We introduce BEACON--Best-Effort Adaptation for Cross-Domain Co-Training--a theory-guided framework for training generative robot policies with abundant source demonstrations and limited target demonstrations. BEACON casts cross-domain co-training as a discrepancy-aware importance-reweighting problem, jointly learning a diffusion-based visuomotor policy and per-sample source weights through an objective inspired by the Best-Effort Adaptation (BEA) bound. To make this approach practical for high-dimensional sequence policies, we develop scalable instance-level discrepancy proxies, stochastic alternating updates for policy and weights, and a multi-source extension that balances heterogeneous source domains. Across sim-to-sim, sim-to-real, and multi-source manipulation settings, BEACON improves robustness and data efficiency over target-only, fixed-ratio co-training, and feature-alignment baselines. Without an explicit alignment objective, BEACON also brings source and target representations closer together as an implicit result of discrepancy-aware co-training.

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