# ES-C51: Expected Sarsa Based C51 Distributional Reinforcement Learning Algorithm

arXiv: https://arxiv.org/abs/2510.15006 · HTML: https://mlsanity.com/paper/2510.15006/

## 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 14 of its 23 references that are in the map.
- First posted on arXiv 16 Oct 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, 14 of 14)

- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [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 Fundamentals of Experience Replay](https://mlsanity.com/paper/2007.06700/) (2020)
- [An Analysis of Categorical Distributional Reinforcement Learning](https://mlsanity.com/paper/1802.08163/) (2018)
- [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)
- [The Phenomenon of Policy Churn](https://mlsanity.com/paper/2206.00730/) (2022)
- [Conservative and Risk-Aware Offline Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2402.08421/) (2024)
- [Distributional Reward Estimation for Effective Multi-Agent Deep Reinforcement Learning](https://mlsanity.com/paper/2210.07636/) (2022)
- [Monte Carlo Tree Search Algorithms for Risk-Aware and Multi-Objective Reinforcement Learning](https://mlsanity.com/paper/2211.13032/) (2022)
- [Revisiting Experience Replayable Conditions](https://mlsanity.com/paper/2402.10374/) (2024)
- [Safety-Aware Reinforcement Learning for Control via Risk-Sensitive Action-Value Iteration and Quantile Regression](https://mlsanity.com/paper/2506.06954/) (2025)

## Abstract (arXiv)

In most value-based reinforcement learning (RL) algorithms, the agent estimates only the expected reward for each action and selects the action with the highest reward. In contrast, Distributional Reinforcement Learning (DRL) estimates the entire probability distribution of possible rewards, providing richer information about uncertainty and variability. C51 is a popular DRL algorithm for discrete action spaces. It uses a Q-learning approach, where the distribution is learned using a greedy Bellman update. However, this can cause problems if multiple actions at a state have similar expected reward but with different distributions, as the algorithm may not learn a stable distribution. This study presents a modified version of C51 (ES-C51) that replaces the greedy Q-learning update with an Expected Sarsa update, which uses a softmax calculation to combine information from all possible actions at a state rather than relying on a single best action. This reduces instability when actions have similar expected rewards and allows the agent to learn higher-performing policies. This approach is evaluated on classic control environments from Gym, and Atari-10 games. For a fair comparison, we modify the standard C51's exploration strategy from e-greedy to softmax, which we refer to as QL-C51 (Q- Learning based C51). The results demonstrate that ES-C51 outperforms QL-C51 across many environments.