LinearARD: Linear-Memory Attention Distillation for RoPE Restoration

The extension of context windows in Large Language Models is typically facilitated by scaling positional encodings followed by Continued Pre-Training (CPT). While effective, this paradigm is notoriously data-hungry and computationally expensive, requiring massive long-text corpora to recalibrate the model to the shifted positional distribution. We propose LinearARD, a self-distillation method that restores Rotary Position Embedding (RoPE)-scaled students through attention-structure consistency with a frozen native-RoPE teacher. Rather than next-token prediction or opaque hidden-state matching, LinearARD aligns row-wise distributions of dense $Q/Q$, $K/K$, and $V/V$ self-relation matrices from the final attention layer to directly supervise attention dynamics. To remove the quadratic memory bottleneck of $n \times n$ relation maps, we introduce a linear-memory kernel that stores only per-token log-sum-exp statistics and recomputes logits in the backward pass to obtain exact Kullback-Leibler divergence gradients. Across LLaMA2-7B, LLaMA3-8B, and Mistral-7B-v0.1 extended to 32K context, LinearARD recovers 93.1\%/94.2\%/94.3\% of native short-context performance and achieves strong long-context robustness on RULER using only \textbf{4.25M} tokens---amounting to just 1.6\% (a $\sim$60$\times$ reduction) of the 256M-token budget required by state-of-the-art baselines. Under this severely constrained budget, CPT and LongReD remain near zero on RULER, demonstrating that relation-level restoration provides a substantially better efficiency-quality tradeoff.