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

2026-04-29

AI Summary

This work aims to eliminate the $\ln \ln T$ term from the data-independent regret bound of the Squint algorithm to achieve a tighter theoretical guarantee. To this end, we uncover an equivalence between the shifted KT potential and the choice of prior in Krichevsky–Trofimov (KT) coding, and for the first time integrate this insight into the analysis framework of Squint. Leveraging this key connection, we successfully remove the $\ln \ln T$ factor from its regret bound, yielding a data-independent bound free of this iterated logarithmic term. This advancement substantially sharpens the theoretical performance guarantee of Squint and extends the analytical toolkit for adaptive algorithms in online learning.

Abstract

In Orabona and Pál [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.

Problem

• Squint algorithm
• ln ln T term
• parameter-free learning
• data-independent bound
• Krichevsky–Trofimov algorithm

Innovation

• shifted KT potentials
• Krichevsky–Trofimov algorithm
• Squint algorithm
• parameter-free learning
• log-log term removal