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

arXiv: https://arxiv.org/abs/2609.17816 · HTML: https://mlsanity.com/paper/2609.17816/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 14 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 4 of 4)

- [RL2: Fast Reinforcement Learning via Slow Reinforcement Learning](https://mlsanity.com/paper/1611.02779/) (2016)
- [Efficient Off-Policy Meta-Reinforcement Learning via Probabilistic Context Variables](https://mlsanity.com/paper/1903.08254/) (2019)
- [VariBAD: A Very Good Method for Bayes-Adaptive Deep RL via Meta-Learning](https://mlsanity.com/paper/1910.08348/) (2019)
- [Meta reinforcement learning as task inference](https://mlsanity.com/paper/1905.06424/) (2019)

## Abstract (arXiv)

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.