Don't Ignore the Tail: Decoupled Distillation Produces Top Maths Students on an Academic Budget

The core learning signal used in language model distillation is the standard Kullback-Leibler (KL) divergence between the student and teacher distributions. Traditional KL divergence tends to be dominated by the teacher’s highest-probability modes, thereby diminishing the influence of less-probable yet potentially informative components of the output distribution. We propose a new tail-aware divergence that decouples the contribution of the teacher model's top-$K$ predicted probabilities from that of lower-probability predictions, while maintaining the same computational profile as the KL Divergence. Our decoupled approach reduces the impact of teacher modes and, consequently, increases the contribution of the distribution's tail. Experimental results demonstrate that our modified distillation method yields competitive performance in both pre-training and supervised distillation for mathematical reasoning of decoder models across various datasets. Furthermore, the distillation process is efficient and can be performed on modest academic budgets for large datasets, drastically reducing the computational costs typically associated with large-scale distillation.