How Instruction and Reasoning Data shape Post-Training: Data Quality
  through the Lens of Layer-wise Gradients
Authors: name: Ming Li, name: Yanhong Li, name: Ziyue Li, name: Tianyi Zhou

As the post-training of large language models (LLMs) advances from
instruction-following to complex reasoning tasks, understanding how different
data affect finetuning dynamics remains largely unexplored. In this paper, we
present a spectral analysis of layer-wise gradients induced by low/high-quality
instruction and reasoning data for LLM post-training. Our analysis reveals that
widely-studied metrics for data evaluation, e.g., IFD, InsTag, Difficulty, and
Reward, can be explained and unified by spectral properties computed from
gradients' singular value decomposition (SVD). Specifically, higher-quality
data are usually associated with lower nuclear norms and higher effective
ranks. Notably, effective rank exhibits better robustness and resolution than
nuclear norm in capturing subtle quality differences. For example, reasoning
data achieves substantially higher effective ranks than instruction data,
implying richer gradient structures on more complex tasks. Our experiments also
highlight that models within the same family share similar gradient patterns
regardless of their sizes, whereas different model families diverge
significantly. Providing a unified view on the effects of data quality across
instruction and reasoning data, this work illuminates the interplay between
data quality and training stability, shedding novel insights into developing
better data exploration strategies for post-training.