A Priori Advantages of Meta-Induction and the No Free Lunch Theorem: A Contradiction?

Abstract

Recently a new account to the problem of induction has been developed [1], based on a priori advantages of regret-weighted meta-induction (RW) in online learning [2]. The claimed a priori advantages seem to contradict the no free lunch (NFL) theorem, which asserts that relative to a state-uniform prior distribution (SUPD) over possible worlds all (non-clairvoyant) prediction methods have the same expected predictive success. In this paper we propose a solution to this problem based on four novel results: RW enjoys free lunches, i.e., its predictive long-run success dominates that of other prediction strategies. Yet the NFL theorem applies to online prediction tasks provided the prior distribution is a SUPD. The SUPD is maximally induction-hostile and assigns a probability of zero to all possible worlds in which RW enjoys free lunches. This dissolves the apparent conflict with the NFL. The a priori advantages of RW can be demonstrated even under the assumption of a SUPD. Further advantages become apparent when a frequency-uniform distribution is considered.