Liquid Networks with Mixture Density Heads for Efficient Imitation Learning

Nikolaus Correll

cs.LGMar 28, 2026

Liquid Networks with Mixture Density Heads for Efficient Imitation Learning

Authors: Nikolaus Correll

Organizations: University of Colorado Boulder

Abstract

We compare liquid neural networks with mixture density heads against diffusion policies on Push-T, RoboMimic Can, and PointMaze under a shared-backbone comparison protocol that isolates policy-head effects under matched inputs, training budgets, and evaluation settings. Across tasks, liquid policies use roughly half the parameters (4.3M vs. 8.6M), achieve 2.4x lower offline prediction error, and run 1.8 faster at inference. In sample-efficiency experiments spanning 1% to 46.42% of training data, liquid models remain consistently more robust, with especially large gains in low-data and medium-data regimes. Closed-loop results on Push-T and PointMaze are directionally consistent with offline rankings but noisier, indicating that strong offline density modeling helps deployment while not fully determining closed-loop success. Overall, liquid recurrent multimodal policies provide a compact and practical alternative to iterative denoising for imitation learning.

Figures & tables

Dataset	Params	NLL $\downarrow$	MSE $\downarrow$	ms $\downarrow$	Params	NLL $\downarrow$	MSE $\downarrow$	ms $\downarrow$
	Liquid + MDN				Diffusion
Push-T	4.34	-6.999	0.000158	195	8.60	-3.768	0.000155	381
RoboMimic Can	4.36	-20.830	0.007	205	8.84	-15.732	0.124	380
PointMaze	4.34	-8.615	0.045	252	8.60	-3.578	0.450	448

Table 1: Offline comparison after 120 epochs under the shared-backbone protocol. Liquid + MDN uses roughly half the parameters (

\approx

4.3 M vs. 8.6 M) and runs 1.8–2

\times

faster (195–252 ms vs. 380–448 ms inference time). It achieves lower NLL on all three tasks , reflecting a better-calibrated density model. MSE is comparable on Push-T (within 2%), but Liquid + MDN is

\approx

\times

lower on RoboMimic Can and 10

\times

lower on PointMaze . Bold entries mark the best value per metric. Params in millions; ms = per-trajectory wall-clock inference time.

Task	Model	Success (%) $\uparrow$	Distance-Success (%) $\uparrow$	Reward $\uparrow$
Push-T	Liquid + MDN	91.0	–	0.9726
Push-T	Diffusion	88.0	–	0.9811
PointMaze	Liquid + MDN	20.0	9.7	7.71
PointMaze	Diffusion	9.5	3.7	6.48

Table 2: Closed-loop action-prediction results on Push-T and PointMaze. Bold entries indicate the better model per metric within each task (higher is better for success, distance-success, and reward). Push-T uses a matched 100-episode comparison; PointMaze reports 50-trial aggregate means. For PointMaze, Success uses the environment success flag, while Distance-Success uses a stricter rollout minimum-distance threshold of

0.20

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Training-Free Imitation Learning with Closed-Form Diffusion Policies

Raghav Mishra, Ian R. Manchester

Australian Center for Robotics, ARIAM Hub, and School of Aerospace, Mechanical and Mechatronic Engineering University of Sydney

Diffusion Model Guidance Visuomotor Policy Learning Diffusion Policy Imitation Learning Policy Learning

Open paper AskPDF arXiv

While diffusion-based policies have impressive performance and expressivity, their long offline training slows down the data collection and policy deployment loop. We introduce Closed-Form Diffusion Policies, a class of training-free diffusion-based policies for imitation learning using the closed-form score derived from the demonstration dataset. We deploy CFDP with real-time inference with a mobile CPU in hardware experiments, showing it can successfully perform imitation directly from the dataset in milliseconds and with faster inference than neural diffusion policies. In experiments on imitation learning benchmarks, we show that CFDP is competitive against neural baselines that require hours of training, providing a favorable tradeoff between training time and performance. Finally, we show how closed-form diffusion policies act as a composable primitive that enables data-driven inference-time editing of pre-trained neural diffusion policies, including policy guidance and novel demonstration augmentation.

Abstract

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