# Ersatz estimation: gradient relationships in complex systems under uncertainty

#### by Associate Professor Felisa Vazquez-Abad

Institution: University of Melbourne
Date: Wed 13th April 2005
Time: 11:00 AM
Location: Russell Love Theatre, Richard Berry Building

Abstract: In this talk we introduce the problem of derivative (or more
generally, gradient) estimation, when the quantity of interest is the
expected cost incurred by a stochastic system, so that only noisy
measurements are observable. For the past three decades researchers have
tried to find ways to build statistics that will measure derivatives or
trends (sensitivities) of the expected cost, when one or several
parameters change values. Gradient estimation can be used for
optimization, estimation, or identification of parameters. To set the
ideas, we will first show simple examples where unbiased estimators of the
derivatives can be easily constructed.

In real systems, such as telecommunication networks, insurance portfolios,
and transportation systems, it is often either very difficult or very
costly to estimate gradients with respect to the control variables of
interest. In the 1990's, several researchers independently proposed ad-hoc
transformations of the derivatives to simplify the problem. Since that
time, I have tried to formulate a general methodology to justify such
transformations, which allow us to express the derivatives with respect to
the control variables of interest in terms of simpler derivatives,
obtained with respect to other parameters in the model. An \"ersatz\"
derivative is the estimator obtained via the inverse transformations. My
approach uses basic methods in Physics, such as time scale changes and
changes of variables, or of reference framework.

We will work out a complete example of a time scale change, discuss its
application to a queueing problem, an insurance problem and a
transportation problem, and then give a counter example where the
transformations do NOT work. For this latter example, we will apply our
general methodology to identify a correction term for the ersatz
derivative, to eliminate the bias. This example provides insight into
the effects of uncertainty and justifies the term \"ersatz\", meaning a
surrogate of lesser quality, because in many cases it is unrealistic to
compute the exact correction terms.

In this talk I will attempt to give a general idea of an on-going
research effort, rather than to present a finished Theory of gradient
estimation.

For More Information: Paul Norbury email: pnorbury@ms.unimelb.edu.au or Ole Warnaar email: warnaar@ms.unimelb.edu.au

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