This report presents a unified approach for the study of constrained Markov decision processes with a countable state space and unbounded costs. We consider a single controller having several objectives; it is desirable to design a controller that minimize one of cost objective, subject to inequality constraints on other cost objectives. The objectives that we study are both the expected average cost, as well as the expected total cost (of which the discounted cost is a special case). We provide two frameworks: the case were costs are bounded below, as well as the contracting framework. We characterize the set of achievable expected occupation measures as well as performance vectors. This allows us to reduce the original control dynamic problem into an infinite Linear Programming. We present a Lagrangian approach that enables us to obtain sensitivity analysis. In particular, we obtain asymptotical results for the constrained control problem: convergence of both the value and the pol...

The main result asserts the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

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. Let f be an unpredictable random function taking (b + c)-bit inputs to b-bit outputs. This paper presents an unpredictable random function f 0 taking variable-length inputs to b-bit outputs. This construction has several advantages over chaining, which was proven unpredictable by Bellare, Kilian, and Rogaway, and cascading, which was proven unpredictable by Bellare, Canetti, and Krawczyk. The highlight here is a very simple proof of security. 1.

This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when M is a homotopy 3-sphere, the width is loosely speaking the

Abstract. Learning probabilities (p-concepts [13]) and other real-valued concepts (regression) is an important role of machine learning. For example, a doctor may need to predict the probability of getting a disease P [y|x], which depends on a number of risk factors. Generalized additive models [9] are a well-studied nonparametric model in the statistics literature, usually with monotonic link functions. However, no known efficient algorithms exist for learning such a general class. We show that regression graphs efficiently learn such real-valued concepts, while regression trees inefficiently learn them. One corollary is that any function E[y|x] = u(w · x) for u monotonic can be learned to arbitrarily small squared error ɛ in time polynomial in 1/ɛ, |w|1, and the Lipschitz constant of u (analogous to a margin). The model includes, as special cases, linear and logistic regression, as well as learning a noisy half-space with a margin [5, 4]. Kearns, Mansour, and McAllester [12, 15], analyzed decision trees and decision graphs as boosting algorithms for classification accuracy. We extend their analysis and the boosting analogy to the case of real-valued predictors, where a small positive correlation coefficient can be boosted to arbitrary accuracy. Viewed as a noisy boosting algorithm [3, 10], the algorithm learns both the target function and the asymmetric noise. 1

. We consider the doubling map T : z 7! z 2 of the circle. For each T -invariant probability measure ¯ we define its barycentre b(¯) = R S 1 zd¯(z), which describes its average weight around the circle. We study the set\Omega of all such barycentres, a compact convex set with non-empty interior. We obtain a numerical approximation of the boundary @\Omega . This appears to have a countable dense set of points of non-differentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency-locking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit. 1991 Mathematics Subject Classification: 58F11, 58F15, 58F03 Section 1. Introduction. A recurring theme in the study of non-linear dynamics is the occurrence of non-smooth phenomena (irregular conjugacies, Cantor-like attractors, intricate Julia sets, fractal descriptions of bifurcations, etc), even when the syste...

Generalized additive models are a powerful generalization of linear and logistic regression models. In this paper we show that a natural regression graph learning algorithm efficiently learns generalized additive models. Efficiency is proven in two senses: the estimator’s future prediction accuracy approaches optimality at rate inverse polynomial in the size of the training data, and its runtime is polynomial in the size of the training data. Furthermore, the guarantees are nearly linear in terms of the dimensionality (number of regressors) of the problem, and hence the algorithm does not suffer from the “curse of dimensionality. ” The algorithm is a simple generalization of Mansour and McAllester’s classification algorithm that generates decision graphs, i.e., decision trees with merges. Our analysis is also viewed as defining a natural extension of the original classification boosting theorems (Schapire, 1990) to the regression setting. Loosely speaking, we define a weak correlator to be a real-valued predictor that has a correlation coefficient with the target function that is bounded from zero. We show how to efficiently boost weak correlators to get predictions with correlation arbitrarily close to 1 (error arbitrarily close to 0). Our boosting analysis is a natural extension of the classification boosting analysis of Kearns and Mansour (1999) and Mansour and McAllester (2002).

Abstract. A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kind � b x(t)+ k(t, s)x(s)ds = y(t), a whose kernel k(t, s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t, s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value decomposition is indicated. Application to integro-differential Schrödinger equations with nonlocal potentials is given. 1.

Abstract. The main result of this paper is to show that a “winding number condition ” which relates to two strictly multivariable linear operators, and which is used in the definition of the Vinnicombe (ν-gap) metric, is equivalent to the existence of a multivariable homotopy in the same metric between the two operators. This allows a characterisation of the Vinnicombe metric which is independent of the linearity of the underlying operators, and suggests possible extension of the metric to nonlinear operators.