# Convergence of Q-value in case of Gaussian rewards

arXiv: https://arxiv.org/abs/2003.03526 · HTML: https://mlsanity.com/paper/2003.03526/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2020.

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

- [Approximation to Deep Q-Network by Stochastic Delay Differential Equations](https://mlsanity.com/paper/2505.00382/) (2025)

## Abstract (arXiv)

In this paper, as a study of reinforcement learning, we converge the Q function to unbounded rewards such as Gaussian distribution. From the central limit theorem, in some real-world applications it is natural to assume that rewards follow a Gaussian distribution , but existing proofs cannot guarantee convergence of the Q-function. Furthermore, in the distribution-type reinforcement learning and Bayesian reinforcement learning that have become popular in recent years, it is better to allow the reward to have a Gaussian distribution. Therefore, in this paper, we prove the convergence of the Q-function under the condition of E[r(s,a)2]<∞, which is much more relaxed than the existing research. Finally, as a bonus, a proof of the policy gradient theorem for distributed reinforcement learning is also posted.