When one observation admits several valid actions, CVAE KL regularization and flow-model Lipschitz smoothness decide whether a policy keeps multimodality, while standard robot simulation benchmarks turn out to be nearly unimodal

The paper gives a precise characterization of multimodal behavioral cloning: for each observation, the support of the expert conditional action distribution is contained in finitely many mode sets with disjoint closures, which yields a mode-assignment function and a mode-label random variable, turning 'several valid actions for the same observation' into an analyzable object. Prior multimodal BC methods largely evaluated multimodality indirectly through rollout performance, with limited guidance on which model- and training-level quantities control it; this work supplies that formalization. Definitions and notation appear in Section 2, with the supporting entropy, conditional mutual information, and Fano-inequality tools collected in Appendix A.

For latent-variable policies, Proposition 1 proves that accurate mode recovery forces the posterior latent to retain action-conditioned information, lower-bounded by the Fano-corrected entropy of the mode label; Corollary 2 further shows that as pointwise posterior-prior regularization strengthens, the certified conditional mutual information is forced down at a corresponding rate, producing multimodality collapse. It links the KL weight in CVAEs, usually treated as an empirical hyperparameter, to whether demonstrated modes can be distinguished, and gives a threshold condition for a desired mode-recovery error. The results are stated as a proposition and corollary with full proofs; on synthetic tasks, PCA visualizations of posterior latents show mode-separated clusters at small regularization and an overlapping cloud at large regularization, and the empirical inequality holds across 20 training runs.

For action-space generative policies, Proposition 2 shows that a base-to-action map with a small Lipschitz constant cannot assign substantial probability to many well-separated modes, so covering many modes requires sharp stretching in base space or bridge regions in action space; any stabilization or regularization that overly restricts the network's Lipschitz constant therefore harms this mechanism. It attributes the multimodal capacity of diffusion and flow-matching policies to the geometry of the sampling map rather than expressiveness alone, and gives an explicit relation between the number of represented modes, the Lipschitz constant, and mode separation. The proof uses the Gaussian isoperimetric inequality to avoid dependence on the base-space dimension; on synthetic tasks, finite-difference estimates of cross-mode transition sensitivity show that runs with high sensitivity and negligible bridge fractions succeed more often than runs with lower sensitivity and substantial bridge fractions.

The authors propose a conditional multimodality estimator based on local joint state-action GMMs and modal clustering for data without ground-truth mode labels; it reconstructs the true mode counts on synthetic benchmarks, but estimates extremely close to 1 on Push-T, UR3, LIBERO, and Meta-World, with Kitchen highest at action horizon 30, while the real-robot bimodal tissue-grasping data is more multimodal. It turns 'how multimodal is this benchmark' from an assumption into a measurable diagnostic, and uses it to explain why deterministic regression remains competitive on those benchmarks. The estimator is validated against ground truth on synthetic tasks; robot dataset fits use stratified query states and a fixed neighborhood radius, with complete results reported across radii.

The analysis applies to action-chunking behavioral cloning policies, covering both latent-variable and action-space generative families, and its conclusions are tested on synthetic multimodal navigation tasks, standard robot simulation benchmarks, and a real-robot bimodal tissue-grasping task. For practitioners choosing a KL weight or deciding whether a multimodal policy is needed, the paper offers a heuristic selection procedure based on a target mode-recovery error and a diagnostic for whether demonstration data are genuinely multimodal; for theorists, it provides testable relations between mode preservation, regularization strength, and the Lipschitz constant.

Corollary 2 is training-independent: it gives a necessary condition for achieving a target mode-recovery error but does not guarantee that training finds such a solution, and the paper reports several runs below the threshold that still fail. Proposition 2 characterizes a structural limitation of low-Lipschitz samplers rather than a specific training outcome. The multimodality estimator depends on neighborhood radius, action horizon, and clustering hyperparameters, and the paper notes that data multimodality is itself horizon-dependent; how to estimate mode counts when only visual features are available remains an open question. Whether the limited conditional multimodality observed in simulation benchmarks also characterizes large-scale real-robot datasets awaits further study.