  The accelerated weight histogram (AWH) algorithm is an iterative extended
ensemble algorithm, developed for statistical physics and computational biology
applications. It is used to estimate free energy differences and expectations
with respect to Gibbs measures. The AWH algorithm is based on iterative updates
of a design parameter, which is closely related to the free energy, obtained by
matching a weight histogram with a specified target distribution. The weight
histogram is constructed from samples of a Markov chain on the product of the
state space and parameter space. In this paper almost sure convergence of the
AWH algorithm is proved, for estimating free energy differences as well as
estimating expectations with adaptive ergodic averages. The proof is based on
identifying the AWH algorithm as a stochastic approximation and studying the
properties of the associated limit ordinary differential equation.
