Exp-Concavity of Proper Composite Losses

TL;DR

This paper characterizes the exp-concavity of proper composite losses and demonstrates how to transform any $eta$-mixable binary proper loss into a $eta$-exp-concave loss, bridging two key concepts in online prediction.

Contribution

It provides a complete characterization of exp-concavity for proper composite losses and shows how to convert $eta$-mixable losses into $eta$-exp-concave ones, enhancing theoretical understanding.

Findings

Complete characterization of exp-concavity for proper composite losses

Transformation method for binary $eta$-mixable proper losses into $eta$-exp-concave losses

Approximate transformation approach for multi-class losses

Abstract

The goal of online prediction with expert advice is to find a decision strategy which will perform almost as well as the best expert in a given pool of experts, on any sequence of outcomes. This problem has been widely studied and $O(\sqrt{T})$ and $O(\log{T})$ regret bounds can be achieved for convex losses (\cite{zinkevich2003online}) and strictly convex losses with bounded first and second derivatives (\cite{hazan2007logarithmic}) respectively. In special cases like the Aggregating Algorithm (\cite{vovk1995game}) with mixable losses and the Weighted Average Algorithm (\cite{kivinen1999averaging}) with exp-concave losses, it is possible to achieve $O(1)$ regret bounds. \cite{van2012exp} has argued that mixability and exp-concavity are roughly equivalent under certain conditions. Thus by understanding the underlying relationship between these two notions we can gain the best of both algorithms (strong theoretical performance guarantees of the Aggregating Algorithm and the computational efficiency of the Weighted Average Algorithm). In this paper we provide a complete characterization of the exp-concavity of any proper composite loss. Using this characterization and the mixability condition of proper losses (\cite{van2012mixability}), we show that it is possible to transform (re-parameterize) any $\beta$-mixable binary proper loss into a $\beta$-exp-concave composite loss with the same $\beta$. In the multi-class case, we propose an approximation approach for this transformation.