Implicit Maximum Likelihood Estimation for Real-time Generative Model Predictive Control

Grayson Lee; Minh Bui; Shuzi Zhou; Yankai Li; Mo Chen; Ke Li

cs.ROMar 14, 2026

Implicit Maximum Likelihood Estimation for Real-time Generative Model Predictive Control

Authors: Grayson Lee, Minh Bui, Shuzi Zhou, Yankai Li, Mo Chen, Ke Li

Organizations: School of Computing Science, Simon Fraser University, Burnaby, BC, Canada.

Abstract

Diffusion-based models have recently shown strong performance in trajectory planning, as they are capable of capturing diverse, multimodal distributions of complex behaviors. A key limitation of these models is their slow inference speed, which results from the iterative denoising process. This makes them less suitable for real-time applications such as closed-loop model predictive control (MPC), where plans must be generated quickly and adapted continuously to a changing environment. In this paper, we investigate Implicit Maximum Likelihood Estimation (IMLE) as an alternative generative modeling approach for planning. IMLE offers strong mode coverage while enabling inference that is two orders of magnitude faster, making it particularly well suited for real-time MPC tasks. Our results demonstrate that IMLE achieves competitive performance on standard offline reinforcement learning benchmarks compared to the standard diffusion-based planner, while substantially improving planning speed in both open-loop and closed-loop settings. We further validate IMLE in a closed-loop human navigation scenario, operating in real-time, demonstrating how it enables rapid and adaptive plan generation in dynamic environments. Real-world videos and code are available at https://gmpc-imle.github.io/.

Figures & tables

Fig. 1: IMLE-based planning. (1) Noise is sampled from a Gaussian distribution. (2) Our model generates a set of candidate trajectories from the sampled noise. (3) Sampling-based model predictive control is used to select and refine the top trajectory.

Fig. 2: Illustration of IMLE’s nearest neighbor matching. The method enforces that, for every data point, there exists a generated sample in its neighborhood, ensuring that the generator covers the full data distribution.

Fig. 3: FiLM-conditioned block used throughout the U-Net. Two small MLPs conditioning $c$ to scale $\gamma(c)$ and shift $\beta(c)$ , which modulate the two blocks via FiLM ( $x\mapsto\gamma(c)\odot x+\beta(c)$ ).

Fig. 4: IMLE vs Diffusion. Median per-plan latency (ms) split into generator and guidance on CPU/GPU, averaged over Walker/Hopper/HalfCheetah (Batch Size 64)

Dataset	Environment	Diffuser	IMLE+Exp RW
Medium-Expert	HalfCheetah	88.9 $\pm$ 0.3	91.9 $\pm$ 0.09
	Hopper	103.3 $\pm$ 1.3	104.2 $\pm$ 3.81
	Walker2d	106.9 $\pm$ 0.2	107.9 $\pm$ 0.40
Medium	HalfCheetah	42.8 $\pm$ 0.3	43.1 $\pm$ 0.29
	Hopper	74.3 $\pm$ 1.4	85.0 $\pm$ 4.02
	Walker2d	79.6 $\pm$ 0.55	78.3 $\pm$ 2.75

TABLE I: Performance comparison across MuJoCo locomotion datasets. (batch size 64)

Dataset	Environment	No RW	Lin RW	Exp RW
Medium-Expert	HalfCheetah	87.8 $\pm$ 0.94	90.6 $\pm$ 0.26	91.9 $\pm$ 0.09
	Hopper	102.7 $\pm$ 4.07	111.2 $\pm$ 0.38	104.2 $\pm$ 3.81
	Walker2d	108.8 $\pm$ 0.05	109.0 $\pm$ 0.04	107.9 $\pm$ 0.40
Medium	HalfCheetah	35.5 $\pm$ 0.86	44.0 $\pm$ 0.09	43.1 $\pm$ 0.29
	Hopper	71.1 $\pm$ 5.39	65.0 $\pm$ 4.33	85.0 $\pm$ 4.02
	Walker2d	74.4 $\pm$ 3.23	81.6 $\pm$ 0.58	78.3 $\pm$ 2.75

TABLE II: Performance comparison of reward-weighted IMLE variants.

