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The need for extracting a small sample from a large amount of real data, possibly streaming, arises routinely in learning problems, e.g., for storage, to cope with computational limitations, obtain good training/test/validation sets, and select minibatches for stochastic gradient neural network training. Unless we have reasons to select the samples in an active way dictated by the specific task and/or model at hand, it is important that the distribution of the selected points is as similar as possible to the original data. This is obvious for unsupervised learning problems, where the goal is to gain insights on the distribution of the data, but it is also relevant for supervised problems, where the theory explains how the training set distribution influences the generalization error. In this paper, we analyze the technique of stratified sampling from the point of view of distances between probabilities. This allows us to introduce an algorithm, based on recursive binary partition of the input space, aimed at obtaining samples that are distributed as much as possible as the original data. A theoretical analysis is proposed, proving the (greedy) optimality of the procedure together with explicit error bounds. An adaptive version of the algorithm is also introduced to cope with streaming data. Simulation tests on various data sets and different learning tasks are also provided.

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.

The traditional extreme learning machine (ELM) approach is based on a random assignment of the hidden weight values, while the linear coefficients of the output layer are determined analytically. This brief presents an analysis based on geometric properties of the sampling points used to assign the weight values, investigating the replacement of random generation of such values with low-discrepancy sequences (LDSs). Such sequences are a family of sampling methods commonly employed for numerical integration, yielding a more efficient covering of multidimensional sets with respect to random sequences, without the need for any computationally intensive procedure. In particular, we prove that the universal approximation property of the ELM is guaranteed when LDSs are employed, and how an efficient covering affects the convergence positively. Furthermore, since LDSs are generated deterministically, the results do not have a probabilistic nature. Simulation results confirm, in practice, the good theoretical properties given by the combination of ELM with LDSs.

Local linear regression models, a kind of nonparametric structures that locally perform a linear estimation of the target function, are analyzed in the context of empirical risk minimization (ERM) for function learning. The analysis is carried out with emphasis on geometric properties of the available data. In particular, the discrepancy of the observation points used both to build the local regression models and compute the empirical risk is considered. This allows to treat indifferently the case in which the samples come from a random external source and the one in which the input space can be freely explored. Both consistency of the ERM procedure and approximating capabilities of the estimator are analyzed, proving conditions to ensure convergence. Since the theoretical analysis shows that the estimation improves as the discrepancy of the observation points becomes smaller, low-discrepancy sequences, a family of sampling methods commonly employed for efficient numerical integration, are also analyzed. Simulation results involving two different examples of function learning are provided.

This brief presents an analysis of the performance of kernel smoothing models used to estimate an unknown target function, addressing the case where the choice of the training set is part of the learning process. In particular, we consider a choice of the points at which the function is observed based on low-discrepancy sequences, which is a family of sampling methods commonly employed for efficient numerical integration. We prove that, under suitable regularity assumptions, consistency of the empirical risk minimization is guaranteed with a good rate of convergence of the estimation error, as well as the convergence of the approximation error. Simulation results confirm, in practice, the good theoretical properties given by the combination of kernel smoothing models with low-discrepancy sampling.

A new efficient technique for estimating probability densities from data through the application of the approximate global maximum likelihood (AGML) approach is proposed. It employs a composition of kernel functions to estimate the correct behavior of parameters involved in the expression of the unknown probability density. Convergence to the optimal solution is guaranteed by a deterministic learning framework when low discrepancy sequences are used to generate the centers of the kernels. Trials on mixture of Gaussians show that the proposed semi-local technique is able to efficiently approximate the maximum likelihood solution even in complex situations where implementations based on standard neural networks require an excessive computational cost., (Copyright 2010 Elsevier Ltd. All rights reserved.)

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.