Proximity Operators of Discrete Information Divergences

TL;DR

This paper introduces new proximity operators for discrete information divergences, enabling more efficient convex optimization involving these measures, with applications demonstrated in database query optimization.

Contribution

The paper derives closed-form proximity operators for two-variable convex divergences, facilitating their use in standard proximal algorithms and broadening their practical applicability.

Findings

Proximity operators enable efficient optimization with divergence measures.

Validated methods improve query selectivity estimation in databases.

Experiments show scalability from small to large datasets.

Abstract

Information divergences allow one to assess how close two distributions are from each other. Among the large panel of available measures, a special attention has been paid to convex $\varphi$-divergences, such as Kullback-Leibler, Jeffreys-Kullback, Hellinger, Chi-Square, Renyi, and I$_{\alpha}$ divergences. While $\varphi$-divergences have been extensively studied in convex analysis, their use in optimization problems often remains challenging. In this regard, one of the main shortcomings of existing methods is that the minimization of $\varphi$-divergences is usually performed with respect to one of their arguments, possibly within alternating optimization techniques. In this paper, we overcome this limitation by deriving new closed-form expressions for the proximity operator of such two-variable functions. This makes it possible to employ standard proximal methods for efficiently solving a wide range of convex optimization problems involving $\varphi$-divergences. In addition, we show that these proximity operators are useful to compute the epigraphical projection of several functions of practical interest. The proposed proximal tools are numerically validated in the context of optimal query execution within database management systems, where the problem of selectivity estimation plays a central role. Experiments are carried out on small to large scale scenarios.