Higher-Order Action Supervision Makes A Strong Policy Class
Abstract
Modern data-driven decision-making methods, such as imitation learning (IL) and reinforcement learning (RL), have achieved great success in solving many complex tasks. However, these methods often suffer from serious control instability and robustness issues when applied in real-world applications such as robotics and autonomous driving, posing notable challenges for their practical deployment. We argue that this instability issue stems largely from their limitations in solely supervising and optimizing zeroth-order actions (i.e., the action labels), failing to account for higher-order action dynamics and temporal consistency. In this paper, we show that simultaneously supervising both zeroth- and first-order actions can dramatically enhance policies' performance and control robustness. To achieve this, we introduce a novel and elegant loss scheme supported by formal theoretical guarantees that can equip any off-the-shelf policy model (e.g., deterministic, stochastic, or flow policies) with the capability for higher-order action supervision, without requiring any structural modifications. Moreover, our proposed method can serve as a lightweight plug-and-play module that seamlessly integrates with a broad spectrum of existing offline RL frameworks. Extensive evaluations on OGBench and D4RL demonstrate that our approach yields substantial performance and robustness improvements across a wide range of continuous control environments. Notably, our method can also enhance policies' out-of-distribution (OOD) generalization capability in the challenging low-data regime, making it an ideal tool in tackling many real-world control problems.
The following analysis is generated by AI and may contain errors. It is separate from the authors’ abstract and is not peer review.
Research summary
The paper introduces a plug-and-play higher-order action supervision scheme that significantly enhances the robustness and performance of offline RL and IL policies by enforcing temporal consistency through first-order action derivatives. By leveraging Jacobian-vector products and integrating with continuous-time HJB frameworks, the method provides a theoretically grounded and empirically validated solution to control instability, demonstrating substantial gains in low-data regimes and complex high-dimensional tasks.
Methodology
The paper proposes a "higher-order action supervision" framework that augments standard zeroth-order action prediction (mimicking $a_t$) with first-order action supervision (mimicking $\dot{a}_t$ or $a_{t+1}-a_t$). The core insight is leveraging the time derivative of the policy function $\pi(s)$, specifically $\nabla_s \pi(s) \dot{s}$, to provide a supervision signal for action dynamics without modifying the network architecture. This is implemented via Jacobian-vector products (JVPs), which are computationally efficient. The method is generalized to deterministic, stochastic (via Itô calculus/OU processes), and flow-matching policies. It also integrates with the Hamilton-Jacobi-Bellman (HJB) equation for value function regularization in continuous-time RL. The theoretical contribution provides error bounds showing that supervising first-order dynamics reduces policy error accumulation in closed-loop rollouts.
Results & evaluation
The experiments are extensive, covering OGBench (high-dimensional, sparse rewards) and D4RL (standard locomotion). The paper demonstrates significant improvements in both performance scores and robustness (reduced variance, smoother action trajectories). Notably, it shows substantial gains in low-data regimes (10k transitions), where standard methods often fail due to OOD extrapolation errors, while the proposed method recovers expert behavior. Ablations confirm the contribution of both the policy-side and value-side first-order terms. Comparisons against smoothness penalties (CAPS, L2C2) and architectural constraints (LipsNet) show the proposed method is more effective and less prone to underfitting or OOD errors.
Reproducibility
The paper provides detailed derivations in the appendix and specifies the use of JVPs, which are standard in modern frameworks (PyTorch 2.0, JAX). Hyperparameters for the new loss terms are discussed (often set to 1.0 or swept). The code is not explicitly linked in the text provided, but the method is described with sufficient mathematical precision for implementation. The reliance on standard offline RL baselines (TD3+BC, IQL, FQL) makes replication feasible.
Limitations
The method relies on finite-difference approximations for state and action derivatives, which can be noisy with coarse sampling. It is primarily evaluated on continuous control tasks; applicability to discrete actions or high-dimensional visual inputs (pixels) is limited without a good state representation. The computational cost increases slightly due to JVP calculations, though it remains lightweight. The theoretical guarantees assume certain smoothness conditions on the policy and environment dynamics.
Broader impact
This work addresses a critical bottleneck in deploying learned policies to real-world robotics and autonomous systems: control instability and jitter. By providing a plug-and-play module that enhances temporal consistency, it has high practical value. It bridges the gap between discrete-time MDP formulations and continuous-time control theory, offering a principled way to incorporate physical priors into neural policy learning. The approach could be widely adopted as a standard regularization term in offline RL pipelines. The paper introduces a plug-and-play higher-order action supervision scheme that significantly enhances the robustness and performance of offline RL and IL policies by enforcing temporal consistency through first-order action derivatives. By leveraging Jacobian-vector products and integrating with continuous-time HJB frameworks, the method provides a theoretically grounded and empirically validated solution to control instability, demonstrating substantial gains in low-data regimes and complex high-dimensional tasks.
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