Stochastic Analysis of Convergence via Dynamic Representation for a Class of Line-search Algorithms

Certain convergent search algorithms can be turned into chaotic dynamic systems by renormalisation back to a standard region at each iteration. This allows the machinery of ergodic theory to be used for a new probabilistic analysis of their behaviour. Rates of convergence can be redefined in terms of various entropies and ergodic characteristics (Kolmogorov and Rényi entropies and Lyapunov exponent). A special class of line-search algorithms, which contains the Golden-Section algorithm, is studied in detail. Their associated dynamic systems exhibit a Markov partition property, from which invariant measures and ergodic characteristics can be computed. A case is made that the Rényi entropy is the most appropriate convergence criterion in this environment.