Symbolic Mechanistic Data Attribution: Tracing Training Influence to Learned Behavioral Policies

Abstract

While existing data attribution methods can identify which training examples build specific mechanistic circuits, they cannot explain how training data shapes the behavioral decisions a model learns. We introduce Symbolic Mechanistic Data Attribution (SMDA), a framework that attributes training pairs to the interpretable symbolic policies governing model behavior. SMDA fits a closed-form Ridge regression over sparse autoencoder (SAE) features to model a behavior, then decomposes how each supervised fine-tuning (SFT) example shifts that policy through feature-activation ($\Delta X$) and output-probability ($\Delta Y$) pathways. We distill a symbolic policy for refusal in Llama-3.2-3B-Instruct and analyze 200 SFT pairs. Our analysis reveals (1)~the symbolic policy's coefficients expose systematic gaps in the base model's safety behavior, such as religious stereotyping; (2)~per-feature $\Delta X/\Delta Y$ decomposition mechanistically explains why harmful and harmless pairs exert qualitatively different influences on certain features; and (3)~individual pairs routinely exhibit cross-feature interference, allowing SMDA to identify pairs whose dominant effect falls on unintended features. Retraining without such pairs improved accuracy by up to 4.5 percentage points over full fine-tuning. These results show that combining mechanistic interpretability with data attribution yields a diagnostic tool both more fine-grained than black-box influence functions and more scalable than manual circuit analysis.