# Reasoning without Regret

arXiv: https://arxiv.org/abs/2504.09777 · HTML: https://mlsanity.com/paper/2504.09777/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 51 of its 76 references that are in the map.
- First posted on arXiv 14 Apr 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 51)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [DeepSeek-Coder: When the Large Language Model Meets Programming -- The Rise of Code Intelligence](https://mlsanity.com/paper/2401.14196/) (2024)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Towards Large Reasoning Models: A Survey of Reinforced Reasoning with Large Language Models](https://mlsanity.com/paper/2501.09686/) (2025)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [Rewarding Progress: Scaling Automated Process Verifiers for LLM Reasoning](https://mlsanity.com/paper/2410.08146/) (2024)
- [RLHF Workflow: From Reward Modeling to Online RLHF](https://mlsanity.com/paper/2405.07863/) (2024)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [RUDDER: Return Decomposition for Delayed Rewards](https://mlsanity.com/paper/1806.07857/) (2018)
- [Secrets of RLHF in Large Language Models Part II: Reward Modeling](https://mlsanity.com/paper/2401.06080/) (2024)
- [Recursive Introspection: Teaching Language Model Agents How to Self-Improve](https://mlsanity.com/paper/2407.18219/) (2024)
- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Optimizing Test-Time Compute via Meta Reinforcement Fine-Tuning](https://mlsanity.com/paper/2503.07572/) (2025)
- [What and How does In-Context Learning Learn? Bayesian Model Averaging, Parameterization, and Generalization](https://mlsanity.com/paper/2305.19420/) (2023)
- [Confronting Reward Model Overoptimization with Constrained RLHF](https://mlsanity.com/paper/2310.04373/) (2023)
- [DoG is SGD's Best Friend: A Parameter-Free Dynamic Step Size Schedule](https://mlsanity.com/paper/2302.12022/) (2023)
- [Process Reward Models for LLM Agents: Practical Framework and Directions](https://mlsanity.com/paper/2502.10325/) (2025)

## Abstract (arXiv)

Chain-of-thought reasoning enables large language models to solve multi-step tasks by framing problem solving as sequential decision problems. Outcome-based rewards, which provide feedback only on final answers, show impressive success, but face challenges with credit assignment and slow convergence. In contrast, procedure-based rewards offer efficient step-level feedback, but typically require costly human supervision. We introduce Backwards Adaptive Reward Shaping (BARS), a no-regret framework that converts sparse outcomes-based rewards into effective procedure-based signals. BARS uses sparse rewards generated from terminal-state priors and cover trees to scale rewards while preventing exploitation. With Bellman contraction and (Delta, ε)-gap rewards, our backward Euler solver achieves ε-accuracy in Oleft((Rmax/Delta)log(1/ε)right) iterations with O(log T) dynamic regret over T rounds. Our analysis, based on generic chaining, continuous scaling limits, and non-linear Feynman-Kac bounds, connects recent outcome-based methods' empirical successes with the benefits of intermediate supervision. Combined, this provides the first rigorous no-regret algorithm for outcome reward shaping, providing a theoretical foundation for the empirical success of DeepSeek's R1.