# SuPLE: Robot Learning with Lyapunov Rewards

arXiv: https://arxiv.org/abs/2411.13613 · HTML: https://mlsanity.com/paper/2411.13613/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 20 Nov 2024.

## Builds on (most cited first, 13 of 13)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Diversity is All You Need: Learning Skills without a Reward Function](https://mlsanity.com/paper/1802.06070/) (2018)
- [Dynamics-Aware Unsupervised Discovery of Skills](https://mlsanity.com/paper/1907.01657/) (2019)
- [Variational Intrinsic Control](https://mlsanity.com/paper/1611.07507/) (2016)
- [Variational Information Maximisation for Intrinsically Motivated Reinforcement Learning](https://mlsanity.com/paper/1509.08731/) (2015)
- [Empowerment -- an Introduction](https://mlsanity.com/paper/1310.1863/) (2013)
- [Adversarial Imitation via Variational Inverse Reinforcement Learning](https://mlsanity.com/paper/1809.06404/) (2018)
- [Variational Intrinsic Control Revisited](https://mlsanity.com/paper/2010.03281/) (2020)
- [Efficient Empowerment Estimation for Unsupervised Stabilization](https://mlsanity.com/paper/2007.07356/) (2020)
- [Intrinsic Motivation in Dynamical Control Systems](https://mlsanity.com/paper/2301.00005/) (2022)
- [An Empowerment-based Solution to Robotic Manipulation Tasks with Sparse Rewards](https://mlsanity.com/paper/2010.07986/) (2020)

## Built on by (most cited first, 1 of 1)

- [Bellman Meets Lyapunov: Unsupervised Reinforcement Learning via Mastering Chaos](https://mlsanity.com/paper/2610.02012/) (2026)

## Abstract (arXiv)

The reward function is an essential component in robot learning. Reward directly affects the sample and computational complexity of learning, and the quality of a solution. The design of informative rewards requires domain knowledge, which is not always available. We use the properties of the dynamics to produce system-appropriate reward without adding external assumptions. Specifically, we explore an approach to utilize the Lyapunov exponents of the system dynamics to generate a system-immanent reward. We demonstrate that the `Sum of the Positive Lyapunov Exponents' (SuPLE) is a strong candidate for the design of such a reward. We develop a computational framework for the derivation of this reward, and demonstrate its effectiveness on classical benchmarks for sample-based stabilization of various dynamical systems. It eliminates the need to start the training trajectories at arbitrary states, also known as auxiliary exploration. While the latter is a common practice in simulated robot learning, it is unpractical to consider to use it in real robotic systems, since they typically start from natural rest states such as a pendulum at the bottom, a robot on the ground, etc. and can not be easily initialized at arbitrary states. Comparing the performance of SuPLE to commonly-used reward functions, we observe that the latter fail to find a solution without auxiliary exploration, even for the task of swinging up the double pendulum and keeping it stable at the upright position, a prototypical scenario for multi-linked robots. SuPLE-induced rewards for robot learning offer a novel route for effective robot learning in typical as opposed to highly specialized or fine-tuned scenarios. Our code is publicly available for reproducibility and further research.