Provedunbiased-attentiongradient-hardnesssampler-lower-bound

Peaked-row gradient bound: if the row puts mass ≥ 1−δ on one key, |Cov_π(u,k)| ≤ 8 F K δ for |u| ≤ F, |k| ≤ K

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{\pi, u, k : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=3}{j_{0} \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ and $\htmlData{mx=1}{\delta, F, K \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $\htmlData{mx=1}{1 - \htmlData{mx=0}{\delta}} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{\htmlData{mx=2}{\pi}(\htmlData{mx=3}{j_{0}})}$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\delta}$ , $\htmlData{mx=0}{\delta} \mathrel{\htmlData{mx=1}{\leq}} 1$ , $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{u}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{F}}$ and $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{k}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{K}}$ , then

\begin{aligned} & \htmlData{mx=31}{\left\lvert \htmlData{mx=30}{\htmlData{mx=12}{\sum_{j} \htmlData{mx=11}{\htmlData{mx=3}{\htmlData{mx=1}{\pi}(\htmlData{mx=2}{j})} \cdot (\htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=4}{u}(\htmlData{mx=5}{j})} \cdot \htmlData{mx=9}{\htmlData{mx=7}{k}(\htmlData{mx=8}{j})}})}} - \htmlData{mx=29}{\left(\htmlData{mx=20}{\sum_{j} \htmlData{mx=19}{\htmlData{mx=15}{\htmlData{mx=13}{\pi}(\htmlData{mx=14}{j})} \cdot \htmlData{mx=18}{\htmlData{mx=16}{u}(\htmlData{mx=17}{j})}}}\right) \cdot \left(\htmlData{mx=28}{\sum_{j} \htmlData{mx=27}{\htmlData{mx=23}{\htmlData{mx=21}{\pi}(\htmlData{mx=22}{j})} \cdot \htmlData{mx=26}{\htmlData{mx=24}{k}(\htmlData{mx=25}{j})}}}\right)}} \right\rvert} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=37}{\htmlData{mx=35}{\htmlData{mx=33}{8\htmlData{mx=32}{F}} \cdot \htmlData{mx=34}{K}} \cdot \htmlData{mx=36}{\delta}} \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
x : TVariables: π, u, k range over Fin n → ℝ: the statement must hold for every choice. ((π u k : Fin n → ℝ))
x ∈ TVariables: δ, F, K range over real numbers: the statement must hold for every choice. ((δ F K : ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ π j)
∑Summation: Add up π(j) for every j. (∑ j, π j)
=Equality: ∑ j, π(j) and 1 are the same. (∑ j, π j = 1)
-Subtraction: 1 minus δ. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 - δ)
≤At most: 0 is less than or equal to π(j). (0 ≤ π j)
|x|Absolute value: The size of u(j) without its sign. (|u j|)
RReal numbers: The real numbers: every point on the number line. (ℝ)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
·Multiplication: u(j) times k(j). (u j * k j)

theorem ug-peaked-gradient-bound {n : ℕ} (π u k : Fin n → ℝ) (j₀ : Fin n) (δ F K : ℝ) (hπ : ∀ j, 0 ≤ π j) (hπs : ∑ j, π j = 1) (hpeak : 1 - δ ≤ π j₀) (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1) (hu : ∀ j, |u j| ≤ F) (hk : ∀ j, |k j| ≤ K) :
    |∑ j, π j * (u j * k j) - (∑ j, π j * u j) * (∑ j, π j * k j)| ≤ 8 * F * K * δ

The query-gradient coordinate of a peaked (retrieval) head is O(δ) in the off-peak mass: covariance is shift-invariant, so subtracting the peak's values makes every remaining term carry the off-peak mass. Consequence: retrieval heads, the expensive ones for the forward pass, have small gradients; with an exact sampler their gradient variance is O(δ) as well, so the sample count for a fixed absolute target scales with δ, while the per-sample cost (e^R under a uniform proposal) scales with 1/δ: no free lunch, the Retrieval–Speed Duality holds for gradients with the same cost. Part of 'Unbiased randomized attention: cost x variance thresholds', gradient-hardness round: the unconditional sampler lower bound transferred to the query gradient.

How it was proved

Proved: 4 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 87 lines of Lean

Peaked-row gradient bound: if the row puts mass ≥ 1−δ on one key, |Cov_π(u,k)| ≤ 8 F K δ for |u| ≤ F, |k| ≤ K

How it was proved

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