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A Differential Calculus for Random Matrices with Applications to (max, +)-Linear Stochastic Systems

EURANDOM, P.O. Box 513, MB Eindhoven, The Netherlands
heidergott{at}eurandom.tue.nl

We introduce the concept of weak differentiability for random matrices and thereby obtain closed-form analytical expressions for derivatives of functions of random matrices. More specifically, we develop a calculus of weak differentiation for random matrices that resembles the standard calculus of differentiation. Our formalism enables us to (algebraically) calculate derivatives of finite-horizon performance measures of stochastic event graphs. More precisely, we develop a theory of weak differentiation for (max, +)-linear systems. The resulting derivatives provide unbiased estimators for gradients of finite-horizon performance measures. For various types of (max, +)-linear systems, we compute these estimators explicitly and state the corresponding gradient estimation algorithm.

Key Words: Gradient estimation; (max, +)-algebra; weak differentiation; Monte Carlo simulation
History: Received: February 1, 1998; revision received: October 5, 2000;revision received: January 25, 2001;