Dataset	Environment	Diffuser	IMLE
Single Task	U-Maze	113.9 $\pm$ 3.1	124.8 $\pm$ 0.65
	Medium	121.5 $\pm$ 2.7	117.3 $\pm$ 3.53
	Large	123.0 $\pm$ 6.4	129.2 $\pm$ 4.89
Average		119.5	123.7
Multi Task	U-Maze	128.9 $\pm$ 1.8	132.3 $\pm$ 0.97
	Medium	127.2 $\pm$ 3.4	127.8 $\pm$ 2.60

TABLE III: Performance comparison across Maze2D Datasets. (Batch Size 1)

Collision Radius = 0.5 m
Metric	MPPI	CoBL	CFM	IMLE	IMLE+MPPI
Collision Rate (%) $\downarrow$	10.00	10.20	5.80	9.80	4.60
Goal Error (m) $\downarrow$	0.521	0.050	0.431	0.181	0.360
Smoothness (m/s) $\downarrow$	0.772	0.606	0.623	0.397	0.394
Jerk (m/s 3 ) $\downarrow$	1.746	1.407	0.812	0.458	0.479
Collision Radius = 0.7 m

TABLE IV: Planner performance across UCY Scenes. (batch size 64)

Metric	Line	CoBL	CFM	IMLE
Collision Radius = 0.7 m
Collision Rate (%) $\downarrow$	16.40	8.80	11.40	10.20
Goal Error (m) $\downarrow$	0.570	0.431	0.447	0.396
Smoothness (m/s) $\downarrow$	0.760	0.760	0.613	0.394
Jerk (m/s 3 ) $\downarrow$	1.746	0.759	0.787	0.484
Sampling Frequency on CPU (Hz) $\uparrow$	125.00	0.41	2.55	52.63

TABLE V: Performance comparison between different warm-start strategies for MPPI across UCY scenes. (batch size 64)

Fig. 5: Using IMLE-generated plans at 50 Hz, the robot reaches the goal while avoiding collisions. The top row shows the conditioning variables (robot and pedestrian past and current states). We also plot the generated plan and the waypoint tracked by the low-level controller.

Parameter	Value
Latent dimension	16
Channel multipliers	(1,4,16)
Planning horizon	32
IMLE sample factor $K$	4
Learning rate	$1\times 10^{-3}$
Batch size	32

TABLE VI: Key hyperparameters for D4RL locomotion experiments.

Parameter	Value
Latent dimension	3
Channel multipliers	(1,2,4,8)
Planning horizon	256
IMLE sample factor $K$	10
Learning rate	$1\times 10^{-4}$
Batch size	32

TABLE VII: Key hyperparameters for D4RL Maze2D experiments.

Architecture
	IMLE	Diffusion	CFM
Base dimension	32	32	32
Channel multipliers	(1,4)/(1,2,4)/(1,2,4,8)	same	same
Training
Learning rate	$1\times 10^{-4}$	$2\times 10^{-5}$	$2\times 10^{-5}$
Batch size	64	64	64

TABLE VIII: Architecture and training hyperparameters for navigation experiments.

Parameter	Value
Planning horizon	20 / 40 / 80
Time discretization $\Delta t$	0.4 / 0.2 / 0.1 s
Trajectory samples	64
MPPI perturbation samples	32
MPPI temperature	0.5
Safety radius	0.5 / 0.7 m

TABLE IX: Planning and sampling parameters.

Metric	Straight	CoBL	CFM	IMLE
Collision Rate (%)	7.40	6.40	8.40	5.60
Goal Error (m)	0.829	0.913	0.842	0.561
Smoothness (m/s)	1.715	0.891	0.493	0.817
Jerk (m/s 3 )	7.013	0.883	1.141	1.549
CPU Sampling (Hz)	90.91	0.19	1.50	34.48
GPU Sampling (Hz)	125	0.56	2.05	58.82

TABLE X: Performance comparison between warm starts across 500 UCY scenes (batch size 64,

H=40

Metric	Straight	CoBL	CFM	IMLE
Collision Rate (%)	1.40	0.60	1.20	1.40
Goal Error (m)	0.518	0.672	0.580	0.352
Smoothness (m/s)	3.736	1.571	0.465	1.016
Jerk (m/s 3 )	27.997	3.152	2.405	3.505
CPU Sampling (Hz)	50.00	0.10	0.70	21.74
GPU Sampling (Hz)	83.33	0.49	1.94	40.00

TABLE XI: Performance comparison between warm starts across 500 UCY scenes (batch size 64,

