What Should SFT Learn? Conditional Information Gain for Generalizable Post-Training

Supervised fine-tuning (SFT) remains a widely used approach for post-training large language models, because fixed demonstrations can directly introduce desired reasoning strategies and behavioral patterns. Yet forcing a model to imitate externally provided trajectories may also interfere with behaviors already supported by pretraining, limiting generalization beyond the training distribution. Recent dynamic SFT methods mitigate this issue by reweighting token-level supervision according to model-side signals such as predictive probability, prior support, or distributional compatibility. However, model compatibility is not equivalent to learning value: a token that is weakly supported by the current model may either be an undesirable mismatch or precisely the task-specific decision that the demonstration is intended to teach. We propose CIG-SFT, an information-theoretic framework that estimates the task-conditioned information carried by each demonstrated token. Under teacher forcing, we compare the token likelihood from a frozen reference model with and without access to the task input while keeping the gold solution prefix fixed. The resulting conditional information gain quantifies how much the task itself reduces uncertainty about the demonstrated decision. We then convert this information into a KL-regularized projection of the current policy and derive an equivalent scalar-weighted SFT objective whose logit gradient exactly matches that of the projected target. Across mathematical reasoning and code-generation benchmarks, CIG-SFT consistently outperforms standard SFT and recent adaptive fine-tuning baselines, with particularly strong improvements in generalization to challenging evaluation distributions, while requiring neither additional training data nor online rollouts.