# Conformal Symplectic Optimization for Stable Reinforcement Learning

arXiv: https://arxiv.org/abs/2412.02291 · HTML: https://mlsanity.com/paper/2412.02291/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 3 Dec 2024.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Improving Generalization Performance by Switching from Adam to SGD](https://mlsanity.com/paper/1712.07628/) (2017)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Distributional Soft Actor-Critic: Off-Policy Reinforcement Learning for Addressing Value Estimation Errors](https://mlsanity.com/paper/2001.02811/) (2020)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives](https://mlsanity.com/paper/2002.12493/) (2020)
- [Integrated Decision and Control: Towards Interpretable and Computationally Efficient Driving Intelligence](https://mlsanity.com/paper/2103.10290/) (2021)
- [On dissipative symplectic integration with applications to gradient-based optimization](https://mlsanity.com/paper/2004.06840/) (2020)
- [On the Optimization Landscape of Dynamic Output Feedback Linear Quadratic Control](https://mlsanity.com/paper/2201.09598/) (2022)
- [Gradient flows and proximal splitting methods: A unified view on accelerated and stochastic optimization](https://mlsanity.com/paper/1908.00865/) (2019)
- [Encoding Distributional Soft Actor-Critic for Autonomous Driving in Multi-lane Scenarios](https://mlsanity.com/paper/2109.05540/) (2021)

## Built on by (most cited first, 4 of 4)

- [Momentum as Residual-Driven Multiplier Correction for Deep Learning Optimization](https://mlsanity.com/paper/2608.12925/) (2026)
- [Jump-Start Reinforcement Learning with Self-Evolving Priors for Extreme Monopedal Locomotion](https://mlsanity.com/paper/2507.01243/) (2025)
- [Real-Time Generative Policy via Langevin-Guided Flow Matching for Autonomous Driving](https://mlsanity.com/paper/2603.02613/) (2026)
- [Augmented Lagrangian Multiplier Network for State-wise Safety in Reinforcement Learning](https://mlsanity.com/paper/2605.00667/) (2026)

## Abstract (arXiv)

Training deep reinforcement learning (RL) agents necessitates overcoming the highly unstable nonconvex stochastic optimization inherent in the trial-and-error mechanism. To tackle this challenge, we propose a physics-inspired optimization algorithm called relativistic adaptive gradient descent (RAD), which enhances long-term training stability. By conceptualizing neural network (NN) training as the evolution of a conformal Hamiltonian system, we present a universal framework for transferring long-term stability from conformal symplectic integrators to iterative NN updating rules, where the choice of kinetic energy governs the dynamical properties of resulting optimization algorithms. By utilizing relativistic kinetic energy, RAD incorporates principles from special relativity and limits parameter updates below a finite speed, effectively mitigating abnormal gradient influences. Additionally, RAD models NN optimization as the evolution of a multi-particle system where each trainable parameter acts as an independent particle with an individual adaptive learning rate. We prove RAD's sublinear convergence under general nonconvex settings, where smaller gradient variance and larger batch sizes contribute to tighter convergence. Notably, RAD degrades to the well-known adaptive moment estimation (ADAM) algorithm when its speed coefficient is chosen as one and symplectic factor as a small positive value. Experimental results show RAD outperforming nine baseline optimizers with five RL algorithms across twelve environments, including standard benchmarks and challenging scenarios. Notably, RAD achieves up to a 155.1% performance improvement over ADAM in Atari games, showcasing its efficacy in stabilizing and accelerating RL training.