# An Information-Theoretic Definition for Open-Ended Learning

arXiv: https://arxiv.org/abs/2606.08369 · HTML: https://mlsanity.com/paper/2606.08369/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 49 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 28 of 28)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Curiosity-driven Exploration by Self-supervised Prediction](https://mlsanity.com/paper/1705.05363/) (2017)
- [Exploration by Random Network Distillation](https://mlsanity.com/paper/1810.12894/) (2018)
- [Darwin Godel Machine: Open-Ended Evolution of Self-Improving Agents](https://mlsanity.com/paper/2505.22954/) (2025)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Learning to Optimize Via Posterior Sampling](https://mlsanity.com/paper/1301.2609/) (2013)
- [Paired Open-Ended Trailblazer (POET): Endlessly Generating Increasingly Complex and Diverse Learning Environments and Their Solutions](https://mlsanity.com/paper/1901.01753/) (2019)
- [Towards End-to-End Automation of AI Research](https://mlsanity.com/paper/2606.15497/) (2026)
- [On Information Gain and Regret Bounds in Gaussian Process Bandits](https://mlsanity.com/paper/2009.06966/) (2020)
- [RIDE: Rewarding Impact-Driven Exploration for Procedurally-Generated Environments](https://mlsanity.com/paper/2002.12292/) (2020)
- [Enhanced POET: Open-Ended Reinforcement Learning through Unbounded Invention of Learning Challenges and their Solutions](https://mlsanity.com/paper/2003.08536/) (2020)
- [Model-Based Active Exploration](https://mlsanity.com/paper/1810.12162/) (2018)
- [Motif: Intrinsic Motivation from Artificial Intelligence Feedback](https://mlsanity.com/paper/2310.00166/) (2023)
- [A Definition of Continual Reinforcement Learning](https://mlsanity.com/paper/2307.11046/) (2023)
- [Autotelic Agents with Intrinsically Motivated Goal-Conditioned Reinforcement Learning: a Short Survey](https://mlsanity.com/paper/2012.09830/) (2020)
- [Open-Endedness is Essential for Artificial Superhuman Intelligence](https://mlsanity.com/paper/2406.04268/) (2024)
- [A Self-Improving Coding Agent](https://mlsanity.com/paper/2504.15228/) (2025)
- [OMNI-EPIC: Open-endedness via Models of human Notions of Interestingness with Environments Programmed in Code](https://mlsanity.com/paper/2405.15568/) (2024)
- [OMNI: Open-endedness via Models of human Notions of Interestingness](https://mlsanity.com/paper/2306.01711/) (2023)
- [Simple regret for infinitely many armed bandits](https://mlsanity.com/paper/1505.04627/) (2015)
- [Satisficing in Time-Sensitive Bandit Learning](https://mlsanity.com/paper/1803.02855/) (2018)
- [MCU: An Evaluation Framework for Open-Ended Game Agents](https://mlsanity.com/paper/2310.08367/) (2023)
- [Deciding What to Learn: A Rate-Distortion Approach](https://mlsanity.com/paper/2101.06197/) (2021)
- [Discovering Temporal Structure: An Overview of Hierarchical Reinforcement Learning](https://mlsanity.com/paper/2506.14045/) (2025)
- [A Definition of Open-Ended Learning Problems for Goal-Conditioned Agents](https://mlsanity.com/paper/2311.00344/) (2023)
- [Boundless Socratic Learning with Language Games](https://mlsanity.com/paper/2411.16905/) (2024)
- [On Creativity and Open-Endedness](https://mlsanity.com/paper/2405.18016/) (2024)
- [Prior Diffusiveness and Regret in the Linear-Gaussian Bandit](https://mlsanity.com/paper/2601.02022/) (2026)

## Abstract (arXiv)

A growing body of work points to the great promise of AI systems that can continually expand their capabilities as they operate in an open-ended environment. But yet there is no coherent definition of open-endedness or theory about how an agent ought to explore an open-ended environment. We introduce an information-theoretic definition based on a new concept -- the textit bit-equivalent -- which quantifies the information required to attain each level of expected reward. We consider an environment to be open-ended if an agent can attain linear growth in the bit-equivalent. We establish that classical bandit environments are not open-ended and formulate a bandit environment that is. We also introduce an algorithm that achieves open-ended learning in this environment.