Modeling the transient behavior of stochastic gradient algorithms

We investigate the transient behavior of a class of stochastic gradient algorithms. Unlike the analysis usually applied to stochastic approximation and simulated annealing which focuses on the rate of convergence and the asymptotic limit, we take a more detailed look at the transient behavior with the goal of better understanding how the global structure of the performance measure influences the behavior of the algorithm. For the sake of tractability, we work with a specific class of problems characterized by gradients with easily characterized stationary points. Our prototype involves stochastic algorithms for ordering a numerical list, a problem which is the subject of a recent paper in the condensed matter physics literature, focusing on hysteretic effects in annealing. These authors raise several questions of interest in studying stochastic dynamics which inspired this paper.