# One Step, One Lead: Mitigating Higher-Order Interference in Multi-Domain Reinforcement Learning via Cross-Step Control

arXiv: https://arxiv.org/abs/2609.06469 · HTML: https://mlsanity.com/paper/2609.06469/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 27 of its 37 references that are in the map.
- First posted on arXiv 6 Sep 2026.

## Placed in

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

## Builds on (most cited first, 27 of 27)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Limitations of the Empirical Fisher Approximation for Natural Gradient Descent](https://mlsanity.com/paper/1905.12558/) (2019)
- [Multi-Task Learning as a Bargaining Game](https://mlsanity.com/paper/2202.01017/) (2022)
- [NVIDIA Nemotron Nano 2: An Accurate and Efficient Hybrid Mamba-Transformer Reasoning Model](https://mlsanity.com/paper/2508.14444/) (2025)
- [FAMO: Fast Adaptive Multitask Optimization](https://mlsanity.com/paper/2306.03792/) (2023)
- [Revisiting Reinforcement Learning for LLM Reasoning from A Cross-Domain Perspective](https://mlsanity.com/paper/2506.14965/) (2025)
- [Imbalanced Gradients in RL Post-Training of Multi-Task LLMs](https://mlsanity.com/paper/2510.19178/) (2025)
- [Can One Domain Help Others? A Data-Centric Study on Multi-Domain Reasoning via Reinforcement Learning](https://mlsanity.com/paper/2507.17512/) (2025)
- [Multi-Task GRPO: Reliable LLM Reasoning Across Tasks](https://mlsanity.com/paper/2602.05547/) (2026)
- [CoBa: Convergence Balancer for Multitask Finetuning of Large Language Models](https://mlsanity.com/paper/2410.06741/) (2024)
- [CDRRM: Contrast-Driven Rubric Generation for Reliable and Interpretable Reward Modeling](https://mlsanity.com/paper/2603.08035/) (2026)
- [ResT: Reshaping Token-Level Policy Gradients for Tool-Use Large Language Models](https://mlsanity.com/paper/2509.21826/) (2025)
- [Advancing General-Purpose Reasoning Models with Modular Gradient Surgery](https://mlsanity.com/paper/2602.02301/) (2026)
- [ResRL: Boosting LLM Reasoning via Negative Sample Projection Residual Reinforcement Learning](https://mlsanity.com/paper/2605.00380/) (2026)
- [When Domains Interact: Asymmetric and Order-Sensitive Cross-Domain Effects in Reinforcement Learning for Reasoning](https://mlsanity.com/paper/2602.01365/) (2026)
- [Transferability for General Reasoning: An Automated Curriculum for Multi-Domain RLVR](https://mlsanity.com/paper/2606.25178/) (2026)
- [A Local Perturbation Theory for Cross-Domain Interference and Recovery in Multi-Domain RL](https://mlsanity.com/paper/2606.02398/) (2026)

## Built on by (most cited first, 1 of 1)

- [A helps B while B hurts A: directed transfer in instruction-tuning mixture](https://mlsanity.com/paper/2609.39702/) (2026)

## Abstract (arXiv)

Reinforcement learning (RL) across multiple domains can broaden the reasoning capabilities of large language models (LLMs), yet joint training often degrades individual-domain performance and can destabilize optimization. Existing work typically diagnoses such interference from a single-step view using first-order gradient alignment or curvature-based proxies. We show that this view can miss a critical form of sequential interference: same-point domain gradients may remain nearly orthogonal even when consecutive realized updates partially reverse one another in output space. We further show that consecutive token log-probability footprints recover this interaction directly from adjacent checkpoints as a local second-order interaction in output space, without explicitly reconstructing same-step curvature. Building on this insight, we propose OSOL, which designates a focus domain at each iteration, uses the preceding checkpoint footprint to rank token-level rebound risk, and applies a drift-ranked, adaptively scaled correction within the standard GRPO update. Our analysis shows that this correction suppresses the targeted cross-step output backtracking component. Controlled studies further show that cross-step backtracking is more strongly associated with subsequent task damage than same-point gradient diagnostics, while the preceding footprint ranks future rebound risk more accurately than Hessian-based proxies. On Qwen3-30B-A3B, OSOL reaches a domain-macro average of 0.4822, improving by 5.7% over the strongest compared baseline, without explicit higher-order differentiation.