generalization error bounds for threshold decision lists in this paper we consider the generalization accuracy of classification methods based on the iterative use of linear classifiers the resulting classifiers which we call threshold decision lists act as follows some points of the data set to be classified are given a particular classification according to a linear threshold function or hyperplane these are then removed from consideration and the procedure is iterated until all points are classified geometrically we can imagine that at each stage points of the same classification are successively chopped off from the data set by a hyperplane we analyse theoretically the generalization properties of data classification techniques that are based on the use of threshold decision lists and on the special subclass of multilevel threshold functions we present bounds on the generalization error in a standard probabilistic learning framework the primary focus in this paper is on obtaining generalization error bounds that depend on the levels of separation or margins achieved by the successive linear classifiers we also improve and extend previously published theoretical bounds on the generalization ability of perceptron decision trees

implementation of dynamic programming for chaos control in discrete systems in this paper the control of discrete chaotic systems by designing linear feedback controllers is presented the linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming for suppressing chaos with minimum control effort the system is stabilized on its first order unstable fixed point ufp the presented method also could be employed to make any desired nth order fixed point of the system stable two different methods for higher order ufps stabilization are suggested afterwards these methods are applied to two well known chaotic discrete systems the logistic and the henon maps for each of them the first and second ufps in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance

scale sensitive dimensions uniform convergence and learnability learnability in valiant s pac learning model has been shown to be strongly related to the existence of uniform laws of large numbers these laws define a distribution free convergence property of means to expectations uniformly over classes of random variables classes of real valued functions enjoying such a property are also known as uniform glivenko cantelli classes in this paper we prove through a generalization of sauer s lemma that may be interesting in its own right a new characterization of uniform glivenko cantelli classes our characterization yields dudley gine and zinn s previous characterization as a corollary furthermore it is the first based on a gine and zinn s previous characterization as a corollary furthermore it is the first based on a simple combinatorial quantity generalizing the vapnik chervonenkis dimension we apply this result to obtain the weakest combinatorial condition known to imply pac learnability in the statistical regression or agnostic framework furthermore we find a characterization of learnability in the probabilistic concept model solving an open problem posed by kearns and schapire these results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner s hypothesis class

analysis of a numerical dynamic programming algorithm applied to economic models in this paper we develop a discretized version of the dynamic programming algorithm and study its convergence and stability properties we show that the computed value function converges quadratically to the true value function and that the computed policy function converges linearly as the mesh size of the discretization converges to zero further the algorithm is stable we also discuss several aspects of the implementation of our procedures as applied to some commonly studied growth models

stable cycles in a cournot duopoly model of kopel we consider a discrete map proposed by m kopel that models a nonlinear cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition the stability of the fixed points of the discrete dynamical system is analyzed synchronization of two dynamics parameters of the cournot duopoly is considered in the computation of stability boundaries formed by parts of codim 1 bifurcation curves we discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim 1 and codim 2 bifurcation points and computing the associated two parameter codim 1 curves rooted in some codim 2 points it enables us to compute the stability domains of the low order iterates of the map we concentrate in particular on the second third and fourth iterates and their relation to the period doubling 1 3 and 1 4 resonant neimark sacker points

numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues in this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues these include all strong resonances as well as the degenerate flip and neimark sacker bifurcations applying our results to n dimensional maps one avoids the computation of the center manifold but one can directly evaluate the critical normal form coefficients in the original basis the formulas remain valid also for n 2 and allow one to avoid the transformation of the linear part of the map into jordan form the developed techniques are tested on two examples 1 a three dimensional map appearing in adaptive control 2 a periodically forced epidemic model we compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models to compute the required derivatives of the poincare map for the epidemic model the automatic differentiation package adol c is used

convergence properties of the likelihood of computed dynamic models this paper studies the econometrics of computed dynamic models since these models generally lack a closed form solution their policy functions are approximated by numerical methods hence the researcher can only evaluate an approximated likelihood associated with the approximated policy function rather than the exact likelihood implied by the exact policy function what are the consequences for inference of the use of approximated likelihoods first we find conditions under which as the approximated policy function converges to the exact policy the approximated likelihood also converges to the exact likelihood second we show that second order approximation errors in the policy function which almost always are ignored by researchers have first order effects on the likelihood function third we discuss convergence of bayesian and classical estimates finally we propose to use a likelihood ratio test as a diagnostic device for problems derived from the use of approximated likelihoods

