Unified Stability Bounds for Structured World Models: Geometry, Equivariance, and Identifiability as Sufficient Conditions

Yanjun Li

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 but raises a critical auditing question: which properties of a learned representation actually govern downstream performance, and how can we verify them without expensive retraining?
\textcolor{blue}{We propose a practical auditing framework based on a sufficient stability bound} that decomposes the suboptimality gap into three verifiable channels: geometric distortion $\kappa$, identifiability (proxied by Total Correlation), and symmetry violation (proxied by Local Equivariance Error).
\textcolor{blue}{Crucially, this bound serves as a safety condition rather than a linear predictor, explicitly anchoring error scaling to MDP Lipschitz constants.}
To interpret these components, we provide two mechanistic perspectives: a quotient-space Johnson–Lindenstrauss argument explains how equivariance reduces effective dimensionality, and a geometry–equivariance trade-off quantifies why non-isometric actions inevitably increase distortion.
Building on this theory, we propose a lightweight diagnostic protocol that audits existing checkpoints.
Using a single calibrated constant $\beta$, our framework consistently covers the performance gap across training trajectories, offering a principled \emph{auditing} tool distinct from architectural \emph{design}.
On DreamerV3 world models, these diagnostics are reproducible, require no retraining, and demonstrate that structural stability bounds can effectively flag failure modes even when simple correlation metrics fail.

Primary Area: learning theory

Submission Number: 2219