Bilinear Flow Policy: Distributional Extrapolation For Goal-conditioned Visuomotor Imitation

Goal-conditioned imitation learning (GCIL) with flow matching is a promising framework that can represent multimodal behaviors while adapting to diverse, user-specified goals, yet often fails when goals lie outside the demonstration support. To extrapolate to such unseen goals without collapsing multimodality—a problem we call distributional extrapolation—we introduce Bilinear Flow Policy (BFP), a generative visuomotor policy that combines transductive retrieval with a bilinear conditional flow. Given an unseen observation-goal pair, BFP retrieves an “anchor” training example and transductively reformulates the unseen pair as this familiar anchor plus a residual term. For this decomposition to guide action prediction, the residual must compactly encode how the current observation-goal pair differs from the anchor, and the anchor must be chosen so that this difference is predictive of the corresponding action distribution. BFP achieves this with pretrained visual features and a novel learned anchor-selection algorithm. The novel bilinear flow then models how the anchor and the residual jointly determine the multimodal action distribution. We prove that, for bilinear flow, action distribution error at unseen goals is bounded by the in-distribution flow-matching error up to problem-dependent factors. Across five manipulation tasks in simulation, BFP achieves 2.63× the out-of-distribution success rate of a GCIL policy and 1.36× that of the strongest extrapolation-targeted baseline. On two real-world tasks, BFP improves over GCIL by 32 percentage points on average. Finally, our theory yields practical, pre-deployment diagnostics for predicting which trained policies will extrapolate well and to which unseen goals. Project website: https://bilinearflow.netlify.app