Provedunbiased-attentionrandomized-algorithmssoftmaxdebiasing

Second moment of the single-term estimator is Σ M_l / p_l

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=3}{M, p : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , if $\htmlData{mx=5}{\forall l < \htmlData{mx=4}{L},\ \htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{l})} \mathrel{\htmlData{mx=3}{\neq}} 0}$ , then

\htmlData{mx=13}{\sum_{l=0}^{\htmlData{mx=0}{L} - 1} \htmlData{mx=12}{\htmlData{mx=3}{\htmlData{mx=1}{p}(\htmlData{mx=2}{l})} \cdot \left(\htmlData{mx=11}{\frac{\htmlData{mx=6}{\htmlData{mx=4}{M}(\htmlData{mx=5}{l})}}{\htmlData{mx=10}{\htmlData{mx=9}{\htmlData{mx=7}{p}(\htmlData{mx=8}{l})}^{2}}}}\right)}} \mathrel{\htmlData{mx=23}{=}} \htmlData{mx=22}{\sum_{l=0}^{\htmlData{mx=14}{L} - 1} \htmlData{mx=21}{\frac{\htmlData{mx=17}{\htmlData{mx=15}{M}(\htmlData{mx=16}{l})}}{\htmlData{mx=20}{\htmlData{mx=18}{p}(\htmlData{mx=19}{l})}}}}

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: M, p range over ℕ → ℝ: the statement must hold for every choice. ((M p : ℕ → ℝ))
∀For all: The statement after the comma holds for every l. (∀ l < L, p l ≠ 0)
∑Summation: Add up p(l) · (M(l)/p(l)^2) for l = 0, 1, …, L − 1. (∑ l ∈ Finset.range L, p l * (M l / p l …)
=Equality: ∑ l, p(l) · (M(l)/p(l)^2) and ∑ l, M(l)/p(l) are the same. (∑ l ∈ Finset.range L, p l * (M 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)
≠Not equal: p(l) and 0 are different. (p l ≠ 0)
·Multiplication: p(l) times M(l)/p(l)^2. (p l * (M l / p l ^ 2))
a/bDivision: M(l) divided by p(l)^2. In Lean, dividing by 0 gives 0. (M l / p l ^ 2)
aᵇPower: p(l) raised to the power 2. (p l ^ 2)

theorem ua-mlmc-second-moment {L : ℕ} (M p : ℕ → ℝ) (hp : ∀ l < L, p l ≠ 0) :
    ∑ l ∈ Finset.range L, p l * (M l / p l ^ 2) = ∑ l ∈ Finset.range L, M l / p l

With M_l = E[(ΔY_l)²], E[Z²] = Σ p_l·M_l/p_l² = Σ M_l/p_l. This is the quantity the level law must control. 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: 13 lines of Lean

Second moment of the single-term estimator is Σ M_l / p_l

How it was proved

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