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

an optimal one way multigrid algorithm for discrete time stochastic control the numerical solution of discrete time stationary infinite horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately within a desired accuracy after a discussion of problem discretization the authors introduce a multigrid version of the successive approximation algorithm that proceeds one way from coarse to fine grids and analyze its computational requirements as a function of the desired accuracy and of the discount factor they also study the effects of a certain mixing ergodicity condition on the algorithm s performance it is shown that the one way multigrid algorithm improves upon the complexity of its single grid variant and is in a certain sense optimal

input to state stabilization of linear systems with positive outputs this paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured it is shown that if a system is controllable and observable then one can find a stabilizing controller which is robust with respect to observation noise in the iss sense

new results on frame proof codes and traceability schemes we derive lower bounds on the maximum number of codewords in a class of frame proof codes and traceability schemes and give constructions for both with more codewords than the best known

a learning result for continuous time recurrent neural networks abstract the following learning problem is considered for continuous time recurrent neural networks having sigmoidal activation functions given a black box representing an unknown system measurements of output derivatives are collected for a set of randomly generated inputs and a network is used to approximate the observed behavior it is shown that the number of inputs needed for reliable generalization the sample complexity of the learning problem is upper bounded by an expression that grows polynomially with the dimension of the network and logarithmically with the number of output derivatives being matched

bounds on the number of examples needed for learning functions we prove general lower bounds on the number of examples needed for learning function classes within different natural learning models which are related to pac learning and coincide with the pac learning model of valiant in the case of 0 1 valued functions the lower bounds are obtained by showing that all nontrivial function classes contain a hard binary valued subproblem although at first glance it seems to be likely that real valued function classes are much harder to learn than their hardest binary valued subproblem we show that these general lower bounds cannot be improved by more than a logarithmic factor this is done by discussing some natural function classes like nondecreasing functions or piecewise smooth functions the function classes that were discussed in m j kearns and r e schapire proc 31st annual symposium on the foundations of computer science ieee computer society press los alamitos ca 1990 pp 382 392 full version j comput system sci 48 1994 pp 464 497 d kimber and p m long proc 5th annual workshop on computational learning theory acm new york 1992 pp 153 160 with certain restrictions concerning their slope

generalization performance of regularization networks and support vector machines via entropy numbers of compact operators we derive new bounds for the generalization error of kernel machines such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers the proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory the hypothesis class is described in terms of a linear operator mapping from a possibly infinite dimensional unit ball in feature space into a finite dimensional space the covering numbers of the class are then determined via the entropy numbers of the operator these numbers which characterize the degree of compactness of the operator can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine as a consequence we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines

on convergence proofs in system identification a general principle using ideas from learning theory it is shown that learning theory offers convergence analysis tools that are useful in system identification problems they allow analysis in a parameter free context which elevates the analysis from parameter sets to model sets and from parameter identification to model identification when a parameterization is eventually introduced this leads to alternative assumptions on the parameterization and parameter set moreover structural identification can be analyzed within the same framework another advantage is that the proofs are technically and conceptually simple

small size quantum automata recognizing some regular languages given a class p i of stochastic events induced by m state 1 way quantum finite automata 1qfa on alphabet we investigate the size number of states of 1qfa s that approximate a convex linear combination of p i and we apply the results to the synthesis of small size 1qfa s we obtain an o md 3 log2 d 2 general size bound where d is the vapnik dimension of p w w for commutative n periodic events p on with h we prove an o h log n 2 size bound for inducing a approximation of p whenever f p 1 nh where f p is the discrete fourier transform of the vector p associated with p if the characteristic function l of an n periodic unary language l satisfies f l 1 n then l is recognized with isolated cut point by a 1qfa with o log n states vice versa if l is recognized with isolated cut point by a 1qfa with o log n state then f l 1 o n log n

complete controllability of continuous time recurrent neural networks this paper studies controllability for the class of control systems commonly called continuous time recurrent neural networks it is shown that under a generic condition on the input matrix the system is controllable for every possible state matrix the result holds when the activation function is the hyperbolic tangent

