# Variance-Reduced Cascade Q-learning: Algorithms and Sample Complexity

arXiv: https://arxiv.org/abs/2408.06544 · HTML: https://mlsanity.com/paper/2408.06544/

## 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 22 of its 55 references that are in the map.
- First posted on arXiv 13 Aug 2024.

## 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, 22 of 22)

- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [End-to-End Training of Deep Visuomotor Policies](https://mlsanity.com/paper/1504.00702/) (2015)
- [Deep Reinforcement Learning for Autonomous Driving: A Survey](https://mlsanity.com/paper/2002.00444/) (2020)
- [Minimax Regret Bounds for Reinforcement Learning](https://mlsanity.com/paper/1703.05449/) (2017)
- [Is Q-learning Provably Efficient?](https://mlsanity.com/paper/1807.03765/) (2018)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [Katyusha: The First Direct Acceleration of Stochastic Gradient Methods](https://mlsanity.com/paper/1603.05953/) (2016)
- [Breaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model](https://mlsanity.com/paper/2005.12900/) (2020)
- [Finite-Time Analysis of Asynchronous Stochastic Approximation and Q-Learning](https://mlsanity.com/paper/2002.00260/) (2020)
- [Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction](https://mlsanity.com/paper/2006.03041/) (2020)
- [Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis](https://mlsanity.com/paper/2102.06548/) (2021)
- [Stochastic approximation with cone-contractive operators: Sharp ℓ_∞-bounds for Q-learning](https://mlsanity.com/paper/1905.06265/) (2019)
- [Variance-reduced Q-learning is minimax optimal](https://mlsanity.com/paper/1906.04697/) (2019)
- [Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis](https://mlsanity.com/paper/2003.07337/) (2020)
- [A Statistical Analysis of Polyak-Ruppert Averaged Q-learning](https://mlsanity.com/paper/2112.14582/) (2021)
- [Instance-optimality in optimal value estimation: Adaptivity via variance-reduced Q-learning](https://mlsanity.com/paper/2106.14352/) (2021)
- [Optimal variance-reduced stochastic approximation in Banach spaces](https://mlsanity.com/paper/2201.08518/) (2022)
- [KL-Entropy-Regularized RL with a Generative Model is Minimax Optimal](https://mlsanity.com/paper/2205.14211/) (2022)
- [Model-Based Reinforcement Learning for Offline Zero-Sum Markov Games](https://mlsanity.com/paper/2206.04044/) (2022)
- [Optimal Control of Logically Constrained Partially Observable and Multi-Agent Markov Decision Processes](https://mlsanity.com/paper/2305.14736/) (2023)

## Abstract (arXiv)

We study the problem of estimating the optimal Q-function of γ-discounted Markov decision processes (MDPs) under the synchronous setting, where independent samples for all state-action pairs are drawn from a generative model at each iteration. We introduce and analyze a novel model-free algorithm called Variance-Reduced Cascade Q-learning (VRCQ). VRCQ comprises two key building blocks: (i) the established direct variance reduction technique and (ii) our proposed variance reduction scheme, Cascade Q-learning. By leveraging these techniques, VRCQ provides superior guarantees in the ℓ_∞-norm compared with the existing model-free stochastic approximation-type algorithms. Specifically, we demonstrate that VRCQ is minimax optimal. Additionally, when the action set is a singleton (so that the Q-learning problem reduces to policy evaluation), it achieves non-asymptotic instance optimality while requiring the minimum number of samples theoretically possible. Our theoretical results and their practical implications are supported by numerical experiments.