Provedunbiased-attentionrandomized-algorithmssoftmaxdebiasing

Single-term randomized MLMC is exactly unbiased: Σ p_l·(ΔY_l/p_l) = Y_L − Y_0

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=3}{Y, p : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , 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=17}{\sum_{l=0}^{\htmlData{mx=0}{L} - 1} \htmlData{mx=16}{\htmlData{mx=3}{\htmlData{mx=1}{p}(\htmlData{mx=2}{l})} \cdot \left(\htmlData{mx=15}{\frac{\htmlData{mx=11}{\htmlData{mx=7}{\htmlData{mx=4}{Y}(\htmlData{mx=6}{\htmlData{mx=5}{l} + 1})} - \htmlData{mx=10}{\htmlData{mx=8}{Y}(\htmlData{mx=9}{l})}}}{\htmlData{mx=14}{\htmlData{mx=12}{p}(\htmlData{mx=13}{l})}}}\right)}} \mathrel{\htmlData{mx=24}{=}} \htmlData{mx=23}{\htmlData{mx=20}{\htmlData{mx=18}{Y}(\htmlData{mx=19}{L})} - \htmlData{mx=22}{\htmlData{mx=21}{Y}(0)}}

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: Y, p range over ℕ → ℝ: the statement must hold for every choice. ((Y p : ℕ → ℝ))
∀For all: The statement after the comma holds for every l. (∀ l < L, p l ≠ 0)
∑Summation: Add up p(l) · ((Y(l + 1) − Y(l))/p(l)) for l = 0, 1, …, L − 1. (∑ l ∈ Finset.range L, p l * ((Y (l + 1)…)
=Equality: ∑ l, p(l) · ((Y(l + 1) − Y(l))/p(l)) and Y(L) − Y(0) are the same. (∑ l ∈ Finset.range L, p l * ((Y (l + 1)…)
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 (Y(l + 1) − Y(l))/p(l). (p l * ((Y (l + 1) - Y l) / p l))
a/bDivision: Y(l + 1) − Y(l) divided by p(l). In Lean, dividing by 0 gives 0. ((Y (l + 1) - Y l) / p l)
-Subtraction: Y(l + 1) minus Y(l). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (Y (l + 1) - Y l)
+Addition: l plus 1. (l + 1)

theorem ua-mlmc-single-term-unbiased {L : ℕ} (Y p : ℕ → ℝ) (hp : ∀ l < L, p l ≠ 0) :
    ∑ l ∈ Finset.range L, p l * ((Y (l + 1) - Y l) / p l) = Y L - Y 0

Draw a level l with probability p_l, return (Y_{l+1} − Y_l)/p_l. Its mean telescopes to Y_L − Y_0 for ANY positive level law: the finite Rhee–Glynn identity behind exact debiasing of a self-normalized attention estimate over nested key sets of size 2^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: 13 lines of Lean

Single-term randomized MLMC is exactly unbiased: Σ p_l·(ΔY_l/p_l) = Y_L − Y_0

How it was proved

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