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

a note on the equilibria of an economic model with local competition a la cournot guirao and rubio 2010 12 introduce an economic model which generalizes the classical duopoly of cournot type where the competitors are located around a circle or a line and each firm competes a la cournot with its right and left neighboring for the case of having three and four players we describe completely the bifurcations of equilibria in terms of the production costs of each firm and we study the stability of them moreover for the case of four players we provide some information on the two periodic orbits of the system

input classes for identifiability of bilinear systems this paper asks what classes of input signals are sufficient in order to completely identify the input output behavior of generic bilinear systems the main results are that step inputs are not sufficient nor are single pulses but the family of all pulses of a fixed amplitude but varying widths do suffice for identification

bifurcation intermittent chaos and multi stability in a two stage cournot game with r d spillover and product differentiation abstract in this paper a dynamical two stage cournot duopoly game with r d spillover effect and product differentiation is established the stability of all the equilibrium points is studied using jury criterion and then the stability condition is given the direction of flip bifurcation is given by using central manifold theorem and norm form theory by numerical simulation two routes to chaos are studied through 2 d bifurcation diagram one of which is flip bifurcation and the other is neimark sacker bifurcation at period 2 point the system will lose its stability as the speed of adjustment increases in addition a common nonlinear phenomenon named intermittent chaos is observed in the built model two types of intermittent chaos are displayed in time series plots and phase diagrams one of which is pm iii intermittent chaos and the other is induced by crisis and then four types of coexistence of attractors are illustrated through basin of attraction which are coexistence of periodic attractor and chaotic attractor coexistence of multiple chaotic attractors coexistence of periodic attractor quasi periodic attractor and chaotic attractor and coexistence of multiple periodic attractors respectively

stabilization of strictly dissipative discrete time systems with discounted optimal control abstract we consider stabilization of an equilibrium point via infinite horizon discounted optimal control in discrete time in addition to applications in economics and social sciences discounted optimal control is a commonly used numerical technique guaranteeing solvability of certain classes of optimal control problems in this paper we present conditions based on strict dissipativity that ensure that the optimally controlled system is asymptotically stable or practically asymptotically stable these conditions are shown to be complementary to recently proposed conditions based on a detectability property illustrative examples are provided

extracting state information form a quantized output record abstract we consider the problem of manipulating the input to a discrete time state space linear system with the goal of obtaining information at each time about the system s current state from a record of past quantized measurements of the system s output we find that if the system is not excessively unstable there exist feedback control strategies that allow one to make an asymtotically perfect determination of the current stage based on the output records that result even if the system is too unstable to apply such strategies there are feedback control laws that make the system s output record more informative about the system s state evolution than one might expect in deriving these control laws we regard quantized measurements of real numbers more as partial observations than as strict approximations and employ techniques from information theory and the theory of markov chains with countable state spaces

error estimation and adaptive discretization for the discrete stochastic hamilton jacobi bellman equation generalizing an idea from deterministic optimal control we construct a posteriori error estimates for the spatial discretization error of the stochastic dynamic programming method based on a discrete hamilton jacobi bellman equation these error estimates are shown to be efficient and reliable furthermore a priori bounds on the estimates depending on the regularity of the approximate solution are derived based on these error estimates we propose an adaptive space discretization scheme whose performance is illustrated by two numerical examples

entropy and the combinatorial dimension we solve talagrand s entropy problem the l2 covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension this extends dudley s theorem on classes of 0 1 valued functions for which the shattering dimension is the vapnik chervonenkis dimension in convex geometry the solution means that the entropy of a convex body k is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of k this has a number of consequences including the optimal elton s theorem and estimates on the uniform central limit theorem in the real valued case

