Provedunbiased-attentionrandomized-algorithmssoftmaxdebiasing

Variance of the single-term estimator ≤ its second moment Σ p (D/p)²

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=3}{D, p : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ :

\begin{aligned} & \htmlData{mx=29}{\htmlData{mx=14}{\sum_{l=0}^{\htmlData{mx=1}{L} - 1} \htmlData{mx=13}{\htmlData{mx=4}{\htmlData{mx=2}{p}(\htmlData{mx=3}{l})} \cdot \htmlData{mx=12}{\left(\htmlData{mx=11}{\frac{\htmlData{mx=7}{\htmlData{mx=5}{D}(\htmlData{mx=6}{l})}}{\htmlData{mx=10}{\htmlData{mx=8}{p}(\htmlData{mx=9}{l})}}}\right)^{2}}}} - \htmlData{mx=28}{\left(\htmlData{mx=27}{\sum_{l=0}^{\htmlData{mx=15}{L} - 1} \htmlData{mx=26}{\htmlData{mx=18}{\htmlData{mx=16}{p}(\htmlData{mx=17}{l})} \cdot \left(\htmlData{mx=25}{\frac{\htmlData{mx=21}{\htmlData{mx=19}{D}(\htmlData{mx=20}{l})}}{\htmlData{mx=24}{\htmlData{mx=22}{p}(\htmlData{mx=23}{l})}}}\right)}}\right)^{2}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=43}{\sum_{l=0}^{\htmlData{mx=30}{L} - 1} \htmlData{mx=42}{\htmlData{mx=33}{\htmlData{mx=31}{p}(\htmlData{mx=32}{l})} \cdot \htmlData{mx=41}{\left(\htmlData{mx=40}{\frac{\htmlData{mx=36}{\htmlData{mx=34}{D}(\htmlData{mx=35}{l})}}{\htmlData{mx=39}{\htmlData{mx=37}{p}(\htmlData{mx=38}{l})}}}\right)^{2}}}} \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)
⟹Implies: If ℕ, then ℝ. (Between types, A → B is the type of functions from A to B.) (ℕ → ℝ)
x : TVariables: D, p range over ℕ → ℝ: the statement must hold for every choice. ((D p : ℕ → ℝ))
∑Summation: Add up p(l) · (D(l)/p(l))^2 for l = 0, 1, …, L − 1. (∑ l ∈ Finset.range L, p l * (D l / p l)…)
-Subtraction: ∑ l, p(l) · (D(l)/p(l))^2 minus (∑ l, p(l) · (D(l)/p(l)))^2. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (∑ l ∈ Finset.range L, p l * (D l / p l)…)
≤At most: (∑ l, p(l) · (D(l)/p(l))^2) − (∑ l, p(l) · (D(l)/p(l)))^2 is less than or equal to ∑ l, p(l) · (D(l)/p(l))^2. (∑ l ∈ Finset.range L, p l * (D l / p l)…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
f(x)Function application: p applied to l. Lean writes f x; papers write f(x). (p l)
·Multiplication: p(l) times (D(l)/p(l))^2. (p l * (D l / p l) ^ 2)
aᵇPower: D(l)/p(l) raised to the power 2. ((D l / p l) ^ 2)
a/bDivision: D(l) divided by p(l). In Lean, dividing by 0 gives 0. (D l / p l)

theorem ua-variance-le-second-moment {L : ℕ} (D p : ℕ → ℝ) :
    ∑ l ∈ Finset.range L, p l * (D l / p l) ^ 2 - (∑ l ∈ Finset.range L, p l * (D l / p l)) ^ 2 ≤ ∑ l ∈ Finset.range L, p l * (D l / p l) ^ 2

Var Z = E Z² − (E Z)² ≤ E Z². Combined with the second-moment identity, Var ≤ Σ M_l/p_l. Part of 'Unbiased randomized attention: cost x variance thresholds' (open theory problem).

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: 10 lines of Lean

Variance of the single-term estimator ≤ its second moment Σ p (D/p)²

How it was proved

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