# Deep reinforcement learning under signal temporal logic constraints using Lagrangian relaxation

arXiv: https://arxiv.org/abs/2201.08504 · HTML: https://mlsanity.com/paper/2201.08504/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 21 Jan 2022.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Deep Reinforcement Learning for Robotic Manipulation with Asynchronous Off-Policy Updates](https://mlsanity.com/paper/1610.00633/) (2016)
- [Applications of Deep Reinforcement Learning in Communications and Networking: A Survey](https://mlsanity.com/paper/1810.07862/) (2018)
- [Deep Reinforcement Learning for Multi-Agent Systems: A Review of Challenges, Solutions and Applications](https://mlsanity.com/paper/1812.11794/) (2018)
- [Learning to Walk in the Real World with Minimal Human Effort](https://mlsanity.com/paper/2002.08550/) (2020)
- [LTL2Action: Generalizing LTL Instructions for Multi-Task RL](https://mlsanity.com/paper/2102.06858/) (2021)
- [Q-Learning for Robust Satisfaction of Signal Temporal Logic Specifications](https://mlsanity.com/paper/1609.07409/) (2016)
- [Logically-Constrained Reinforcement Learning](https://mlsanity.com/paper/1801.08099/) (2018)
- [Modular Deep Reinforcement Learning for Continuous Motion Planning with Temporal Logic](https://mlsanity.com/paper/2102.12855/) (2021)
- [Modular Deep Reinforcement Learning with Temporal Logic Specifications](https://mlsanity.com/paper/1909.11591/) (2019)
- [Tractable Reinforcement Learning of Signal Temporal Logic Objectives](https://mlsanity.com/paper/2001.09467/) (2020)
- [Learning from Demonstrations using Signal Temporal Logic](https://mlsanity.com/paper/2102.07730/) (2021)
- [Model-Free Reinforcement Learning for Optimal Control of MarkovDecision Processes Under Signal Temporal Logic Specifications](https://mlsanity.com/paper/2109.13377/) (2021)
- [Model Predictive Control for Signal Temporal Logic Specification](https://mlsanity.com/paper/1703.09563/) (2017)
- [Deep Reinforcement Learning Based Networked Control with Network Delays for Signal Temporal Logic Specifications](https://mlsanity.com/paper/2108.01317/) (2021)

## Built on by (most cited first, 8 of 8)

- [LTL-Constrained Policy Optimization with Cycle Experience Replay](https://mlsanity.com/paper/2404.11578/) (2024)
- [Funnel-based Reward Shaping for Signal Temporal Logic Tasks in Reinforcement Learning](https://mlsanity.com/paper/2212.03181/) (2022)
- [Zero-Shot Trajectory Planning for Signal Temporal Logic Tasks](https://mlsanity.com/paper/2501.13457/) (2025)
- [TGPO: Temporal Grounded Policy Optimization for Signal Temporal Logic Tasks](https://mlsanity.com/paper/2510.00225/) (2025)
- [Neural Controller Synthesis for Signal Temporal Logic Specifications Using Encoder-Decoder Structured Networks](https://mlsanity.com/paper/2212.05200/) (2022)
- [Concurrent Learning of Policy and Unknown Safety Constraints in Reinforcement Learning](https://mlsanity.com/paper/2402.15893/) (2024)
- [Specification-Aware Distribution Shaping for Robotics Foundation Models](https://mlsanity.com/paper/2603.17969/) (2026)
- [Reward Machines for Signal Temporal Logic](https://mlsanity.com/paper/2608.13625/) (2026)

## Abstract (arXiv)

Deep reinforcement learning (DRL) has attracted much attention as an approach to solve optimal control problems without mathematical models of systems. On the other hand, in general, constraints may be imposed on optimal control problems. In this study, we consider the optimal control problems with constraints to complete temporal control tasks. We describe the constraints using signal temporal logic (STL), which is useful for time sensitive control tasks since it can specify continuous signals within bounded time intervals. To deal with the STL constraints, we introduce an extended constrained Markov decision process (CMDP), which is called a τ-CMDP. We formulate the STL-constrained optimal control problem as the τ-CMDP and propose a two-phase constrained DRL algorithm using the Lagrangian relaxation method. Through simulations, we also demonstrate the learning performance of the proposed algorithm.