Probability Collectives for Optimization of Computer Simulations

A significant body of research under the rubric of blackbox optimization addresses the problem of optimization in a design space where the designs are evaluated by a computer simulation. For many such simulations, conventional local optimization methods prove inadequate, owing to the presence of local minima, local non-smoothness, and so on. A more effective approach would learn the global characteristics of the design space, and sample from disparate regions as it progresses. Several probabilistic approaches to this problem are based on the following idea: during the course of any search process, our partial knowledge of the design space grows as we evaluate a growing number of designs, and this knowledge can be elegantly expressed in terms of probability distributions. This is usually done using surrogate models, or data fits. We would like to use this partial knowledge to lead us to promising new designs, and this involves searching the data fit for points where the uncertainty of the fit, in conjunction with the fit itself, indicates a good chance of finding a improved design. This is the basis for the well-known Efficient Global Optimization (EGO) algorithm. This task can be regarded as an auxiliary optimization problem, but one that has some undesirable characteristics, including large flat regions and multiple local optima. Previous solution approaches have included branch and bound (the EGO algorithm), and local optimization with multi-start. In this paper, we present a new approach, itself based on probability distributions. Probability Collectives (PC) is an optimization framework in which a given optimization problem is transformed to one over probability distributions. This transformation often enables us to overcome problems such as local non-smoothness and multiple local optima. One particular PC approach is closely related to function smoothing and filtering. Our formulation is a type of Monte Carlo Optimization, often encountered in machine learning algorithms. We briefly introduce this approach, and apply it to the auxiliary optimization problem described above. Finally, we compare this technique with other algorithms, including a heuristic sampling algorithm, and a genetic algorithm.