AI summaryⓘ
The authors study a problem in reinforcement learning where models learn from reward feedback, but the way this feedback updates the model is not well understood. They discover that models keep certain core weight patterns (the spectrum) from training and mainly change related parts (frames) to learn new tasks. They introduce a method called Isospectral Optimization (ISO) that fixes these core patterns and updates only the frames, which works well both when combining models without extra data and when training faster online. Their approach improves accuracy on reasoning and coding tasks with fewer training steps compared to standard methods. Overall, the authors provide a new way to update models after training by focusing on changing specific components while keeping the main structure fixed.
Reinforcement LearningReward FeedbackModel WeightsSpectral InheritanceSingular Value DecompositionIsospectral OptimizationOffline Model MergingOnline OptimizationAdamWLanguage Models
Authors
Hanqing Zhu, Wenyan Cong, Zhizhou Sha, Sagnik Mukherjee, Xinyuan Song, David González-Martínez, Xiaoxia Wu, Yuandong Tian, Shiwei Liu, David Z. Pan, Zhangyang "Atlas" Wang
Abstract
Reinforcement learning with verifiable rewards (RLVR) is rapidly advancing the reasoning capabilities of language models, yet the optimization layer that converts reward feedback into weight-space updates remains poorly understood. Building on our prior analysis (Zhu et al., 2025), we study this missing layer through the singular structure of model weights and identify spectral inheritance: RLVR can reuse the base model's weight spectra while acquiring new behavior through changes in the associated input and output singular frames. We operationalize spectral inheritance as Isospectral Optimization (ISO), an RLVR-native, fixed-spectrum optimization framework with complementary offline and online instantiations. Offline, ISO-Merger combines the frame changes of shared-base specialists into a single fixed-spectrum model, requiring no post-merge data, rollouts, gradient updates, or on-policy distillation (OPD). It recovers complementary specialist capabilities and achieves the strongest aggregate performance among the compared data-free merging methods. Online, ISO-Optimizer applies a chosen base optimizer, including AdamW and Muon, to the frame variables while keeping the base spectra fixed. Across reasoning and coding tasks ranging from 1.5B to 8B parameters, ISO-Optimizer improves accuracy in the reported runs and reaches matched scores with substantially fewer training steps. On Qwen3-8B-Base, AdamW reaches an aggregate accuracy of 0.495 after 270 training steps. ISO-AdamW reaches the same accuracy after only 100 training steps and improves further to 0.509 after 210 training steps. Together, ISO offers a concrete answer to RLVR's missing optimization layer: rather than inheriting pre-training optimization wholesale, design post-training around the structure of reward-driven adaptation: inherit the spectrum, optimize the frames.