# Rethinking Model-based, Policy-based, and Value-based Reinforcement Learning via the Lens of Representation Complexity

arXiv: https://arxiv.org/abs/2312.17248 · HTML: https://mlsanity.com/paper/2312.17248/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 40 of its 65 references that are in the map.
- First posted on arXiv 28 Dec 2023.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [When to Trust Your Model: Model-Based Policy Optimization](https://mlsanity.com/paper/1906.08253/) (2019)
- [MOPO: Model-based Offline Policy Optimization](https://mlsanity.com/paper/2005.13239/) (2020)
- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [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)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [Contextual Decision Processes with Low Bellman Rank are PAC-Learnable](https://mlsanity.com/paper/1610.09512/) (2016)
- [Bellman-consistent Pessimism for Offline Reinforcement Learning](https://mlsanity.com/paper/2106.06926/) (2021)
- [Model-Based Reinforcement Learning with Value-Targeted Regression](https://mlsanity.com/paper/2006.01107/) (2020)
- [Bellman Eluder Dimension: New Rich Classes of RL Problems, and Sample-Efficient Algorithms](https://mlsanity.com/paper/2102.00815/) (2021)
- [The Statistical Complexity of Interactive Decision Making](https://mlsanity.com/paper/2112.13487/) (2021)
- [Provably Efficient Exploration in Policy Optimization](https://mlsanity.com/paper/1912.05830/) (2019)
- [Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes](https://mlsanity.com/paper/2012.08507/) (2020)
- [Tighter Problem-Dependent Regret Bounds in Reinforcement Learning without Domain Knowledge using Value Function Bounds](https://mlsanity.com/paper/1901.00210/) (2019)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [Bilinear Classes: A Structural Framework for Provable Generalization in RL](https://mlsanity.com/paper/2103.10897/) (2021)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [Pessimistic Model-based Offline Reinforcement Learning under Partial Coverage](https://mlsanity.com/paper/2107.06226/) (2021)
- [Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes](https://mlsanity.com/paper/2102.00135/) (2021)
- [Is a Good Representation Sufficient for Sample Efficient Reinforcement Learning?](https://mlsanity.com/paper/1910.03016/) (2019)
- [On the Convergence Rates of Policy Gradient Methods](https://mlsanity.com/paper/2201.07443/) (2022)
- [Is Reinforcement Learning More Difficult Than Bandits? A Near-optimal Algorithm Escaping the Curse of Horizon](https://mlsanity.com/paper/2009.13503/) (2020)
- [PC-PG: Policy Cover Directed Exploration for Provable Policy Gradient Learning](https://mlsanity.com/paper/2007.08459/) (2020)
- [Optimistic Policy Optimization with Bandit Feedback](https://mlsanity.com/paper/2002.08243/) (2020)
- [GEC: A Unified Framework for Interactive Decision Making in MDP, POMDP, and Beyond](https://mlsanity.com/paper/2211.01962/) (2022)
- [Settling the Sample Complexity of Online Reinforcement Learning](https://mlsanity.com/paper/2307.13586/) (2023)

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

- [How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs](https://mlsanity.com/paper/2410.13857/) (2024)
- [Bringing Value Models Back: Generative Critics for Value Modeling in LLM Reinforcement Learning](https://mlsanity.com/paper/2604.10701/) (2026)

## Abstract (arXiv)

Reinforcement Learning (RL) encompasses diverse paradigms, including model-based RL, policy-based RL, and value-based RL, each tailored to approximate the model, optimal policy, and optimal value function, respectively. This work investigates the potential hierarchy of representation complexity -- the complexity of functions to be represented -- among these RL paradigms. We first demonstrate that, for a broad class of Markov decision processes (MDPs), the model can be represented by constant-depth circuits with polynomial size or Multi-Layer Perceptrons (MLPs) with constant layers and polynomial hidden dimension. However, the representation of the optimal policy and optimal value proves to be mathsfNP-complete and unattainable by constant-layer MLPs with polynomial size. This demonstrates a significant representation complexity gap between model-based RL and model-free RL, which includes policy-based RL and value-based RL. To further explore the representation complexity hierarchy between policy-based RL and value-based RL, we introduce another general class of MDPs where both the model and optimal policy can be represented by constant-depth circuits with polynomial size or constant-layer MLPs with polynomial size. In contrast, representing the optimal value is mathsfP-complete and intractable via a constant-layer MLP with polynomial hidden dimension. This accentuates the intricate representation complexity associated with value-based RL compared to policy-based RL. In summary, we unveil a potential representation complexity hierarchy within RL -- representing the model emerges as the easiest task, followed by the optimal policy, while representing the optimal value function presents the most intricate challenge.