# Quantile Geometry Regularization for Distributional Reinforcement Learning

arXiv: https://arxiv.org/abs/2605.08182 · HTML: https://mlsanity.com/paper/2605.08182/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 15 of its 25 references that are in the map.
- First posted on arXiv 5 May 2026.

## 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, 15 of 15)

- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Rainbow: Combining Improvements in Deep Reinforcement Learning](https://mlsanity.com/paper/1710.02298/) (2017)
- [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)
- [Revisiting the Arcade Learning Environment: Evaluation Protocols and Open Problems for General Agents](https://mlsanity.com/paper/1709.06009/) (2017)
- [Dopamine: A Research Framework for Deep Reinforcement Learning](https://mlsanity.com/paper/1812.06110/) (2018)
- [Fully Parameterized Quantile Function for Distributional Reinforcement Learning](https://mlsanity.com/paper/1911.02140/) (2019)
- [Statistics and Samples in Distributional Reinforcement Learning](https://mlsanity.com/paper/1902.08102/) (2019)
- [Atari-5: Distilling the Arcade Learning Environment down to Five Games](https://mlsanity.com/paper/2210.02019/) (2022)
- [Robust Quadrupedal Locomotion via Risk-Averse Policy Learning](https://mlsanity.com/paper/2308.09405/) (2023)
- [Distributional Reinforcement Learning via Moment Matching](https://mlsanity.com/paper/2007.12354/) (2020)
- [Robust Unmanned Surface Vehicle Navigation with Distributional Reinforcement Learning](https://mlsanity.com/paper/2307.16240/) (2023)
- [Perturbation-mitigated USV Navigation with Distributionally Robust Reinforcement Learning](https://mlsanity.com/paper/2512.00030/) (2025)
- [Wasserstein Distributionally Robust Quantile Regression](https://mlsanity.com/paper/2603.14991/) (2026)

## Abstract (arXiv)

Quantile-based distributional reinforcement learning methods learn return distributions through sampled quantile regression, but their bootstrapped target quantiles may induce distorted or degenerate distribution estimates. We propose Robust Quantile-based Implicit Quantile Networks (RQIQN), a lightweight Wasserstein distributionally robust enhancement boosted from a quantile estimation perspective. We first reinterpret a snapshot of IQN loss as a collection of local empirical quantile estimation problems over sampled current fractions. We then robustify each local slot with a Wasserstein distributionally robust quantile estimation formulation, yielding a closed-form, fraction-dependent correction to the Bellman target. This correction directly addresses distributional degeneration: its median antisymmetry preserves the risk-neutral quantile average, while its monotonicity enlarges upper-lower quantile gaps and counteracts collapsed distributional spread. RQIQN thus regularizes quantile geometry without changing the underlying value objective or requiring additional sample set reconstruction. Finally, we empirically show that the proposed RQIQN outperforms other existing quantile-based distributional reinforcement learning algorithms in risk-sensitive navigation and Atari games.