What to Reuse in Generative Robot Policies? A Solver-Aware Mathematical Analysis of Training-Free Output Caching for Action Generation
- 1Department of Computer Science, Kookmin University, Seoul 02707, Republic of Korea
- 2Department of Applied Artificial Intelligence, Hansung University, Seoul 02876, Republic of Korea
Abstract
Diffusion- and flow-matching-based generative robot policies, including recent vision–language–action (VLA) models, generate continuous action chunks by repeatedly invoking an action generator over multiple generation steps, which constitutes a major bottleneck for real-time robot control. Training-free output caching—recomputing the generator only at selected steps and reusing its output in between—is an attractive remedy, but its mathematical properties and stability criteria have not been rigorously examined for robot action generation. This paper provides a solver-aware mathematical analysis of what quantity should be reused. We first prove a numerical degeneracy result: for explicit Euler integrators applied to deterministic linear flow-matching heads, blockwise velocity output reuse is exactly equivalent to coarse-grid Euler integration, and hence to native few-step inference; output caching in this regime is a reparameterization of step reduction rather than a distinct numerical algorithm. In contrast, for stochastic ancestral samplers, the equivalence fails and the choice of reuse space becomes mathematically decisive. We prove that the staleness error of a cached noise estimate is amplified into endpoint (clean-action) error by the inverse root signal-to-noise ratio SNR(τ)−1/2=στ/ατ, which diverges at high-noise steps (τ→0), whereas caching the posterior-mean endpoint propagates staleness non-expansively: an exact error recursion shows that the cumulative deviation of the generated action is uniformly bounded by the largest staleness error under endpoint-space caching, while under noise-space caching the recursion is expansive and staleness incurred at high-noise steps is transferred to the final action with unbounded gain. A Tweedie-type sensitivity identity further establishes that the spectral norm of the endpoint Jacobian is directly controlled by the conditional posterior covariance of actions given observation conditioning, explaining why endpoints form a mathematically stable target when conditioned on dense observations. The theoretical framework is instantiated as EndCache, a training-free endpoint-space caching protocol, and validated empirically without any policy retraining: in a controlled study over ~60 Diffusion Policy checkpoints, endpoint-space caching preserves task success (85.9% vs. an 87.2% baseline) while noise-space caching collapses to 12.2%—a 73.7 percentage-point performance gap predicted by our amplification analysis; iso-budget numerical evaluations on two flow-matching VLA models confirm the degeneracy proposition to within 0.5 percentage points; and across diffusion/flow-matching policies and robotic simulation benchmarks, endpoint-space reuse reduces action-generator calls with up to 2.7× end-to-end speedup while keeping task success near baseline.
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BibTeX
@article{kim2026what,
title = {{What to Reuse in Generative Robot Policies? A Solver-Aware Mathematical Analysis of Training-Free Output Caching for Action Generation}},
author = {Jiwon Kim and Jaekwon Lee and Jaekoo Lee},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203677},
url = {https://www.mdpi.com/2227-7390/14/20/3677},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - What to Reuse in Generative Robot Policies? A Solver-Aware Mathematical Analysis of Training-Free Output Caching for Action Generation
AU - Jiwon Kim
AU - Jaekwon Lee
AU - Jaekoo Lee
PY - 2026
DA - 2026/10/10
JO - Mathematics
DO - 10.3390/math14203677
UR - https://www.mdpi.com/2227-7390/14/20/3677
AB - Diffusion- and flow-matching-based generative robot policies, including recent vision–language–action (VLA) models, generate continuous action chunks by repeatedly invoking an action generator over multiple generation steps, which constitutes a major bottleneck for real-time robot control. Training-free output caching—recomputing the generator only at selected steps and reusing its output in between—is an attractive remedy, but its mathematical properties and stability criteria have not been rigorously examined for robot action generation. This paper provides a solver-aware mathematical analysis of what quantity should be reused. We first prove a numerical degeneracy result: for explicit Euler integrators applied to deterministic linear flow-matching heads, blockwise velocity output reuse is exactly equivalent to coarse-grid Euler integration, and hence to native few-step inference; output caching in this regime is a reparameterization of step reduction rather than a distinct numerical algorithm. In contrast, for stochastic ancestral samplers, the equivalence fails and the choice of reuse space becomes mathematically decisive. We prove that the staleness error of a cached noise estimate is amplified into endpoint (clean-action) error by the inverse root signal-to-noise ratio SNR(τ)−1/2=στ/ατ, which diverges at high-noise steps (τ→0), whereas caching the posterior-mean endpoint propagates staleness non-expansively: an exact error recursion shows that the cumulative deviation of the generated action is uniformly bounded by the largest staleness error under endpoint-space caching, while under noise-space caching the recursion is expansive and staleness incurred at high-noise steps is transferred to the final action with unbounded gain. A Tweedie-type sensitivity identity further establishes that the spectral norm of the endpoint Jacobian is directly controlled by the conditional posterior covariance of actions given observation conditioning, explaining why endpoints form a mathematically stable target when conditioned on dense observations. The theoretical framework is instantiated as EndCache, a training-free endpoint-space caching protocol, and validated empirically without any policy retraining: in a controlled study over ~60 Diffusion Policy checkpoints, endpoint-space caching preserves task success (85.9% vs. an 87.2% baseline) while noise-space caching collapses to 12.2%—a 73.7 percentage-point performance gap predicted by our amplification analysis; iso-budget numerical evaluations on two flow-matching VLA models confirm the degeneracy proposition to within 0.5 percentage points; and across diffusion/flow-matching policies and robotic simulation benchmarks, endpoint-space reuse reduces action-generator calls with up to 2.7× end-to-end speedup while keeping task success near baseline.
SN - 2227-7390
ER -