Zs Vágó

The mathematics of noise-free SPSA

We consider discrete-time fixed gain stochastic approximation processes that are defined in terms of a random field that is identically zero at some point /spl theta/*. The boundedness of the estimator process is enforced by a resetting mechanism. Under appropriate technical conditions the estimator sequence is shown to converge to /spl theta/* with geometric rate almost surely. This result is in striking contrast to classical stochastic approximation theory where the typical convergence rate is n/sup -1/2/. For the proof a discrete-time version of the ODE-method is developed and used, and the techniques of Gerencser (1996) are extended. The paper is motivated by the study of simultaneous perturbation stochastic approximation (SPSA) methods applied to noise-free problems and to direct adaptive control.

SPSA for non-smooth optimization with application in ECG analysis

It is shown that nonsmooth optimization problems can be solved by a suitable extension of the simultaneous perturbation stochastic approximation (SPSA) method. The new optimization method has been tested in a min-max classification problem using both simulated and real data. The latter are ECG signals which were collected for the detection of so-called late potentials.

Controlled Lyapunov-exponents with applications

Let X = (X/sub n/) be a stationary process of k /spl times/ k real-valued matrices, depending on some vector-valued parameter /spl theta//spl epsiv//spl Ropf//sup p/, satisfying E log/sup +/ /spl par/X0(/spl theta/)/spl par/ < /spl infin/ for all /spl theta/. The top-Lyapunov exponent of X is defined as /spl lambda/(/spl theta/) = /sub n//sup lim/ 1/n E log /spl par/X/sub n//spl middot/X/sub n-1/.../spl middot/X/sub 0//spl par/. Top-Lyapunov exponents play a prominent role in randomization procedures for optimization, such as SPSA, and in finance, giving the growth-rate of a self-financing currency-portfolio with a fixed strategy. We develop an iterative procedure for the optimization of /spl lambda/(/spl theta/). In the case when X is a Markov-process, the proposed procedure is formally within the class defined in (Beneviste, 1990). However the analysis of the general case requires different techniques: an ODE method defined in terms of asymptotically stationary random fields. The verification of some standard technical conditions, such as a uniform law of large numbers for the error process is hard. For this we need some auxiliary results which are interesting in their own right. These are given in the Appendix. Simulation results are also presented.

Risk-sensitive identification of ARMA processes

A new definition of risk-sensitive recursive identification is given, which is applicable to ARMA-system and even to general linear stochastic systems. The new identification criterion is defined in terms of a suitably transformed estimation-error process, and an exponential cost function. Applying stochastic realization theory and the bounded real lemma we derive an alternative expression for the LEQG cost function. A key design parameter is the weight matrix that is used in the recursive estimation. Its optimal value is found in explicit form. These results generalize the results of Stoorvogel and van Schuppen (1995) for risk-sensitive identification of AR-processes.

Excitation and Performance in Continuous-time Stochastic Adaptive LQ-control

The purpose of the paper is to present a new methodology for the analysis of the interaction of identification and control. Earlier works on this interaction were presented in [AW, A2, G3, Ke]. Our approach is based on techniques of stochastic complexity (cf. [GV1]).

Stochastic complexity in identification of continuous time systems

The purpose of this paper is to present a continuous time identification method which can be used for high accuracy prediction and control. We consider continuous time systems with quasi-periodic inputs and white observation noise. These investigations have been motivated by control problems in microrobotics, where sampling rate and accuracy requirements are very high. It is shown that continuous time identification methods lead to numerically well conditioned prediction. The key tool in showing this is a general result of the theory of stochastic complexity. Also, we give an explanation on why discrete time methods break down.