Ruoxi Jia

In reasoning supervised fine-tuning, candidate responses for the same instruction can differ substantially in how well they match the student’s current distribution. Recent likelihood-based response selection methods suggest that responses closer to the student distribution provide more effective supervision, motivating the hypothesis that high-likelihood responses may generally be preferable for fine-tuning. In this paper, we revisit this intuition and show that the value of likelihood-based data selection depends critically on model capacity and training duration. Through controlled experiments on mathematical reasoning, using students ranging from 1.5B to 8B parameters and supervision generated by stronger teacher models, we observe a clear capacity-dependent `Fast-Fit / Slow-Gain' pattern. High-likelihood data provides faster and more stable early improvements, especially for smaller models, but low-likelihood data becomes increasingly beneficial for larger models when training is allowed to continue longer. To explain this phenomenon, we analyze learning dynamics, showing that small models often fail to absorb low-likelihood supervision and instead fall into shallow or repetitive behaviors, while larger models are better able to move toward the teacher distribution under such data. We further provide a capacity-constrained theoretical view of distillation that clarifies how data difficulty, data span, and student capacity jointly govern transfer. Overall, our findings show that effective data selection for reasoning should be aware of model capacity and computing budget rather than based on a single universal preference for high-likelihood supervision.

Skills are a natural unit for describing what a language model can do and how its behavior can be changed. However, existing characterizations rely on human-written taxonomies, textual descriptions, or manual profiling pipelines---all external hypotheses about what matters that need not align with the model's internal representations. We argue that when the goal is to intervene on model behavior, skill characterization should be \emph{model-native}: grounded in the model's own representations rather than imposed through external ontologies. We instantiate this view by recovering a compact orthogonal basis from sequence-level activations. The resulting basis is semantically interpretable but need not correspond to any predefined human ontology; instead, it captures axes of behavioral variation that the model itself organizes around. We validate this characterization on reasoning post-training, using the recovered basis for both supervised fine-tuning data selection and inference-time steering. We develop lightweight proxy interventions to identify which directions are most useful for a given model. Across Llama3-8B and Qwen2.5-3B, selecting data along those directions improves Pass@1 by up to 20\% on MATH-500 and 41\% on AMC, outperforming data selection based on human-characterized skills. Because the basis lives in activation space, the same directions also serve as steering vectors at inference time, improving Pass@8 by up to 4.8\% on MATH-500---an intervention that human-characterized skills cannot support. We further validate the characterization on safety alignment, where selecting adversarial training data for model-native skill coverage rather than textual diversity yields more sample-efficient learning. These results suggest that recovering skills from the model's own representations, rather than imposing them externally, provides a more effective foundation for intervening on model behavior. Codes are open-sourced.