Universal Individual-Sequence Prediction with a Primitive-Recursive Superpredictor

2026-07-28

AI Summary

This work addresses the problem of universally predicting arbitrary individual binary sequences under zero–one loss, circumventing the fundamental limitation that no computable master predictor can outperform all computable predictors on every sequence. The authors construct a computable probabilistic predictor that achieves an explicit sublinear regret bound against all primitive recursive predictors. Key contributions include the first proof that the Prediction by Partial Matching (PPM) predictor belongs to the primitive recursive class; the construction of a superpredictor that incurs infinite past Bayesian regret on Martin-Löf random sequences, thereby strictly dominating any finite-state or fixed primitive recursive predictor; and optimality on Martin-Löf random realizations of any computable stationary ergodic binary source, with extensions to scenarios involving multiple sources interleaved according to a primitive recursive schedule.

Abstract

We study sequential prediction of individual binary sequences under zero-one loss. No computable master can compete on every sequence with all total computable predictors. We therefore consider rational-valued primitive-recursive forecasters, a broad syntactically enumerable class containing finite-state, context-based, and Prediction by Partial Matching (PPM) rules. We construct a computable probabilistic predictor with an explicit sublinear regret bound relative to every primitive-recursive forecaster on every individual sequence. We further prove that the PPM predictor is primitive recursive. Consequently, our predictor attains the infinite-past Bayes error on every Martin-Löf random realization of every computable stationary ergodic binary source. This optimality extends to finitely many independent such sources interleaved according to an arbitrary primitive-recursive schedule. Finally, we establish strict separations from finite-state prediction and from every fixed primitive-recursive predictor. keywords: Universal prediction, individual sequences, prediction with expert advice, primitive recursive functions, Kolmogorov complexity, PPM.

Problem

• Universal prediction
• individual sequences
• primitive recursive functions
• PPM
• prediction with expert advice

Innovation

• Universal prediction
• primitive recursive functions
• individual sequences
• sublinear regret
• PPM