# Trajectory-wise Control Variates for Variance Reduction in Policy Gradient Methods

arXiv: https://arxiv.org/abs/1908.03263 · HTML: https://mlsanity.com/paper/1908.03263/

## Key facts

- Cited by 11 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 8 Aug 2019.

## Built on by (most cited first, 11 of 11)

- [Offline Reinforcement Learning: Tutorial, Review, and Perspectives on Open Problems](https://mlsanity.com/paper/2005.01643/) (2020)
- [On Proximal Policy Optimization's Heavy-tailed Gradients](https://mlsanity.com/paper/2102.10264/) (2021)
- [Dealing with the Unknown: Pessimistic Offline Reinforcement Learning](https://mlsanity.com/paper/2111.05440/) (2021)
- [Rapidly Adapting Policies to the Real World via Simulation-Guided Fine-Tuning](https://mlsanity.com/paper/2502.02705/) (2025)
- [Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions](https://mlsanity.com/paper/2208.04055/) (2022)
- [Offline Policy Optimization with Eligible Actions](https://mlsanity.com/paper/2207.00632/) (2022)
- [Variance Reduction for Policy-Gradient Methods via Empirical Variance Minimization](https://mlsanity.com/paper/2206.06827/) (2022)
- [Constrained Policy Gradient Method for Safe and Fast Reinforcement Learning: a Neural Tangent Kernel Based Approach](https://mlsanity.com/paper/2107.09139/) (2021)
- [Coordinate-wise Control Variates for Deep Policy Gradients](https://mlsanity.com/paper/2107.04987/) (2021)
- [A Multi-Fidelity Control Variate Approach for Policy Gradient Estimation](https://mlsanity.com/paper/2503.05696/) (2025)
- [ABC: Advantage-Based Control Variates for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2609.36058/) (2026)

## Abstract (arXiv)

Policy gradient methods have demonstrated success in reinforcement learning tasks that have high-dimensional continuous state and action spaces. However, policy gradient methods are also notoriously sample inefficient. This can be attributed, at least in part, to the high variance in estimating the gradient of the task objective with Monte Carlo methods. Previous research has endeavored to contend with this problem by studying control variates (CVs) that can reduce the variance of estimates without introducing bias, including the early use of baselines, state dependent CVs, and the more recent state-action dependent CVs. In this work, we analyze the properties and drawbacks of previous CV techniques and, surprisingly, we find that these works have overlooked an important fact that Monte Carlo gradient estimates are generated by trajectories of states and actions. We show that ignoring the correlation across the trajectories can result in suboptimal variance reduction, and we propose a simple fix: a class of "trajectory-wise" CVs, that can further drive down the variance. We show that constructing trajectory-wise CVs can be done recursively and requires only learning state-action value functions like the previous CVs for policy gradient. We further prove that the proposed trajectory-wise CVs are optimal for variance reduction under reasonable assumptions.