Beyond Attention Entropy: Predicting the Damage of Approximating Attention Heads

Sparse and local attention approximations (top-k attention, attention sinks plus a sliding window, KV-cache eviction) are cheap, but their cost depends on which heads they are applied to. A common heuristic decides this from the shape of the attention distribution: a head with low entropy is "focused" and therefore safe to sparsify. We test this heuristic directly. For every (prompt, head) pair we approximate one head at a time and measure the damage to the next-token distribution, for three open models with different attention layouts and sizes (Qwen2.5-1.5B with grouped-query attention, SmolLM2-1.7B with multi-head attention, and Mistral-7B-v0.3 with grouped-query attention), 137 prompts across 13 task types and up to eight approximation budgets, about 1.5 million single-head interventions in total. Attention entropy is a weak predictor of damage (AUROC 0.58-0.81), and when used to choose which heads to approximate it is at best no better than random: on the two smaller models it is worse than random (at half the heads approximated it changes the top-1 answer in 40-53% of prompts versus 31-39% for random choice), while the more approximation-robust 7B model avoids that reversal but is still beaten by the forward-only score below. Instead, damage factorises into three forward-pass quantities: the local output error ||aV - a'V||, an exact identity computable inside the attention kernel; a static per-head gain calibrated offline on unrelated tasks; and the entropy of the model's output. This forward-only score reaches AUROC 0.93-0.96 in every setting, including the 7B model, and recovers 86-107% of the gap between the best local signal and a backward-pass gradient oracle. We further show that 94-98% of the variance in gradient sensitivity is explained by prompt-level output confidence, consistently across all three models, and we document a failure mode: allocating approximations across prompts by fragility fails under aggressive approximation unless each prompt's share is capped.