StreamKL: Fast and Memory-Efficient KL Divergence for Boosting Attention Distillation

Attention distillation, which trains one attention distribution to match another by minimizing their Kullback-Leibler (KL) divergence, is widely used in knowledge distillation, model compression, continual learning, and sparse-attention LLM training. However, existing approaches materialize both attention distributions before computing the KL reduction, incurring O(NQNK) memory and IO costs that become prohibitive at long context lengths. We present StreamKL, the first fused GPU primitive for attention KL divergence that eliminates this quadratic materialization. StreamKL derives a novel online formulation for the coupled two-distribution KL reduction, enabling a single one-pass forward kernel that streams query-key tiles through on-chip SRAM. For the backward pass, StreamKL recomputes attention probabilities tile-by-tile, avoiding storage of quadratic intermediates. We further design and implement efficient GPU kernels with dedicated optimizations. Experiments show StreamKL delivers up to 43× and 14× speedups over baseline methods in the forward and backward passes, respectively. Most importantly, StreamKL reduces the extra HBM footprint of attention distillation from O(NQNK) to O(1), enabling long-context distillation on a single GPU.