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 $O(N_QN_K)$ 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 $43\times$ and $14\times$ speedups over baseline methods in the forward and backward passes, respectively. Most importantly, StreamKL reduces the extra HBM footprint of attention distillation from $O(N_QN_K)$ to $O(1)$, enabling long-context distillation on a single GPU.

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