Divergence and Sufficiency for Convex Optimization

TL;DR

This paper explores the relationship between divergence measures and optimization, showing that under certain sufficiency conditions, regret functions are proportional to information divergence, with implications across various fields.

Contribution

It establishes the equivalence of sufficiency, locality, and monotonicity conditions for regret functions, clarifying when results can be transferred between different application areas.

Findings

Sufficiency implies proportionality to information divergence.

Sufficiency is equivalent to locality and monotonicity.

Different fields vary in the relevance of these conditions.

Abstract

Logarithmic score and information divergence appear in information theory, statistics, statistical mechanics, and portfolio theory. We demonstrate that all these topics involve some kind of optimization that leads directly to regret functions and such regret functions are often given by a Bregman divergence. If the regret function also fulfills a sufficiency condition it must be proportional to information divergence. We will demonstrate that sufficiency is equivalent to the apparently weaker notion of locality and it is also equivalent to the apparently stronger notion of monotonicity. These sufficiency conditions have quite different relevance in the different areas of application, and often they are not fulfilled. Therefore sufficiency conditions can be used to explain when results from one area can be transferred directly to another and when one will experience differences.