Sequential knowledge editing in large language models often causes catastrophic collapse of the model’s general abilities, especially for parameter-modifying methods. Existing approaches mitigate this issue through heuristic constraints on parameter updates, the mechanisms underlying such degradation remain insufficiently understood. In this work, we present a systematic spectral analysis of sequential knowledge editing and show that a model’s general abilities are closely associated with dominant singular directions of pretrained weight matrices. These directions are highly sensitive to perturbations and are progressively disrupted by repeated edits, closely tracking the collapse in both editing efficacy and general performance. Building on this insight, we propose REVIVE, a plug-and-play framework that prevents model collapse by explicitly preserving this dominant subspace. REVIVE analyzes parameter updates in the spectral basis of the original weights and filters out components that would interfere with the dominant subspace. Extensive experiments across multiple models and benchmarks show that REVIVE consistently improves editing efficacy while substantially preserving general abilities under long-horizon sequential editing, including extreme settings with up to 20,000 edits.

Self-taught reasoners (STaRs) enhance the mathematical reasoning abilities of large language models (LLMs) by leveraging self-generated responses for self-training. Recent studies have incorporated reward models to guide response selection or decoding, aiming to obtain higher-quality data. However, they typically allocate a uniform sampling budget across all problems, overlooking the varying utility of problems at different difficulty levels. In this work, we conduct an empirical study and find that problems near the boundary of the LLM’s reasoning capability offer significantly greater learning utility than both easy and overly difficult ones. To identify and exploit such problems, we propose HS-STaR, a Hierarchical Sampling framework for Self-Taught Reasoners. Given a fixed sampling budget, HS-STaR first performs lightweight pre-sampling with a reward-guided difficulty estimation strategy to efficiently identify boundary-level problems. Subsequently, it dynamically reallocates the remaining budget toward these high-utility problems during a re-sampling phase, maximizing the generation of valuable training data. Extensive experiments across multiple reasoning benchmarks and backbone LLMs demonstrate that HS-STaR significantly outperforms other baselines without requiring additional sampling budget.