A Classification of Phi-Divergences for Data-Driven Stochastic Optimization

摘要

Phi-divergences (e.g., Kullback-Leibler, chi-squared distance, etc.) provide a measure of distance between two probability distributions. They can be used in data-driven stochastic optimization to create an ambiguity set of distributions that are centered around a nominal distribution and this ambiguity set of distributions is used for optimization. The nominal distribution is often determined by collected observations, expert opinions, simulations, etc. Given that there are many phi-divergences, a decision maker is left with the question of how each divergence behaves for his/her problem and which one to choose. In this paper, we present a classification of phi-divergences to elucidate their use for models with different properties and different sources of data. We provide examples of phi-divergences that result in commonly used risk models and discuss their behavior with respect to our classification. We refine our classification for a class of problems and further analyze each category in more detail. We illustrate the behavior of several commonly used phi-divergences in each classification category on a small power capacity expansion problem.