples accelerates and stabilizes the convergence. The term semi-online is referred to the fact that the algorithm uses the current but also past learning samples. However, the algorithm is able to learn in real-time while the robot is interacting with the environment. The paper shows simulated results with the "mountain-car" benchmark and, also, real results with an underwater robot in a target following behavior

Investigações sobre o papel da generalização funcional em uma situação de resolução súbita de problemas (insight) em Rattus norvegicus

A psicologia da aprendizagem foi marcada pelo debate sobre se a aprendizagem seria um processo gradual ou súbito. Enquanto os associacionistas defendiam a primeira proposta, os gestaltistas afirmavam a existência de situações de aprendizagem abruptas. Dentre os principais autores a defenderem esta possibilidade estava Wolfgang Köhler. Os trabalhos deste autor têm sido apontados como evidência de que a aprendizagem seria um processo súbito. Apesar da relevância destes trabalhos em demonstrar a existência de situações em que uma forma súbita de aprendizagem ocorra, muito se tem questionado sobre as conclusões apresentadas por ele para o porquê da ocorrência deste tipo de fenômeno comportamental. Dentre as muitas críticas feitas, a que tem recebido mais atenção refere-se à ausência de controle da história dos seus sujeitos experimentais, bem como à desconsideração do papel que esta história teria nos resultados encontrados. Estudos que investigaram este papel (Epstein et al., 1984 e Epstein & Medalie 1983, 1985) demonstraram que a resposta típica de insight podia ser o resultado da junção de repertórios aprendidos previamente. Os trabalhos de Epstein foram importantes em demonstrar que o insight seria uma junção de repertórios que se combinam em situações apropriadas pelo possível envolvimento de um processo conhecido como Generalização Funcional. O presente trabalho se propôs a investigar se a Generalização Funcional era realmente responsável pela interconexão dos repertórios que culminariam na resolução da tarefa de um modo considerado como súbito. Para tal foi feita uma replicação dos experimentos de Epstein, utilizando-se ratos como sujeitos. Os resultados mostraram que a Generalização Funcional parece ser um requisito necessário, mas não suficiente para a resolução súbita de problemas, de um modo considerado como insight".

Learning generalization in problem solving by a blue-fronted parrot (Amazona aestiva)

Pepperberg (The Alex studies: cognitive and communicative abilities of gray parrots. Harvard University Press, Cambridge;1999) showed that some of the complex cognitive capabilities found in primates are also present in psittacine birds. Through the replication of an experiment performed with cotton-top tamarins (Saguinus oedipus oedipus) by Hauser et al. (Anim Behav 57:565-582; 1999), we examined a blue-fronted parrot`s (Amazona aestiva) ability to generalize the solution of a particular problem in new but similar cases. Our results show that, at least when it comes to solving this particular problem, our parrot subject exhibited learning generalization capabilities resembling the tamarins`.

On a generalization of the Oberwolfach problem

Let e(1),e(2),... e(n) be a sequence of nonnegative integers Such that the first non-zero term is not one. Let Sigma(i=1)(n) e(i) = (q - 1)/2, where q = p(n) and p is an odd prime. We prove that the complete graph on q vertices can be decomposed into e(1) C-pn-factors, e(2) C-pn (1)-factors,..., and e(n) C-p-factors. (C) 2004 Elsevier Inc. All rights reserved.

A more efficient and effective heuristic algorithm for the MapReduce placement problem in cloud computing

The placement of the mappers and reducers on the machines directly affects the performance and cost of the MapReduce computation in cloud computing. From the computational point of view, the mappers/reducers placement problem is a generalization of the classical bin packing problem, which is NP-complete. Thus, in this paper we propose a new heuristic algorithm for the mappers/reducers placement problem in cloud computing and evaluate it by comparing with other several heuristics on solution quality and computation time by solving a set of test problems with various characteristics. The computational results show that our heuristic algorithm is much more efficient than the other heuristics. Also, we verify the effectiveness of our heuristic algorithm by comparing the mapper/reducer placement for a benchmark problem generated by our heuristic algorithm with a conventional mapper/reducer placement. The comparison results show that the computation using our mapper/reducer placement is much cheaper while still satisfying the computation deadline.

Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

Analysis,Insights and Generalization of a Fast Decentralized Relay Selection Mechanism

Relay selection for cooperative communications has attracted considerable research interest recently. While several criteria have been proposed for selecting one or more relays and analyzed, mechanisms that perform the selection in a distributed manner have received relatively less attention. In this paper, we analyze a splitting algorithm for selecting the single best relay amongst a known number of active nodes in a cooperative network. We develop new and exact asymptotic analysis for computing the average number of slots required to resolve the best relay. We then propose and analyze a new algorithm that addresses the general problem of selecting the best Q >= 1 relays. Regardless of the number of relays, the algorithm selects the best two relays within 4.406 slots and the best three within 6.491 slots, on average. Our analysis also brings out an intimate relationship between multiple access selection and multiple access control algorithms.

Solution of a Boundary-Value Problem associated with Diffraction of Water Waves by a Nearly Vertical Barrier

An exact solution is derived for a boundary-value problem for Laplace's equation which is a generalization of the one occurring in the course of solution of the problem of diffraction of surface water waves by a nearly vertical submerged barrier. The method of solution involves the use of complex function theory, the Schwarz reflection principle, and reduction to a system of two uncoupled Riemann-Hilbert problems. Known results, representing the reflection and transmission coefficients of the water wave problem involving a nearly vertical barrier, are derived in terms of the shape function.

Thermomechanical generalization of pin joints in finite plates

The understanding of thermoelastic behaviour of joints is significant in order to ensure the integrity of large and complex structures exposed to a thermal environment, particularly in fields such as aerospace and nuclear engineering. Thermomechanical generalization of partial contact behaviour of a pin joint under combined in-plane mechanical loading and on-axis unidirectional heat flow has already been established by the authors for the analytically simpler domains of large plates. This paper successfully extends the on-going investigation to a single pin in a finite rectangular isotropic plate as a two-dimensional abstraction from a practical situation of a multipin fastener joint. The finite element method is used to analyse the joint problem under on-axis thermomechanical loading and unified load-contact relationships are established for a class of loading conditions.

Generalization of the collision cone approach for motion safety in 3-D environments

Avoidance of collision between moving objects in a 3-D environment is fundamental to the problem of planning safe trajectories in dynamic environments. This problem appears in several diverse fields including robotics, air vehicles, underwater vehicles and computer animation. Most of the existing literature on collision prediction assumes objects to be modelled as spheres. While the conservative spherical bounding box is valid in many cases, in many other cases, where objects operate in close proximity, a less conservative approach, that allows objects to be modelled using analytic surfaces that closely mimic the shape of the object, is more desirable. In this paper, a collision cone approach (previously developed only for objects moving on a plane) is used to determine collision between objects, moving in 3-D space, whose shapes can be modelled by general quadric surfaces. Exact collision conditions for such quadric surfaces are obtained and used to derive dynamic inversion based avoidance strategies.

Higher-Dimensional Analogues of the Map Coloring Problem

After a brief discussion of the history of the problem, we propose a generalization of the map coloring problem to higher dimensions.

Time-optimal control of a 3-level quantum system and its generalization to an n-level system

We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimal control problem for a general n-level quantum system. For the three-level system we find all optimal controls by finding two types of symmetry in the problem: ℤ2 × S3 discrete symmetry and S1 continuous symmetry, and exploiting them to solve the problem through discrete reduction and symplectic reduction. We then study the geometry, in the same framework, which occurs in the time-optimal control of a general n-level quantum system. © 2007 IEEE.

Time-optimal control of a 3-level quantum system and its generalization

We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimal control problem for a general n-level quantum system. For the three-level system we find all optimal controls by finding two types of symmetry in the problems: ℤ × S3 discrete symmetry and 51 continuous symmetry, and exploiting them to solve the problem through discrete reduction and symplectic reduction. We then study the geometry, in the same framework, which occurs in the time-optimal control of a general n-level quantum system. Copyright ©2007 Watam Press.