Sample-Efficient Pareto Front Modeling for Energy-Aware Reinforcement Learning Using Bayesian Optimization

Industrial automation increasingly demands control strategies that balance operational performance with strict energy efficiency requirements. A common approach to solving this multi-objective problem, particularly within the framework of reinforcement learning (RL), is to formulate a single, scalar reward function that linearly combines the competing objectives. However, the manual weighting of these different objectives is heavily reliant on domain intuition, incredibly time-consuming, prone to human bias, and frequently fails to uncover optimal trade-off solutions. This work addresses the critical challenge of automating the weight selection process to systematically and efficiently discover the Pareto front of optimal trade-off policies. We formulate the weight selection process as a multi-objective Bayesian optimization (MOBO) problem and evaluate its sample efficiency against a standard uniform grid search baseline. Using a physical Quanser Aero 2 testbed configured for 1-DoF pitch control, our results demonstrate that the MOBO approach, utilizing the expected hypervolume improvement (qEHVI) acquisition function, consistently outperforms uniform grid sampling. MOBO achieves superior hypervolume and maximum spread, successfully identifying high-quality, diverse trade-off policies with a reduced evaluation budget, thereby enabling highly efficient energy-aware control in complex mechatronic systems.

Objective-Behavior Alignment: Diagnostics for MORL Policy Selection

Real-world decision-making often requires optimizing multiple competing objectives simultaneously. In reinforcement learning (RL), this is typically addressed by combining reward signals into a single scalar objective via a scalarization function, which can be fragile: small changes in the weights can induce drastically different policies. Multi-objective reinforcement learning (MORL) instead produces sets of policies that explicitly represent trade-offs between objectives. However, these policies are typically presented to the decision maker only through their value vectors, which can obscure substantial behavioral variation: policies that induce distinct trajectories may appear indistinguishable when evaluated solely by expected returns. We propose an exploratory diagnostic workflow that automatically highlights behavioral variation along the Pareto front that objective values alone do not reveal, providing both quantitative and visual tools to support policy inspection. We validate our approach on simple grid examples and scale it to continuous control benchmarks, demonstrating that it remains effective as problem complexity increases.

SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front

Scalarization is widely used in multi-objective optimization owing to its simplicity and scalability. In many applications, the goal is to generate solutions that represent diverse user preferences, ideally with uniform coverage of the Pareto front (PF). However, uniformly sampling scalarization weights usually induces non-uniform coverage of the PF. We explain this mismatch through a geometric analysis of the scalarization path. As the scalarization weight varies, the corresponding solutions trace the PF with a generally non-uniform traversal speed. This speed induces an arc-length cumulative distribution function (CDF); inverting this CDF map yields a principled rule for selecting weights that produce uniform PF coverage. Building on this insight, we propose SURF (Sampling Uniformly along the PaReto Front). For structured problems, including bi-objective bandits, we derive closed-form expressions for this CDF map and the resulting PF-aware weight sampling rule. For general problems, SURF alternates between CDF reconstruction and weight sampling. Theoretically, we show that under provable conditions, SURF converges linearly to an unavoidable finite-sampling floor. Empirically, experiments on bandits, multi-objective-gymnasium, and multi-objective LLM alignment demonstrate that SURF efficiently achieves more uniform PF coverage than baselines.