The Free Inference Dimension: Complexity Measure for Zero-Collision Navigation under Hypothesis Mixtures

Abstract:Solomonoff induction frames prediction as a mixture over computable hypotheses, typically leading to identification of the true environment. In a finite meta-reinforcement learning setting with nested constraint families, in our previous work, we observe a different regime: a value-mixture (VM) agent achieves near-optimal, zero-collision navigation without identifying the true environment, a phenomenon we call Free Inference. This regime persists up to a sharp density threshold, beyond which performance degrades and posterior-mode selection (PMS) becomes preferable. We formalize this behavior via the Free Inference dimension dFI(S,N), a combinatorial measure of the environmental complexity a VM agent can handle while preserving trajectory coherence. We prove dFI is strictly smaller than the VC-dimension and relates to the Natarajan dimension up to a path-length factor, capturing the cost of non-decomposable loss. A PAC-style relaxation yields generalization bounds driven by dFI^(epsilon,delta). We also define a complementary PMS identification dimension and show that a hybrid strategy---averaging until the first collision, then switching to selection---is optimal, with links to Littlestone-type dimensions supported by grid-world experiments.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2609.17816 [cs.LG]

(or arXiv:2609.17816v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.17816 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

Proceedings of BRACIS 2026. LNCS/LNAI, Springer (2026)

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