Title: Unified Stability Bounds for Structured World Models: Geometry, Equivariance, and Identifiability as Sufficient Conditions
Primary area: learning theory
Keywords: world models, DreamerV3, representation learning, equivariance, Johnson–Lindenstrauss, bi-Lipschitz embedding, Bellman operator, nonlinear ICA

Abstract: Representation learning for model-based RL offers sample efficiency yet raises a practical question, namely which properties of a learned representation govern downstream performance, and how to test them without re-running large-scale training? We present a \emph{unified stability bound} that decomposes the suboptimality gap into three verifiable channels, consisting of geometric distortion $\kappa$, an identifiability gap proxied by total correlation, and an equivalence or equivariance defect proxied by local equivariance error. The bound is \emph{sufficient} and makes explicit how MDP constants and metric choices scale error through a simple, natural-scale right-hand side. Two mechanism theorems explain when and why proxies can appear unstable in practice. For instance, a quotient-space Johnson–Lindenstrauss result accounts for dimension reduction under symmetry, and a quantitative geometry–equivariance trade-off shows that non-isometric actions increase distortion. Building on this theory, we propose a lightweight diagnostic protocol that uses existing checkpoints, which includes a bound--vs--gap plot with a single calibration constant $\beta$ that is calibrated on early checkpoints and held fixed for validation, and rank consistency with block bootstrap. On DreamerV3 world models, these diagnostics are reproducible, require no retraining, and are consistent with the sufficient-bound view that weak correlations are expected and not counterexamples. This framework provides a principled, low-overhead path to interpretable and auditable representation learning in RL.