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

learning complexity dimensions for a continuous time control system this paper takes a computational learning theory approach to a problem of linear systems identification it is assumed that inputs are generated randomly from a known class consisting of linear combinations of k sinusoidals the output of the system is classified at some single instant of time the main result establishes that the number of samples needed for identification with small error and high probability independently from the distribution of inputs scales polynomially with n the system dimension and logarithmically with k

observability of linear systems with saturated outputs we present necessary and sufficient conditions for observability of the class of output saturated systems these are linear systems whose output passes through a saturation function before it can be measured

output feedback stabilization of a class of wiener systems a globally stabilizing output feedback controller is designed for a class of continuous time wiener systems the wiener systems we consider consist of a linear dynamical block and an output polynomial nonlinearity connected in series the hybrid controller consists of three modes of operation which are periodically applied to the system the controller achieves a dead beat response of the closed loop system

a note on observability tests for general polynomial and simple wiener hammerstein systems we propose an algorithm based on symbolic computation packages for testing observability conditions of general polynomial systems which were formulated in sontag siam j control optim 17 1979 139 151 computational complexity of the observability test can be reduced and the test simplified for classes of polynomial systems we illustrate this by considering the class of simple wiener hammerstein systems which consist of a series of two linear dynamic blocks between which a static nonlinearity is sandwiched we consider the case when the nonlinearity is a monomial n q q 1 simple necessary and sufficient conditions for observability are given and they resemble but are subtly different from the well known result on observability for the series connection of two linear systems c 1998 elsevier science b v all rights reserved

on the dynamics of kopel s cournot duopoly model abstract in this paper we study the duopoly model proposed by m kopel 26 where two firms compete a la cournot and the reaction curves have a higher degree of coupling in the sense that firms have to make their choices simultaneously we will make a descriptive analysis of the two dimensional model making an approach through particular situations on the other hand when the firms are homogeneous a one dimensional invariant subset is present in the model we will give an analytical proof of the existence of a topological chaotic behavior for a wide range of parameter values and we will study when chaotic synchronization and collusion occur

combinatorial properties of frameproof and traceability codes in order to protect copyrighted material codes may be embedded in the content or codes may be associated with the keys used to recover the content codes can offer protection by providing some form of traceability ta for pirated data several researchers have studied different notions of ta and related concepts in previous years strong versions of ta allow at least one member of a coalition that constructs a pirate decoder to be traced weaker versions of this concept ensure that no coalition can frame a disjoint user or group of users all these concepts can be formulated as codes having certain combinatorial properties we study the relationships between the various notions and we discuss equivalent formulations using structures such as perfect hash families we use methods from combinatorics and coding theory to provide bounds necessary conditions and constructions sufficient conditions for the objects of interest

vc dimension of an integrate and fire neuron model we compute the vc dimension of a leaky integrate and fire neuron model the vc dimension quantifies the ability of a function class to partition an input pattern space and can be considered a measure of computational capacity in this case the function class is the class of integrate and fire models generated by varying the integration time constant t and the threshold the input space they partition is the space of continuous time signals and the binary partition is specified by whether or not the model reaches threshold at some specified time we show that the vc dimension diverges only logarithmically with the input signal bandwidth n we also extend this approach to arbitrary passive dendritic trees the main contributions of this work are 1 it offers a novel treatment of computational capacity of this class of dynamic system and 2 it provides a framework for analyzing the computational capabilities of the dynamic systems defined by networks of spiking neurons

linear systems with sign observations this paper deals with systems that are obtained from linear time invariant continuous or discrete time devices followed by a function that just provides the sign of each output such systems appear naturally in the study of quantized observations as well as in signal processing and neural network theory results are given on observability minimal realizations and other system theoretic concepts certain major differences exist with the linear case and other results generalize in a surprisingly straightforward manner

