# Bayesian Risk-Sensitive Policy Optimization For MDPs With General Loss Functions

arXiv: https://arxiv.org/abs/2509.15509 · HTML: https://mlsanity.com/paper/2509.15509/

## 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 11 of its 35 references that are in the map.
- First posted on arXiv 19 Sep 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, 11 of 11)

- [Variational Policy Gradient Method for Reinforcement Learning with General Utilities](https://mlsanity.com/paper/2007.02151/) (2020)
- [Stochastic Variance-Reduced Policy Gradient](https://mlsanity.com/paper/1806.05618/) (2018)
- [Algorithms for CVaR Optimization in MDPs](https://mlsanity.com/paper/1406.3339/) (2014)
- [Policy Gradient for Coherent Risk Measures](https://mlsanity.com/paper/1502.03919/) (2015)
- [On the Convergence and Sample Efficiency of Variance-Reduced Policy Gradient Method](https://mlsanity.com/paper/2102.08607/) (2021)
- [Beyond Confidence Regions: Tight Bayesian Ambiguity Sets for Robust MDPs](https://mlsanity.com/paper/1902.07605/) (2019)
- [Reinforcement Learning with General Utilities: Simpler Variance Reduction and Large State-Action Space](https://mlsanity.com/paper/2306.01854/) (2023)
- [Bayesian Distributionally Robust Optimization](https://mlsanity.com/paper/2112.08625/) (2021)
- [Scalable Primal-Dual Actor-Critic Method for Safe Multi-Agent RL with General Utilities](https://mlsanity.com/paper/2305.17568/) (2023)
- [Bayesian Risk Markov Decision Processes](https://mlsanity.com/paper/2106.02558/) (2021)
- [Risk averse stochastic programming: time consistency and optimal stopping](https://mlsanity.com/paper/1808.10807/) (2018)

## Abstract (arXiv)

Motivated by many application problems, we consider Markov decision processes (MDPs) with a general loss function and unknown parameters. To mitigate the epistemic uncertainty associated with unknown parameters, we take a Bayesian approach to estimate the parameters from data and impose a coherent risk functional (with respect to the Bayesian posterior distribution) on the loss. Since this formulation usually does not satisfy the interchangeability principle, it does not admit Bellman equations and cannot be solved by approaches based on dynamic programming. Therefore, We propose a policy gradient optimization method, leveraging the dual representation of coherent risk measures and extending the envelope theorem to continuous cases. We then show the stationary analysis of the algorithm with a convergence rate of O(T-1/2+r-1/2), where T is the number of policy gradient iterations and r is the sample size of the gradient estimator. We further extend our algorithm to an episodic setting, and establish the global convergence of the extended algorithm and provide bounds on the number of iterations needed to achieve an error bound O(ε) in each episode.