# Risk-Sensitive Deep RL: Variance-Constrained Actor-Critic Provably Finds Globally Optimal Policy

arXiv: https://arxiv.org/abs/2012.14098 · HTML: https://mlsanity.com/paper/2012.14098/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2020.

## Built on by (most cited first, 7 of 7)

- [Demystifying Group Relative Policy Optimization: Its Policy Gradient is a U-Statistic](https://mlsanity.com/paper/2603.01162/) (2026)
- [A Risk-Sensitive Approach to Policy Optimization](https://mlsanity.com/paper/2208.09106/) (2022)
- [RVPO: Risk-Sensitive Alignment via Variance Regularization](https://mlsanity.com/paper/2605.05750/) (2026)
- [STOPS: Short-Term-based Volatility-controlled Policy Search and its Global Convergence](https://mlsanity.com/paper/2201.09857/) (2022)
- [Generalized Fitted Q-Iteration with Clustered Data](https://mlsanity.com/paper/2510.03912/) (2025)
- [Counterfactually Safe Reinforcement Learning](https://mlsanity.com/paper/2605.25114/) (2026)
- [Nonparametric Variance-Penalized Actor-Critic: Statistical Inference for Risk-Sensitive Reinforcement Learning](https://mlsanity.com/paper/2609.14327/) (2026)

## Abstract (arXiv)

While deep reinforcement learning has achieved tremendous successes in various applications, most existing works only focus on maximizing the expected value of total return and thus ignore its inherent stochasticity. Such stochasticity is also known as the aleatoric uncertainty and is closely related to the notion of risk. In this work, we make the first attempt to study risk-sensitive deep reinforcement learning under the average reward setting with the variance risk criteria. In particular, we focus on a variance-constrained policy optimization problem where the goal is to find a policy that maximizes the expected value of the long-run average reward, subject to a constraint that the long-run variance of the average reward is upper bounded by a threshold. Utilizing Lagrangian and Fenchel dualities, we transform the original problem into an unconstrained saddle-point policy optimization problem, and propose an actor-critic algorithm that iteratively and efficiently updates the policy, the Lagrange multiplier, and the Fenchel dual variable. When both the value and policy functions are represented by multi-layer overparameterized neural networks, we prove that our actor-critic algorithm generates a sequence of policies that finds a globally optimal policy at a sublinear rate. Further, We provide numerical studies of the proposed method using two real datasets to back up the theoretical results.