sample complexity for learning recurrent perceptron mappings recurrent perceptron classifiers generalize the usual perceptron model they correspond to linear transformations of input vectors obtained by means of autoregressive moving average schemes or infinite impulse response filters and take into account those correlations and dependences among input coordinates which arise from linear digital filtering this paper provides tight bounds on the sample complexity associated to the fitting of such models to experimental data the results are expressed in the context of the theory of probably approximately correct pac learning

comments on new results on frame proof codes and traceability schemes the paper new results on frame proof codes and traceability schemes claims to give results for two code classes frame proof codes and traceability schemes in the form of lower bounds on the maximum code size and explicit code constructions we will here briefly review the four claims of noting that the proofs and constructions presented in fail and that the claims also contradict previously published upper bounds

approximate dynamic programming with a fuzzy parameterization dynamic programming dp is a powerful paradigm for general nonlinear optimal control computing exact dp solutions is in general only possible when the process states and the control actions take values in a small discrete set in practice it is necessary to approximate the solutions therefore we propose an algorithm for approximate dp that relies on a fuzzy partition of the state space and on a discretization of the action space this fuzzy q iteration algorithm works for deterministic processes under the discounted return criterion we prove that fuzzy q iteration asymptotically converges to a solution that lies within a bound of the optimal solution a bound on the suboptimality of the solution obtained in a finite number of iterations is also derived under continuity assumptions on the dynamics and on the reward function we show that fuzzy q iteration is consistent i e that it asymptotically obtains the optimal solution as the approximation accuracy increases these properties hold both when the parameters of the approximator are updated in a synchronous fashion and when they are updated asynchronously the asynchronous algorithm is proven to converge at least as fast as the synchronous one the performance of fuzzy q iteration is illustrated in a two link manipulator control problem

on the periodicity of symbolic observations of piecewise smooth discrete time systems a study is made of the behavior of discrete time systems composed of a set of smooth transition maps coupled by a quantized feedback function the feedback function partitions the state space into disjoint regions and assigns a smooth transition function to each region the main result is that under a constraint on the norm of the derivative of the transition maps a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic under these assumptions it is shown that eventually the feedback function is determined by a finite state automaton a similar result is proved in the case of finite state dynamic feedback

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

the once but not twice differentiability of the policy function the goal of this paper is to study the differentiability of the policy function or in other words the twice differentiability of the value function in onedimensional dynamic programming problems here we treat the case commonly used in economics where the objective function is a concave function the need for the twice differentiability of the value function has been noticed a long time ago pontryagin et al 1962 p 73 last paragraph shows it to be necessary to derive the maximum principle from the bellman equation and gives a counterexample for nonconcave problems the twice differentiability of the value function is also important if one wants to do comparative statics or if one wants to have smooth coefficients in bellman s partial differential equation another important application has been given recently by kehoe levine and romer 1989 to obtain the finiteness of the number of equilibria in an economy with an infinite number of goods and a finite number of consumers for other motivations as well as background for this paper see stokey and lucas 1989 more formally in the second section of this paper we introduce the basic facts and notations to be used latter the third section is devoted to showing that the policy function is c1 and therefore that the value function is c2 if the policy function is increasing for a multidimensional generalization the reader is referred to santos 1991 in section 4 we show by means of a counterexample that the policy function might not be a twice differentiable function and hence that the value function might not be three times differentiable even if the objective is three times continuously differentiable and strongly concave and the

accuracy of numerical solutions using the euler equation residuals in this paper we derive sorne asymptotic properties on the accuracy of numerical solutions we silow tilat the approximation error of the policy function is of the same order of magnitude as the size of the euler equation residuals moreover for bounding this approximation error tile most relevant parameters are the discount factor and the curvature of the return function these findings provide theoretical foundations for the construction of tests that can assess the performance of alternative computational methods

detecting simple dynamics in cournot like models we introduced the so called cournot like models i e n dimensional discrete dynamical systems which constitute the mathematical environment for modeling some economic and biological processes the main aim of this work is to present a characterization of the dynamical simplicity for these types of systems through the property to have zero topological entropy cournot like systems generalize the well known economic situation of competition in a duopolistic market introduces by cournot in 1838