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

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.