Kolmogorov Regression for Robust Diffusion Policies

Diffusion models, exemplified by algorithms like DDPM (Ho et al., 2020) and score-based generative models (Song & Ermon, 2020), have achieved remarkable success in high-dimensional data synthesis. These models learn the probability distribution of data by gradually corrupting the observable with noise in a forward process, then denoising in reverse, effectively sampling from complex distributions. In control and robotics, recent efforts have adapted diffusion strategies to generate continuous action trajectories, leveraging their ability to handle high-dimensional visual inputs and produce smooth outputs. However, these methods predominantly operate in finite-dimensional Euclidean spaces, where discretization artifacts and density estimation challenges limit their long-term stability and scalability. Theoretical analyses (De Bortoli, 2022; Chen et al., 2023) have established convergence bounds in Wasserstein and KL-divergence metrics, but these bounds degrade with increasing action dimension and horizon length. Consequently, practical deployment in safety-critical systems remains problematic, as small errors accumulate over time, leading to trajectory drift and instability. To address these issues, recent research has explored measure-theoretic and PDE-based approaches, aiming to formulate the problem in a way that is independent of discretization and dimension, thereby providing more robust guarantees and diagnostics.

Despite the success of diffusion models in generative tasks, their direct application to control systems faces significant hurdles. The core issues include the reliance on density estimation in high-dimensional spaces, which is infeasible in infinite-dimensional function spaces, and the accumulation of discretization errors over long horizons. These problems manifest as trajectory drift, instability, and reduced safety in real-world deployments. Existing control-oriented diffusion methods lack rigorous convergence guarantees that are independent of the action dimension, limiting their scalability. Furthermore, there is a lack of effective, real-time diagnostics for detecting model failure or unsafe trajectories during inference, which is critical for safety-critical applications such as robotic manipulation and industrial automation. Addressing these challenges requires a new mathematical framework that can provide dimension-independent guarantees, ensure trajectory regularity, and enable unsupervised failure detection.

This paper introduces a novel infinite-dimensional diffusion policy framework based on the backward Kolmogorov PDE. The key innovations include:

• �� Replacing white Gaussian noise with colored noise derived from the Cholesky factor of the Matérn kernel’s Gram matrix, ensuring sample regularity and physical plausibility.
• �� Employing a precision-weighted Cameron-Martin loss, which aligns the model’s distribution with the Gaussian measure in the function space, guaranteeing convergence independent of the action dimension.
• �� Formulating the policy evolution as a PDE, with the value function u(x,s) satisfying the backward Kolmogorov equation. During inference, the residual R(ˆu) measures PDE violation, serving as a deterministic diagnostic for model fidelity.
• �� Demonstrating that the convergence rate depends only on the effective rank of the kernel, not on the action dimension or discretization resolution, thus providing a dimension-independent guarantee.
• �� Validating the approach on robotic manipulation and manufacturing tasks, showing improvements in reward, stability, and safety metrics, and enabling real-time failure detection without reward signals.

• �� Define the action trajectory space as a Hilbert space H = L2([0,T], R^d_a), with a Gaussian prior μ_0 = N(0, C_μ), where C_μ is constructed via a Matérn kernel (e.g., 3/2 kernel) to encode sample regularity.
• �� Model the forward diffusion process as an Ornstein-Uhlenbeck (OU) semigroup, dX_t = -1/2 X_t dt + dW_{C_μ,t}, where W_{C_μ,t} is a C_μ-Wiener process with spectral decomposition.
• �� Replace standard white noise η ∼ N(0, I) with colored noise η = LN ξ, where LN = chol(K) is the Cholesky factor of the Gram matrix K of the kernel, ensuring the noise respects the covariance structure.
• �� During training, minimize the Cameron-Martin loss L_CM = E[∥C^(-1/2)_μ (η_θ - η)∥^2_H], which aligns the model’s noise distribution with the Gaussian measure, ensuring measure-theoretic consistency.
• �� During inference, generate samples by solving the PDE backward in time, with the value function u(x,s) satisfying the backward Kolmogorov PDE: ∂u/∂s + (-1/2 x, ∇_x u) + 1/2 Tr[C_μ · ∇^2 u] = 0, with boundary condition u(x,t) = f(x).
• �� Compute the residual R(ˆu) at each step as a measure of PDE violation, using automatic differentiation for derivatives and Hutchinson trace estimation for Hessian terms.
• �� Use the residual as a real-time diagnostic to detect deviations from the PDE, indicating potential model failure or unsafe trajectories.
• �� Validate the approach on robotic manipulation (PushT) and manufacturing flow control, comparing convergence, reward, residuals, and safety metrics against baseline methods.

在PushT操控任务中，采用RGB-D图像输入，训练了基于ResNet-18特征的条件Unet模型，使用不同损失函数（MSE、Cameron-Martin和混合损失）进行训练。训练过程中在5个随机种子上进行，评估最大奖励、轨迹漂移和偏微分方程残差。制造线任务中，采用六站CONWIP流控制，验证RMSE、异常检测能力和信噪比。所有实验均对比了不同损失方案的收敛速度、轨迹平滑性和故障检测能力，验证了理论中的维度无关收敛保证。超参数如Matérn核的长度尺度和方差经过调优，确保模型在实际场景中的表现。

实验显示，采用Cameron-Martin损失的模型在推理中最大奖励达0.95，优于传统MSE模型的0.78，提升17%。轨迹漂移指标降低67.6%，表明轨迹平滑性显著增强。在制造线任务中，RMSE降低28.4%，信噪比提升13倍，检测异常事件的召回率达到1.0。偏微分方程残差作为故障预警指标，CM损失的残差最低，验证了其在理论上的优越性。结合Hamilton-Jacobi可达性理论，有效降低死锁事件，系统安全性大幅提升。这些结果验证了方法在长时域控制和工业应用中的实用性和鲁棒性。

该方法适用于机器人操控、制造调度、自动驾驶等需要长时域连续控制的场景。只需定义核函数的参数和噪声结构，即可训练出具有维度无关收敛保证的策略。模型能在复杂环境中实现平滑、稳定的动作生成，增强系统的安全性和可靠性。未来，结合强化学习和自适应核参数调节，有望在无人机、智能制造等更复杂系统中实现更高水平的自主控制和安全保障。

尽管在长时域控制中表现优异，但在极端非线性或非高斯噪声环境下，偏微分方程的适用性和准确性可能受到影响。此外，数值解算复杂度较高，尤其在大规模系统中，计算成本较大。核参数的调优依赖于经验，可能影响模型的泛化能力。未来需开发自适应核参数调节机制和高效数值算法，以提升模型的实用性和扩展性。