On Sequences with Non-Learnable Subsequences

TL;DR

This paper demonstrates the existence of sequences with subsequences that cannot be learned by any computationally efficient randomized forecasting algorithms, challenging previous universal learnability results.

Contribution

It introduces a probabilistic algorithm that constructs sequences with non-learnable subsequences for all partially weakly computable randomized forecasting algorithms.

Findings

Sequences with non-learnable subsequences exist for efficient algorithms.

Universal randomized forecasting algorithms cannot learn all sequences.

The probabilistic construction works with high probability.

Abstract

The remarkable results of Foster and Vohra was a starting point for a series of papers which show that any sequence of outcomes can be learned (with no prior knowledge) using some universal randomized forecasting algorithm and forecast-dependent checking rules. We show that for the class of all computationally efficient outcome-forecast-based checking rules, this property is violated. Moreover, we present a probabilistic algorithm generating with probability close to one a sequence with a subsequence which simultaneously miscalibrates all partially weakly computable randomized forecasting algorithms. %subsequences non-learnable by each randomized algorithm. According to the Dawid's prequential framework we consider partial recursive randomized algorithms.