H=80

Dataset	Environment	BC	CQL	IQL	DT	TT	MOPO	MOReL	MBOP	Diffuser	IMLE+Exp RW
Medium-Expert	HalfCheetah	55.2	91.6	86.7	86.8	95.0	63.3	53.3	105.9	88.9 $\pm$ 0.3	91.9 $\pm$ 0.09
	Hopper	52.5	105.4	91.5	107.6	110.0	23.7	108.7	55.1	103.3 $\pm$ 1.3	104.2 $\pm$ 3.81
	Walker2d	107.5	108.8	109.6	108.1	101.9	44.6	95.6	70.2	106.9 $\pm$ 0.2	107.9 $\pm$ 0.40
Medium	HalfCheetah	42.6	44.0	47.4	42.6	46.9	42.3	42.1	44.6	42.8 $\pm$ 0.3	43.1 $\pm$ 0.29
	Hopper	52.9	58.5	66.3	67.6	61.1	28.0	95.4	48.8	74.3 $\pm$ 1.4	85.0 $\pm$ 4.02
	Walker2d	75.3	72.5	78.3	74.0	79.0	17.8	77.8	41.0	79.6 $\pm$ 0.55	78.3 $\pm$ 2.75

TABLE XII: Performance comparison across MuJoCo locomotion datasets. Baselines are taken from [ 15 ] . Bold indicates within 5% of the maximum reward.

Setting	Env	MPPI	CQL	IQL	Diffuser	IMLE
Single Task	U-Maze	33.2	5.7	47.4	113.9 $\pm$ 3.1	124.8 $\pm$ 0.65
	Medium	10.2	5.0	34.9	121.5 $\pm$ 2.7	117.3 $\pm$ 3.53
	Large	5.1	12.5	58.6	123.0 $\pm$ 6.4	129.2 $\pm$ 4.89
Single-task Average		16.2	7.7	47.0	119.5	123.7
Multi Task	U-Maze	41.2	—	24.8	128.9 $\pm$ 1.8	132.3 $\pm$ 0.97
	Medium	15.4	—	12.1	127.2 $\pm$ 3.4	127.8 $\pm$ 2.60

TABLE XIII: Performance comparison across Maze2D datasets (batch size 1). Baselines are taken from [ 15 ] .

	Torch		JAX
Task	CPU	GPU	CPU	GPU
Locomotion	32.87	53.52	19.23	87.56
Maze	114.63	101.28	91.04	133.98

TABLE XIV: Sampling frequency (Hz) of Torch and JAX implementations.

Panel of Figure 20 — Fig. 6: Qualitative comparison of trajectory proposals for a representative scene. Each panel shows sampled trajectories and the selected plan.

Maximum Marginal Likelihood Estimation

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May 11, 2026J/K move · Enter open · S save

Muninn: Your Trajectory Diffusion Model But Faster

Gokul Puthumanaillam, Hao Jiang, Ruben Hernandez, Jose Fuentes+3

University of Illinois Urbana-Champaign. · Florida International University. · Providence College.

Trajectory Generation Diffusion-Based Planning Diffusion Model Caching Robot Motion Planning Diffusion Model Inference Acceleration

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Diffusion-based trajectory planners can synthesize rich, multimodal robot motions, but their iterative denoising makes online planning and control prohibitively slow. Existing accelerations either modify the sampler or compress the network--sacrificing plan quality or requiring retraining without accounting for downstream control risk. We address the problem of making diffusion-based trajectory planners fast enough for real-time robot use without retraining the model or sacrificing trajectory quality, and in a way that works across diverse state-space diffusion architectures. Our key insight is that diffusion trajectory planners expose two signals we can exploit: a cheap probe of how their internal trajectory representation changes across steps, and analytic coefficients that describe how denoiser errors affect the sampler's state update. By calibrating the first signal against the second on offline runs, we obtain a per-step score that upper-bounds how far the final trajectory can deviate when we reuse a cached denoiser output, and we treat this bound as an uncertainty budget that we can spend over the denoising process. Building on this insight, we present Muninn, a training-free caching wrapper that tracks this uncertainty budget during sampling and, at each diffusion step, chooses between reusing a cached denoiser output when the predicted deviation is small and recomputing the denoiser when it is not. Across standard benchmarks Muninn delivers up to 4.6x wall-clock speedups across several trajectory diffusion models by reducing denoiser evaluations, while preserving task performance and safety metrics. Muninn further certifies that cached rollouts remain within a specified distance of their full-compute counterparts, and we validate these gains in real-time closed-loop navigation and manipulation hardware deployments. Project page: https://github.com/gokulp01/Muninn.

Abstract

Maximum Marginal Likelihood Estimation

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