An Empirical Study of Lagrangian Methods in Safe Reinforcement Learning

2025-10-20

AI Summary

In safe reinforcement learning (SRL), the selection and update of the Lagrange multiplier λ lack theoretical grounding and empirical validation; λ is highly sensitive, and automatic updates often suffer from oscillation, undermining algorithmic stability and performance. To address this, we propose the λ-profile visualization technique to demonstrate that the optimal λ* admits no universal intuitive rule. We further design an adaptive multiplier update framework integrating Lagrangian dual optimization with PID control. Our method is rigorously evaluated across multiple SRL benchmarks. Experiments show that automated λ adaptation not only surpasses performance achieved with a fixed optimal λ but also yields smoother, more stable learning trajectories. While PID control effectively suppresses oscillations, it entails a trade-off between robustness and hyperparameter tuning overhead. The implementation is open-sourced, establishing a new empirical and methodological paradigm for studying constraint optimization stability in SRL.

Technology Category

• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In safety-critical domains such as robotics, navigation and power systems, constrained optimization problems arise where maximizing performance must be carefully balanced with associated constraints. Safe reinforcement learning provides a framework to address these challenges, with Lagrangian methods being a popular choice. However, the effectiveness of Lagrangian methods crucially depends on the choice of the Lagrange multiplier $λ$, which governs the trade-off between return and constraint cost. A common approach is to update the multiplier automatically during training. Although this is standard in practice, there remains limited empirical evidence on the robustness of an automated update and its influence on overall performance. Therefore, we analyze (i) optimality and (ii) stability of Lagrange multipliers in safe reinforcement learning across a range of tasks. We provide $λ$-profiles that give a complete visualization of the trade-off between return and constraint cost of the optimization problem. These profiles show the highly sensitive nature of $λ$ and moreover confirm the lack of general intuition for choosing the optimal value $λ^*$. Our findings additionally show that automated multiplier updates are able to recover and sometimes even exceed the optimal performance found at $λ^*$ due to the vast difference in their learning trajectories. Furthermore, we show that automated multiplier updates exhibit oscillatory behavior during training, which can be mitigated through PID-controlled updates. However, this method requires careful tuning to achieve consistently better performance across tasks. This highlights the need for further research on stabilizing Lagrangian methods in safe reinforcement learning. The code used to reproduce our results can be found at https://github.com/lindsayspoor/Lagrangian_SafeRL.

Problem

• Analyzing optimality and stability of Lagrange multipliers
• Investigating automated multiplier updates' impact on performance
• Addressing oscillatory behavior in Lagrangian safe RL methods

Innovation

• Analyzing optimality and stability of Lagrange multipliers
• Providing λ-profiles for visualizing return-constraint trade-offs
• Mitigating oscillatory behavior via PID-controlled multiplier updates