Hybrid Energy-Aware Reward Shaping: A Unified Lightweight Physics-Guided Methodology for Policy Optimization

Reward shaping is fundamental to modern robotic control with deep reinforcement learning (RL), yet practitioners still rely heavily on heuristic principles borrowed from classical optimal control and trajectory optimization. Existing methods rarely distinguish reward terms that are intrinsic to the control objective from numerical regularizers, leading to brittle hyperparameter tuning. To determine which quantities a reward must contain, we study the stabilization control problem with a focus on zeroth-order (configuration) and first-order (velocity) information. We theoretically and empirically demonstrate that policy gradient methods can successfully solve stabilization tasks without first-order reward terms, adding such terms can instead introduce severe sensitivity as their scale grows. Conversely, our findings confirm that reward functions must be zeroth-order complete over goal-relevant coordinates, while the first-order state remains necessary in the policy observation under our low-dissipation assumptions. Overall, these results provide actionable and principled guidance for reward design in robotic RL.

We study reinforcement learning in hybrid discrete-continuous action spaces, such as settings where the discrete component selects a regime (or index) and the continuous component optimizes within it -- a structure common in robotics, control, and operations problems. Standard model-free policy gradient methods rely on score-function (SF) estimators and suffer from severe credit-assignment issues in high-dimensional settings, leading to poor gradient quality. On the other hand, differentiable simulation largely sidesteps these issues by backpropagating through a simulator, but the presence of discrete actions or non-smooth dynamics yields biased or uninformative gradients. To address this, we propose Hybrid Policy Optimization (HPO), which backpropagates through the simulator wherever smoothness permits, using a mixed gradient estimator that combines pathwise and SF gradients while maintaining unbiasedness. We also show how problems with action discontinuities can be reformulated in hybrid form, further broadening its applicability. Empirically, HPO substantially outperforms PPO on inventory control and switched linear-quadratic regulator problems, with performance gaps increasing as the continuous action dimension grows. Finally, we characterize the structure of the mixed gradient, showing that its cross term -- which captures how continuous actions influence future discrete decisions -- becomes negligible near a discrete best response, thereby enabling approximate decentralized updates of the continuous and discrete components and reducing variance near optimality. All resources are available at github.com/MatiasAlvo/hybrid-rl.

Sparse, delayed, and weakly informative rewards remain central obstacles to efficient reinforcement learning. Reward shaping addresses these limitations by supplementing the task reward with an auxiliary signal that can accelerate learning while, in the classical setting, the original objective remains the evaluation criterion. Established theory guarantees safety for fixed shaping signals: potential-based reward shaping preserves optimal policies when the auxiliary term is the discounted difference of a time-invariant potential. In contemporary reinforcement learning systems, however, both the learner and the information available for guidance evolve during training: value estimates improve, novelty diminishes, feedback shifts, and predictive models are refined. Adaptive reward mechanisms occur across exploration, Bayesian inference, human-in-the-loop learning, automated reward design, and foundation-model-based approaches. This study introduces a unified analytical framework for comparing dynamic reward shaping and neighbouring adaptive reward mechanisms. The proposed framework distinguishes parametric revision from state-dependent variation, separates additive shaping from reward replacement and reward-adjacent guidance, and organises existing methods along temporal, informational, and theoretical dimensions. Using this framework, twelve method families are comparatively analysed. The framework further highlights the conditions under which optimality guarantees survive contemporary deep reinforcement learning pipelines, replay buffers, bootstrapped critics, and reward normalisation, while exposing the unresolved relationship between adaptation rate and learner stability.