aiqbench analysis
This study demonstrates that enforcing exact equivariance throughout training preserves mathematical symmetry in latent world models, enabling zero-shot generalization across entire symmetry groups rather than just restricted slices. The results show that while standard augmentation or soft-equivariance approaches fail to maintain float-floor exactness, this method achieves invariant prediction errors and significantly smaller model sizes even under complex optimizers like Muon. For practitioners, the key takeaway is that hard symmetry constraints offer provable robustness to out-of-distribution shifts in orientation that standard scaling laws cannot replicate.
Exact equivariance, kept through training, buys zero-shot generalisation across the symmetry group
arXiv:2606.03003v1 Announce Type: cross Abstract: A latent world model built from an equivariant encoder $E$ and an equivariant predictor $f$ inherits a provable symmetry of its training loss: when the world's dynamics genuinely carries a group $G$ acting on latents by an orthogonal representation $\rho(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting the dynamics on a restricted slice of orientations mathematically determines it on the entire orbit (j\v{u} y\=i f\v{a}n s\=an). We verify this end-to-end at laptop scale (CPU/MPS, fully seeded). [A] The symmetry survives a real Muon/AdamW + EMA + VICReg run -- composed encode-then-predict residual $\sim 10^{-6}$ after optimisation, not just at initialisation, and under any optimiser. [B] One-step error is flat to five digits across the group, while a same-hypothesis-class non-equivariant baseline fits the slice but breaks out-of-distribution (VN $\times 1.00$ vs baseline $\times 13.8$ in 2D, $\times 17.2$ in 3D, $\times 157$ over the full $\mathrm{SE}(3)$ ladder), with the equivariant model $4.5$-$7.4\times$ smaller. [C] The same isometry argument lifts to closed loop: under a matching equivariant planner the control trajectory at orientation $g$ is exactly $\rho(g)$ applied to the seen one, so closed-loop error is invariant across the group -- float-floor-exact in 2D/$\mathrm{SO}(2)$ on real PushT and statistically flat in 3D/$\mathrm{SE}(3)$ (disjoint 95% CIs). We stress-test the prior against Sutton's Bitter Lesson: augmentation, brute-force scale, and soft-equivariance each close at most the across-group task metric, never the float-floor exactness. Because equivariance is closed under composition, the $H$-fold rollout stays flat ($\times 1.00$, $\le 2\times 10^{-7}$) at every horizon, while the baseline's residual compounds with $H$. Out of scope: task-success sweeps, planner-free invariance, and scaling.
Abstract:A latent world model built from an equivariant encoder $E$ and an equivariant predictor $f$ inherits a provable symmetry of its training loss: when the world's dynamics genuinely carries a group $G$ acting on latents by an orthogonal representation $\rho(g)$, the one-step prediction relMSE is exactly invariant across the whole group, so fitting the dynamics on a restricted slice of orientations mathematically determines it on the entire orbit (jǔ yī fǎn sān). We verify this end-to-end at laptop scale (CPU/MPS, fully seeded). [A] The symmetry survives a real Muon/AdamW + EMA + VICReg run -- composed encode-then-predict residual $\sim 10^{-6}$ after optimisation, not just at initialisation, and under any optimiser. [B] One-step error is flat to five digits across the group, while a same-hypothesis-class non-equivariant baseline fits the slice but breaks out-of-distribution (VN $\times 1.00$ vs baseline $\times 13.8$ in 2D, $\times 17.2$ in 3D, $\times 157$ over the full $\mathrm{SE}(3)$ ladder), with the equivariant model $4.5$-$7.4\times$ smaller. [C] The same isometry argument lifts to closed loop: under a matching equivariant planner the control trajectory at orientation $g$ is exactly $\rho(g)$ applied to the seen one, so closed-loop error is invariant across the group -- float-floor-exact in 2D/$\mathrm{SO}(2)$ on real PushT and statistically flat in 3D/$\mathrm{SE}(3)$ (disjoint 95% CIs). We stress-test the prior against Sutton's Bitter Lesson: augmentation, brute-force scale, and soft-equivariance each close at most the across-group task metric, never the float-floor exactness. Because equivariance is closed under composition, the $H$-fold rollout stays flat ($\times 1.00$, $\le 2\times 10^{-7}$) at every horizon, while the baseline's residual compounds with $H$. Out of scope: task-success sweeps, planner-free invariance, and scaling.
92 pages, 11 figures. Core paper plus an extended results-log appendix and a forward-looking theory supplement. All experiments are laptop-scale (CPU/MPS), fully seeded and deterministic
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Robotics (cs.RO)
arXiv:2606.03003 [cs.LG]
(or arXiv:2606.03003v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.03003 arXiv-issued DOI via DataCite (pending registration)
Full text under the publisher's open licence · 317 words
