# Bilevel reinforcement learning via the development of hyper-gradient without lower-level convexity

arXiv: https://arxiv.org/abs/2405.19697 · HTML: https://mlsanity.com/paper/2405.19697/

## Key facts

- Cited by 10 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 68 of its 85 references that are in the map.
- First posted on arXiv 30 May 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 68)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Dota 2 with Large Scale Deep Reinforcement Learning](https://mlsanity.com/paper/1912.06680/) (2019)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Learning robust perceptive locomotion for quadrupedal robots in the wild](https://mlsanity.com/paper/2201.08117/) (2022)
- [PEBBLE: Feedback-Efficient Interactive Reinforcement Learning via Relabeling Experience and Unsupervised Pre-training](https://mlsanity.com/paper/2106.05091/) (2021)
- [A Survey of Inverse Reinforcement Learning: Challenges, Methods and Progress](https://mlsanity.com/paper/1806.06877/) (2018)
- [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)
- [Extrapolating Beyond Suboptimal Demonstrations via Inverse Reinforcement Learning from Observations](https://mlsanity.com/paper/1904.06387/) (2019)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Bridging the Gap Between Value and Policy Based Reinforcement Learning](https://mlsanity.com/paper/1702.08892/) (2017)
- [On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift](https://mlsanity.com/paper/1908.00261/) (2019)
- [Understanding the impact of entropy on policy optimization](https://mlsanity.com/paper/1811.11214/) (2018)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)

## Built on by (most cited first, 10 of 10)

- [On The Sample Complexity Bounds In Bilevel Reinforcement Learning](https://mlsanity.com/paper/2503.17644/) (2025)
- [BiSSL: Enhancing the Alignment Between Self-Supervised Pretraining and Downstream Fine-Tuning via Bilevel Optimization](https://mlsanity.com/paper/2410.02387/) (2024)
- [Sample-Efficient Hypergradient Estimation for Decentralized Bi-Level Reinforcement Learning](https://mlsanity.com/paper/2603.14867/) (2026)
- [Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems](https://mlsanity.com/paper/2605.08006/) (2026)
- [On the Convergence of Self-Improving Online LLM Alignment](https://mlsanity.com/paper/2606.31524/) (2026)
- [From Data to Rewards: a Bilevel Optimization Perspective on Maximum Likelihood Estimation](https://mlsanity.com/paper/2510.07624/) (2025)
- [Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning](https://mlsanity.com/paper/2602.00282/) (2026)
- [Hypergradient-based Bilevel Reinforcement Learning with Improved Sample Complexity](https://mlsanity.com/paper/2607.28849/) (2026)
- [Efficient Hypergradient Descent for Inverse Reinforcement Learning](https://mlsanity.com/paper/2608.11052/) (2026)
- [Learning the Robustness Mechanism with Bilevel Optimization](https://mlsanity.com/paper/2609.35541/) (2026)

## Abstract (arXiv)

Bilevel reinforcement learning (RL), which features intertwined two-level problems, has attracted growing interest recently. The inherent non-convexity of the lower-level RL problem is, however, to be an impediment to developing bilevel optimization methods. By employing the fixed point equation associated with the regularized RL, we characterize the hyper-gradient via fully first-order information, thus circumventing the assumption of lower-level convexity. This, remarkably, distinguishes our development of hyper-gradient from the general AID-based bilevel frameworks since we take advantage of the specific structure of RL problems. Moreover, we design both model-based and model-free bilevel reinforcement learning algorithms, facilitated by access to the fully first-order hyper-gradient. Both algorithms enjoy the convergence rate O(ε-1). To extend the applicability, a stochastic version of the model-free algorithm is proposed, along with results on its iteration and sample complexity. In addition, numerical experiments demonstrate that the hyper-gradient indeed serves as an integration of exploitation and exploration.