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Approximate dynamic programming (ADP) is the standard tool to solve Markovian decision problems under general hypotheses on the system and the cost equations. It is known that one of the key issues of the procedure is how to generate an efficient sampling of the state space, needed for the approximation of the value function, in order to cope with the well-known phenomenon of the curse of dimensionality. The most common approaches in the literature are either aimed at a uniform covering of the state space or driven by the actual evolution of the system trajectories. Concerning the latter approach, F -discrepancy, a quantity closely related to the Kolmogorov-Smirnov statistic, that measures how strictly a set of random points represents a probability distribution, has been recently proposed for an efficient ADP framework in the finite-horizon case. In this paper, we extend this framework to infinite-horizon discounted problems, providing a constructive algorithm to generate efficient sampling points driven by the system behavior. Then, the algorithm is refined with the aim of acquiring a more balanced covering of the state space, thus addressing possible drawbacks of a pure system-driven sampling approach to obtain, in fact, an efficient hybrid between the latter and the pure uniform design. A theoretical analysis is provided through the introduction of an original notion of the F -discrepancy and the proof of its properties. Simulation tests are provided to showcase the behavior of the proposed sampling method.

In this paper, we discuss how the extreme learning machine (ELM) framework can be effectively employed in the unsupervised context of multivariate density estimation. In particular, two algorithms are introduced, one for the estimation of the cumulative distribution function underlying the observed data, and one for the estimation of the probability density function. The algorithms rely on the concept of F -discrepancy, which is closely related to the Kolmogorov-Smirnov criterion for goodness of fit. Both methods retain the key feature of the ELM of providing the solution through random assignment of the hidden feature map and a very light computational burden. A theoretical analysis is provided, discussing convergence under proper hypotheses on the chosen activation functions. Simulation tests show how ELMs can be successfully employed in the density estimation framework, as a possible alternative to other standard methods.

In this paper, we address the problem of generating efficient state sample points for the solution of continuous-state finite-horizon Markovian decision problems through approximate dynamic programming. It is known that the selection of sampling points at which the value function is observed is a key factor when such function is approximated by a model based on a finite number of evaluations. A standard approach consists in generating these points through a random or deterministic procedure, aiming at a balanced covering of the state space. Yet, this solution may not be efficient if the state trajectories are not uniformly distributed. Here, we propose to exploit F -discrepancy, a quantity that measures how closely a set of random points represents a probability distribution, and introduce an example of an algorithm based on such concept to automatically select point sets that are efficient with respect to the underlying Markovian process. An error analysis of the approximate solution is provided, showing how the proposed algorithm enables convergence under suitable regularity hypotheses. Then, simulation results are provided concerning an inventory forecasting test problem. The tests confirm in general the important role of F -discrepancy, and show how the proposed algorithm is able to yield better results than uniform sampling, using sets even 50 times smaller.

In this brief, the use of lattice point sets (LPSs) is investigated in the context of general learning problems (including function estimation and dynamic optimization), in the case where the classic empirical risk minimization (ERM) principle is considered and there is freedom to choose the sampling points of the input space. Here it is proved that convergence of the ERM principle is guaranteed when LPSs are employed as training sets for the learning procedure, yielding up to a superlinear convergence rate under some regularity hypotheses on the involved functions. Preliminary simulation results are also provided.

In this paper, a general method for the numerical solution of maximum-likelihood estimation (MLE) problems is presented; it adopts the deterministic learning (DL) approach to find close approximations to ML estimator functions for the unknown parameters of any given density. The method relies on the choice of a proper neural network and on the deterministic generation of samples of observations of the likelihood function, thus avoiding the problem of generating samples with the unknown density. Under mild assumptions, consistency and convergence with favorable rates to the true ML estimator function can be proved. Simulation results are provided to show the good behavior of the algorithm compared to the corresponding exact solutions.

A methodology to design state estimators for a class of nonlinear continuous-time dynamic systems that is based on neural networks and nonlinear programming is proposed. The estimator has the structure of a Luenberger observer with a linear gain and a parameterized (in general, nonlinear) function, whose argument is an innovation term representing the difference between the current measurement and its prediction. The problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error. Toward this end, if a neural network is used to take on this function, the parameters (i.e., the neural weights) are chosen, together with the gain, by constraining the derivative of a quadratic Lyapunov function for the estimation error to be negative definite on a given compact set. It is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points. The gain is determined by solving a Lyapunov equation. The neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid-point constraints that are not satisfied. Techniques based on low-discrepancy sequences are applied to deal with a small number of sampling points, and, hence, to reduce the computational burden required to optimize the parameters. Numerical results are reported and comparisons with those obtained by the extended Kalman filter are made.

The general problem of reconstructing an unknown function from a finite collection of samples is considered, in case the position of each input vector in the training set is not fixed beforehand but is part of the learning process. In particular, the consistency of the empirical risk minimization (ERM) principle is analyzed, when the points in the input space are generated by employing a purely deterministic algorithm (deterministic learning). When the output generation is not subject to noise, classical number-theoretic results, involving discrepancy and variation, enable the establishment of a sufficient condition for the consistency of the ERM principle. In addition, the adoption of low-discrepancy sequences enables the achievement of a learning rate of O(1/L), with L being the size of the training set. An extension to the noisy case is provided, which shows that the good properties of deterministic learning are preserved, if the level of noise at the output is not high. Simulation results confirm the validity of the proposed approach.