Kolmogorov Regression for Robust Diffusion Policies

Problem
This work addresses the limitations of finite-dimensional (FD) diffusion policies, which suffer from temporal drift due to discretization artifacts, particularly affecting long-horizon performance in real-world applications. The authors highlight that existing methods do not adequately mitigate these issues, leading to suboptimal deployment in physical systems. This paper is a preprint and has not undergone peer review.

Method
The authors introduce a backward Kolmogorov equation that elevates diffusion policies into a Cameron-Martin space, a specific subset of Hilbert space. This approach replaces traditional stochastic score matching with a deterministic boundary-value partial differential equation (PDE) problem. The core innovation leverages Gaussian measure theory, where the diffusion noise covariance operator is derived from a colored noise distribution, imposing a regularity constraint on model samples during inference. The diffusion model is trained using a precision-weighted Cameron-Martin loss, while a Kolmogorov residual is employed as a diagnostic tool during inference. This methodology yields convergence guarantees, with bounds dependent on the effective rank of the kernel rather than the action dimension, enhances trajectory regularity through spectral weighting, and introduces a deterministic failure detection mechanism independent of reward signals.

Results
The proposed method was validated across two application domains. In the PushT manipulation benchmark, the Cameron-Martin loss achieved a 17% improvement in maximum episode reward (0.95 vs. 0.78 for mean squared error (MSE)) and a 67.6% reduction in inter-step drifts during inference, as indicated by the introduced residual magnitude. In a 6-station manufacturing line with constant work-in-process (CONWIP) flow control, the method demonstrated a 28.4% reduction in root mean square error (RMSE) compared to classical LSTM baselines, achieving a perfect starvation-event recall (1.0) and effective bottleneck identification (Precision@1 = 1.0 in the test set, with a 13x signal-to-noise ratio). Furthermore, the dispatch policies were certified using Hamilton-Jacobi reachability theory, resulting in a 96% reduction in deadlock events compared to uncontrolled dispatch over 100 simulated runs, preventing 351 events.

Limitations
The authors acknowledge that their approach may require extensive computational resources for training due to the complexity of the backward Kolmogorov equation and the associated loss function. Additionally, while the method shows promise in the tested domains, its generalizability to other tasks and environments remains to be fully explored. The paper does not address potential scalability issues when applied to larger state spaces or more complex environments.

Why it matters
This work has significant implications for the development of robust reinforcement learning policies, particularly in environments where long-horizon decision-making is critical. The introduction of a deterministic framework for diffusion policies could lead to more reliable and interpretable models in robotics and manufacturing, enhancing their deployment in real-world scenarios. The findings contribute to the ongoing discourse on improving policy robustness and performance in reinforcement learning, as published in arXiv.