How robustly can you predict the future?

TL;DR

This paper investigates the limits of predicting real-valued functions based on past data, exploring robustness under complex time distortions and improving existing theoretical bounds using group-theoretic results.

Contribution

It advances the understanding of predictor robustness by refining bounds on how well predictions can withstand complex time transformations, building on and extending prior theoretical work.

Findings

Improved bounds on predictor robustness under complex time distortions.

Connections established between predictor robustness and properties of Archimedean groups.

Extensions of previous results using group-theoretic principles, especially H"older's Theorem.

Abstract

Hardin and Taylor \cite{MR2384262} proved that any function on the reals -- even a nowhere continuous one -- can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed in \cite{MR3100500} that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman \cite{MR3552748}, who provided upper and lower frontiers (in the subgroup lattice of $\text{Homeo}^+(\mathbb{R})$) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of H\"older's Theorem (that every Archimedean group is abelian).