Loss Bounds and Time Complexity for Speed Priors

TL;DR

This paper analyzes the predictive capabilities and computational complexity of speed priors, showing their ability to predict polynomial-time sequences and establishing bounds on their computability.

Contribution

It introduces a variant of the speed prior, providing new complexity bounds and insights into its predictive performance on different classes of sequences.

Findings

The proposed speed prior predicts sequences from polynomial-time estimable measures.

It is computable in doubly-exponential time, not polynomial time.

Schmidhuber's original speed prior predicts polynomial-time computable deterministic sequences.

Abstract

This paper establishes for the first time the predictive performance of speed priors and their computational complexity. A speed prior is essentially a probability distribution that puts low probability on strings that are not efficiently computable. We propose a variant to the original speed prior (Schmidhuber, 2002), and show that our prior can predict sequences drawn from probability measures that are estimable in polynomial time. Our speed prior is computable in doubly-exponential time, but not in polynomial time. On a polynomial time computable sequence our speed prior is computable in exponential time. We show better upper complexity bounds for Schmidhuber's speed prior under the same conditions, and that it predicts deterministic sequences that are computable in polynomial time; however, we also show that it is not computable in polynomial time, and the question of its predictive properties for stochastic sequences remains open.