# A Differential Perspective on Distributional Reinforcement Learning

arXiv: https://arxiv.org/abs/2506.03333 · HTML: https://mlsanity.com/paper/2506.03333/

## Key facts

- Cited by 5 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 8 of its 24 references that are in the map.
- First posted on arXiv 3 Jun 2025.

## 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, 8 of 8)

- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Distributional Reinforcement Learning with Quantile Regression](https://mlsanity.com/paper/1710.10044/) (2017)
- [Implicit Quantile Networks for Distributional Reinforcement Learning](https://mlsanity.com/paper/1806.06923/) (2018)
- [An Analysis of Categorical Distributional Reinforcement Learning](https://mlsanity.com/paper/1802.08163/) (2018)
- [Learning and Planning in Average-Reward Markov Decision Processes](https://mlsanity.com/paper/2006.16318/) (2020)
- [An Analysis of Quantile Temporal-Difference Learning](https://mlsanity.com/paper/2301.04462/) (2023)
- [Discounted Reinforcement Learning Is Not an Optimization Problem](https://mlsanity.com/paper/1910.02140/) (2019)
- [Burning RED: Unlocking Subtask-Driven Reinforcement Learning and Risk-Awareness in Average-Reward Markov Decision Processes](https://mlsanity.com/paper/2410.10578/) (2024)

## Built on by (most cited first, 5 of 5)

- [Maximum Entropy Exploration Without the Rollouts](https://mlsanity.com/paper/2603.12325/) (2026)
- [Quotient-Categorical Representations for Bellman-Compatible Average-Reward Distributional Reinforcement Learning](https://mlsanity.com/paper/2605.11289/) (2026)
- [Ergodic Risk Measures: Towards a Risk-Aware Foundation for Continual Reinforcement Learning](https://mlsanity.com/paper/2510.02945/) (2025)
- [Thermodynamics of Reinforcement Learning Curricula](https://mlsanity.com/paper/2603.12324/) (2026)
- [Deep Reinforcement Learning and The Tale of Two Temporal Difference Errors](https://mlsanity.com/paper/2603.21921/) (2026)

## Abstract (arXiv)

To date, distributional reinforcement learning (distributional RL) methods have exclusively focused on the discounted setting, where an agent aims to optimize a discounted sum of rewards over time. In this work, we extend distributional RL to the average-reward setting, where an agent aims to optimize the reward received per time step. In particular, we utilize a quantile-based approach to develop the first set of algorithms that can successfully learn and/or optimize the long-run per-step reward distribution, as well as the differential return distribution of an average-reward MDP. We derive proven-convergent tabular algorithms for both prediction and control, as well as a broader family of algorithms that have appealing scaling properties. Empirically, we find that these algorithms yield competitive and sometimes superior performance when compared to their non-distributional equivalents, while also capturing rich information about the long-run per-step reward and differential return distributions.