# Regularized Q-Learning with Linear Function Approximation

arXiv: https://arxiv.org/abs/2401.15196 · HTML: https://mlsanity.com/paper/2401.15196/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 43 of its 69 references that are in the map.
- First posted on arXiv 26 Jan 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, 30 of 43)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [On the Global Convergence Rates of Softmax Policy Gradient Methods](https://mlsanity.com/paper/2005.06392/) (2020)
- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [A Theoretical Analysis of Deep Q-Learning](https://mlsanity.com/paper/1901.00137/) (2019)
- [Understanding the impact of entropy on policy optimization](https://mlsanity.com/paper/1811.11214/) (2018)
- [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://mlsanity.com/paper/2103.06257/) (2021)
- [Neural Policy Gradient Methods: Global Optimality and Rates of Convergence](https://mlsanity.com/paper/1909.01150/) (2019)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [A Finite Time Analysis of Two Time-Scale Actor Critic Methods](https://mlsanity.com/paper/2005.01350/) (2020)
- [Adaptive Trust Region Policy Optimization: Global Convergence and Faster Rates for Regularized MDPs](https://mlsanity.com/paper/1909.02769/) (2019)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning](https://mlsanity.com/paper/1905.11425/) (2019)
- [A Finite-Time Analysis of Q-Learning with Neural Network Function Approximation](https://mlsanity.com/paper/1912.04511/) (2019)
- [Breaking the Deadly Triad with a Target Network](https://mlsanity.com/paper/2101.08862/) (2021)
- [Twice regularized MDPs and the equivalence between robustness and regularization](https://mlsanity.com/paper/2110.06267/) (2021)
- [Non-asymptotic Convergence Analysis of Two Time-scale (Natural) Actor-Critic Algorithms](https://mlsanity.com/paper/2005.03557/) (2020)
- [Maximum-Likelihood Inverse Reinforcement Learning with Finite-Time Guarantees](https://mlsanity.com/paper/2210.01808/) (2022)
- [Sparse Markov Decision Processes with Causal Sparse Tsallis Entropy Regularization for Reinforcement Learning](https://mlsanity.com/paper/1709.06293/) (2017)
- [Two Time-scale Off-Policy TD Learning: Non-asymptotic Analysis over Markovian Samples](https://mlsanity.com/paper/1909.11907/) (2019)
- [Target Network and Truncation Overcome The Deadly Triad in Q-Learning](https://mlsanity.com/paper/2203.02628/) (2022)
- [Reanalysis of Variance Reduced Temporal Difference Learning](https://mlsanity.com/paper/2001.01898/) (2020)

## Built on by (most cited first, 2 of 2)

- [Optimistic Training and Convergence of Q-Learning -- Extended Version](https://mlsanity.com/paper/2602.06146/) (2026)
- [Fast Rates for Offline Q-Learning with KL Regularization](https://mlsanity.com/paper/2610.18839/) (2026)

## Abstract (arXiv)

Regularized Markov Decision Processes serve as models of sequential decision making under uncertainty wherein the decision maker has limited information processing capacity and/or aversion to model ambiguity. With functional approximation, the convergence properties of learning algorithms for regularized MDPs (e.g. soft Q-learning) are not well understood because the composition of the regularized Bellman operator and a projection onto the span of basis vectors is not a contraction with respect to any norm. In this paper, we consider a bi-level optimization formulation of regularized Q-learning with linear functional approximation. The em lower level optimization problem aims to identify a value function approximation that satisfies Bellman's recursive optimality condition and the em upper level aims to find the projection onto the span of basis vectors. This formulation motivates a single-loop algorithm with finite time convergence guarantees. The algorithm operates on two time-scales: updates to the projection of state-action values are `slow' in that they are implemented with a step size that is smaller than the one used for `faster' updates of approximate solutions to Bellman's recursive optimality equation. We show that, under certain assumptions, the proposed algorithm converges to a stationary point in the presence of Markovian noise. In addition, we provide a performance guarantee for the policies derived from the proposed algorithm.