161 research outputs found

    A companion for the Kiefer--Wolfowitz--Blum stochastic approximation algorithm

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    A stochastic algorithm for the recursive approximation of the location θ\theta of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462--466] in the univariate framework, and by Blum [Ann. Math. Statist. 25 (1954) 737--744] in the multivariate case. The aim of this paper is to provide a companion algorithm to the Kiefer--Wolfowitz--Blum algorithm, which allows one to simultaneously recursively approximate the size μ\mu of the maximum of the regression function. A precise study of the joint weak convergence rate of both algorithms is given; it turns out that, unlike the location of the maximum, the size of the maximum can be approximated by an algorithm which converges at the parametric rate. Moreover, averaging leads to an asymptotically efficient algorithm for the approximation of the couple (θ,μ)(\theta,\mu).Comment: Published in at http://dx.doi.org/10.1214/009053606000001451 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Estimation of the parameters of sampled-data systems by stochastic approximation

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    Parameter estimation of sampled data control systems by stochastic approximatio

    Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials

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    In 1951 Robbins and Monro published the seminal article on stochastic approximation and made a specific reference to its application to the "estimation of a quantal using response, nonresponse data." Since the 1990s, statistical methodology for dose-finding studies has grown into an active area of research. The dose-finding problem is at its core a percentile estimation problem and is in line with what the Robbins--Monro method sets out to solve. In this light, it is quite surprising that the dose-finding literature has developed rather independently of the older stochastic approximation literature. The fact that stochastic approximation has seldom been used in actual clinical studies stands in stark contrast with its constant application in engineering and finance. In this article, I explore similarities and differences between the dose-finding and the stochastic approximation literatures. This review also sheds light on the present and future relevance of stochastic approximation to dose-finding clinical trials. Such connections will in turn steer dose-finding methodology on a rigorous course and extend its ability to handle increasingly complex clinical situations.Comment: Published in at http://dx.doi.org/10.1214/10-STS334 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

    A control algorithm for autonomous optimization of extracellular recordings

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    This paper develops a control algorithm that can autonomously position an electrode so as to find and then maintain an optimal extracellular recording position. The algorithm was developed and tested in a two-neuron computational model representative of the cells found in cerebral cortex. The algorithm is based on a stochastic optimization of a suitably defined signal quality metric and is shown capable of finding the optimal recording position along representative sampling directions, as well as maintaining the optimal signal quality in the face of modeled tissue movements. The application of the algorithm to acute neurophysiological recording experiments and its potential implications to chronic recording electrode arrays are discussed

    On the Determination of the Step Size in Stochastic Quasigradient Methods

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    For algorithms of the Robbins-Monro type, the best choice (from the asymptotic point of view) for the step-size constants a_n is known to be a/n. From the practical point of view, however, adaptive step-size rules seem more likely to produce quick convergence. In this paper a new adaptive rule for controlling the stepsize is presented and its behavior is studied

    Estimating the minimizer and the minimum value of a regression function under passive design

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    We propose a new method for estimating the minimizer x\boldsymbol{x}^* and the minimum value ff^* of a smooth and strongly convex regression function ff from the observations contaminated by random noise. Our estimator zn\boldsymbol{z}_n of the minimizer x\boldsymbol{x}^* is based on a version of the projected gradient descent with the gradient estimated by a regularized local polynomial algorithm. Next, we propose a two-stage procedure for estimation of the minimum value ff^* of regression function ff. At the first stage, we construct an accurate enough estimator of x\boldsymbol{x}^*, which can be, for example, zn\boldsymbol{z}_n. At the second stage, we estimate the function value at the point obtained in the first stage using a rate optimal nonparametric procedure. We derive non-asymptotic upper bounds for the quadratic risk and optimization error of zn\boldsymbol{z}_n, and for the risk of estimating ff^*. We establish minimax lower bounds showing that, under certain choice of parameters, the proposed algorithms achieve the minimax optimal rates of convergence on the class of smooth and strongly convex functions.Comment: 35 page

    Event-triggered Learning

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    The efficient exchange of information is an essential aspect of intelligent collective behavior. Event-triggered control and estimation achieve some efficiency by replacing continuous data exchange between agents with intermittent, or event-triggered communication. Typically, model-based predictions are used at times of no data transmission, and updates are sent only when the prediction error grows too large. The effectiveness in reducing communication thus strongly depends on the quality of the prediction model. In this article, we propose event-triggered learning as a novel concept to reduce communication even further and to also adapt to changing dynamics. By monitoring the actual communication rate and comparing it to the one that is induced by the model, we detect a mismatch between model and reality and trigger model learning when needed. Specifically, for linear Gaussian dynamics, we derive different classes of learning triggers solely based on a statistical analysis of inter-communication times and formally prove their effectiveness with the aid of concentration inequalities
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