# The Central Role of the Loss Function in Reinforcement Learning

arXiv: https://arxiv.org/abs/2409.12799 · HTML: https://mlsanity.com/paper/2409.12799/

## Key facts

- Cited by 8 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 49 of its 77 references that are in the map.
- First posted on arXiv 19 Sep 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 49)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Conservative Q-Learning for Offline Reinforcement Learning](https://mlsanity.com/paper/2006.04779/) (2020)
- [A Distributional Perspective on Reinforcement Learning](https://mlsanity.com/paper/1707.06887/) (2017)
- [Implicit Quantile Networks for Distributional Reinforcement Learning](https://mlsanity.com/paper/1806.06923/) (2018)
- [Information-Theoretic Considerations in Batch Reinforcement Learning](https://mlsanity.com/paper/1905.00360/) (2019)
- [Bridging Offline Reinforcement Learning and Imitation Learning: A Tale of Pessimism](https://mlsanity.com/paper/2103.12021/) (2021)
- [Bellman-consistent Pessimism for Offline Reinforcement Learning](https://mlsanity.com/paper/2106.06926/) (2021)
- [FLAMBE: Structural Complexity and Representation Learning of Low Rank MDPs](https://mlsanity.com/paper/2006.10814/) (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)
- [Pessimistic Model-based Offline Reinforcement Learning under Partial Coverage](https://mlsanity.com/paper/2107.06226/) (2021)
- [Hybrid RL: Using Both Offline and Online Data Can Make RL Efficient](https://mlsanity.com/paper/2210.06718/) (2022)
- [Stop Regressing: Training Value Functions via Classification for Scalable Deep RL](https://mlsanity.com/paper/2403.03950/) (2024)
- [Nemotron-4 340B Technical Report](https://mlsanity.com/paper/2406.11704/) (2024)
- [Kinematic State Abstraction and Provably Efficient Rich-Observation Reinforcement Learning](https://mlsanity.com/paper/1911.05815/) (2019)
- [Adversarially Trained Actor Critic for Offline Reinforcement Learning](https://mlsanity.com/paper/2202.02446/) (2022)
- [When Is Partially Observable Reinforcement Learning Not Scary?](https://mlsanity.com/paper/2204.08967/) (2022)
- [Practical Contextual Bandits with Regression Oracles](https://mlsanity.com/paper/1803.01088/) (2018)
- [The Role of Coverage in Online Reinforcement Learning](https://mlsanity.com/paper/2210.04157/) (2022)
- [Model-free Representation Learning and Exploration in Low-rank MDPs](https://mlsanity.com/paper/2102.07035/) (2021)
- [Conservative Offline Distributional Reinforcement Learning](https://mlsanity.com/paper/2107.06106/) (2021)
- [Is Behavior Cloning All You Need? Understanding Horizon in Imitation Learning](https://mlsanity.com/paper/2407.15007/) (2024)
- [On Oracle-Efficient PAC RL with Rich Observations](https://mlsanity.com/paper/1803.00606/) (2018)
- [Exponential Lower Bounds for Planning in MDPs With Linearly-Realizable Optimal Action-Value Functions](https://mlsanity.com/paper/2010.01374/) (2020)
- [Offline Reinforcement Learning: Fundamental Barriers for Value Function Approximation](https://mlsanity.com/paper/2111.10919/) (2021)
- [REBEL: Reinforcement Learning via Regressing Relative Rewards](https://mlsanity.com/paper/2404.16767/) (2024)
- [Improving Regression Performance with Distributional Losses](https://mlsanity.com/paper/1806.04613/) (2018)
- [Efficient First-Order Contextual Bandits: Prediction, Allocation, and Triangular Discrimination](https://mlsanity.com/paper/2107.02237/) (2021)
- [Fast Rates for Contextual Linear Optimization](https://mlsanity.com/paper/2011.03030/) (2020)
- [Conditional Language Policy: A General Framework for Steerable Multi-Objective Finetuning](https://mlsanity.com/paper/2407.15762/) (2024)

## Built on by (most cited first, 8 of 8)

- [Bigger, Regularized, Categorical: High-Capacity Value Functions are Efficient Multi-Task Learners](https://mlsanity.com/paper/2505.23150/) (2025)
- [Diffusing States and Matching Scores: A New Framework for Imitation Learning](https://mlsanity.com/paper/2410.13855/) (2024)
- [Qsharp: Provably Optimal Distributional RL for LLM Post-Training](https://mlsanity.com/paper/2502.20548/) (2025)
- [Second Order Bounds for Contextual Bandits with Function Approximation](https://mlsanity.com/paper/2409.16197/) (2024)
- [Catoni Contextual Bandits are Robust to Heavy-tailed Rewards](https://mlsanity.com/paper/2502.02486/) (2025)
- [SquarechiPO: Differentially Private and Robust chi2-Preference Optimization in Offline Direct Alignment](https://mlsanity.com/paper/2505.21395/) (2025)
- [Eluder dimension: localise it!](https://mlsanity.com/paper/2601.09825/) (2026)
- [Efficient Controllable Diffusion via Optimal Classifier Guidance](https://mlsanity.com/paper/2505.21666/) (2025)

## Abstract (arXiv)

This paper illustrates the central role of loss functions in data-driven decision making, providing a comprehensive survey on their influence in cost-sensitive classification (CSC) and reinforcement learning (RL). We demonstrate how different regression loss functions affect the sample efficiency and adaptivity of value-based decision making algorithms. Across multiple settings, we prove that algorithms using the binary cross-entropy loss achieve first-order bounds scaling with the optimal policy's cost and are much more efficient than the commonly used squared loss. Moreover, we prove that distributional algorithms using the maximum likelihood loss achieve second-order bounds scaling with the policy variance and are even sharper than first-order bounds. This in particular proves the benefits of distributional RL. We hope that this paper serves as a guide analyzing decision making algorithms with varying loss functions, and can inspire the reader to seek out better loss functions to improve any decision making algorithm.