A Note on How to Remove the $\ln\ln T$ Term from the Squint Bound

TL;DR

This paper demonstrates how to eliminate the $ ln n$ term from the Squint algorithm's bound by modifying the prior, building on previous work with shifted KT potentials.

Contribution

It shows that removing the $ ln n$ factor is equivalent to changing the prior in the Krichevsky--Trofimov algorithm and applies this idea to the Squint algorithm.

Findings

The $ ln n$ term can be removed by prior modification.

The approach is based on shifted KT potentials.

The method applies to data-independent bounds for Squint.

Abstract

In Orabona and P\'al [2016], we introduced the shifted KT potentials, to remove the $\ln \ln T$ factor in the parameter-free learning with expert bound. In this short technical note, I show that this is equivalent to changing the prior in the Krichevsky--Trofimov algorithm. Then, I show how to use the same idea to remove the $\ln \ln T$ factor in the data-independent bound for the Squint algorithm.