Provedunbiased-attentionend-to-endsoftmaxdebiasing

END-TO-END debiasing: single-term MLMC Z = Y₀ + (Y_{ℓ+1} − Y_ℓ)/p_ℓ (level ℓ ~ p independent of the sample draw) is exactly unbiased for m_L and E(Z − m_L)² ≤ 2E(Y₀ − m₀)² + 2Σ_ℓ D_ℓ/p_ℓ

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{\Omega \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{w : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , $\htmlData{mx=3}{p : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , $\htmlData{mx=5}{Y : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{\Omega} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=3}{m, D : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall \omega,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})}}$ , $\htmlData{mx=3}{\sum_{\omega} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $\htmlData{mx=8}{\forall l,\ \htmlData{mx=0}{l} \mathrel{\htmlData{mx=2}{<}} \htmlData{mx=1}{L} \mathrel{\htmlData{mx=7}{\implies}} 0 \mathrel{\htmlData{mx=6}{<}} \htmlData{mx=5}{\htmlData{mx=3}{p}(\htmlData{mx=4}{l})}}$ , $\htmlData{mx=4}{\sum_{l=0}^{\htmlData{mx=0}{L} - 1} \htmlData{mx=3}{\htmlData{mx=1}{p}(\htmlData{mx=2}{l})}} \mathrel{\htmlData{mx=5}{=}} 1$ , $\htmlData{mx=17}{\forall l,\ \htmlData{mx=0}{l} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{L} \mathrel{\htmlData{mx=16}{\implies}} \htmlData{mx=11}{\sum_{\omega} \htmlData{mx=10}{\htmlData{mx=5}{\htmlData{mx=3}{w}(\htmlData{mx=4}{\omega})} \cdot \htmlData{mx=9}{\htmlData{mx=6}{Y}(\htmlData{mx=7}{l}, \htmlData{mx=8}{\omega})}}} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=12}{m}(\htmlData{mx=13}{l})}}$ and $\htmlData{mx=24}{\forall l,\ \htmlData{mx=0}{l} \mathrel{\htmlData{mx=2}{<}} \htmlData{mx=1}{L} \mathrel{\htmlData{mx=23}{\implies}} \htmlData{mx=18}{\sum_{\omega} \htmlData{mx=17}{\htmlData{mx=5}{\htmlData{mx=3}{w}(\htmlData{mx=4}{\omega})} \cdot \htmlData{mx=16}{(\htmlData{mx=15}{\htmlData{mx=10}{\htmlData{mx=6}{Y}(\htmlData{mx=8}{\htmlData{mx=7}{l} + 1}, \htmlData{mx=9}{\omega})} - \htmlData{mx=14}{\htmlData{mx=11}{Y}(\htmlData{mx=12}{l}, \htmlData{mx=13}{\omega})}})^{2}}}} \mathrel{\htmlData{mx=22}{\leq}} \htmlData{mx=21}{\htmlData{mx=19}{D}(\htmlData{mx=20}{l})}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega}}$ ), then

\begin{aligned} & \left(\right. \htmlData{mx=30}{\sum_{l=0}^{\htmlData{mx=2}{L} - 1} \htmlData{mx=29}{\sum_{\omega} \htmlData{mx=28}{\htmlData{mx=9}{\htmlData{mx=5}{\htmlData{mx=3}{p}(\htmlData{mx=4}{l})} \cdot \htmlData{mx=8}{\htmlData{mx=6}{w}(\htmlData{mx=7}{\omega})}} \cdot \left(\htmlData{mx=27}{\htmlData{mx=12}{\htmlData{mx=10}{Y}(0, \htmlData{mx=11}{\omega})} + \htmlData{mx=26}{\frac{\htmlData{mx=22}{\htmlData{mx=17}{\htmlData{mx=13}{Y}(\htmlData{mx=15}{\htmlData{mx=14}{l} + 1}, \htmlData{mx=16}{\omega})} - \htmlData{mx=21}{\htmlData{mx=18}{Y}(\htmlData{mx=19}{l}, \htmlData{mx=20}{\omega})}}}{\htmlData{mx=25}{\htmlData{mx=23}{p}(\htmlData{mx=24}{l})}}}}\right)}}} \\ & \quad \htmlData{mx=1}{=} \htmlData{mx=33}{\htmlData{mx=31}{m}(\htmlData{mx=32}{L})} \left.\right) \\ & \htmlData{mx=0}{\land} \left(\right. \htmlData{mx=68}{\sum_{l=0}^{\htmlData{mx=35}{L} - 1} \htmlData{mx=67}{\sum_{\omega} \htmlData{mx=66}{\htmlData{mx=42}{\htmlData{mx=38}{\htmlData{mx=36}{p}(\htmlData{mx=37}{l})} \cdot \htmlData{mx=41}{\htmlData{mx=39}{w}(\htmlData{mx=40}{\omega})}} \cdot \htmlData{mx=65}{\left(\htmlData{mx=64}{\htmlData{mx=60}{\htmlData{mx=45}{\htmlData{mx=43}{Y}(0, \htmlData{mx=44}{\omega})} + \htmlData{mx=59}{\frac{\htmlData{mx=55}{\htmlData{mx=50}{\htmlData{mx=46}{Y}(\htmlData{mx=48}{\htmlData{mx=47}{l} + 1}, \htmlData{mx=49}{\omega})} - \htmlData{mx=54}{\htmlData{mx=51}{Y}(\htmlData{mx=52}{l}, \htmlData{mx=53}{\omega})}}}{\htmlData{mx=58}{\htmlData{mx=56}{p}(\htmlData{mx=57}{l})}}}} - \htmlData{mx=63}{\htmlData{mx=61}{m}(\htmlData{mx=62}{L})}}\right)^{2}}}}} \\ & \quad \htmlData{mx=34}{\leq} \htmlData{mx=92}{\htmlData{mx=81}{2 \cdot \htmlData{mx=80}{\sum_{\omega} \htmlData{mx=79}{\htmlData{mx=71}{\htmlData{mx=69}{w}(\htmlData{mx=70}{\omega})} \cdot \htmlData{mx=78}{(\htmlData{mx=77}{\htmlData{mx=74}{\htmlData{mx=72}{Y}(0, \htmlData{mx=73}{\omega})} - \htmlData{mx=76}{\htmlData{mx=75}{m}(0)}})^{2}}}}} + \htmlData{mx=91}{2 \cdot \htmlData{mx=90}{\sum_{l=0}^{\htmlData{mx=82}{L} - 1} \htmlData{mx=89}{\frac{\htmlData{mx=85}{\htmlData{mx=83}{D}(\htmlData{mx=84}{l})}}{\htmlData{mx=88}{\htmlData{mx=86}{p}(\htmlData{mx=87}{l})}}}}}} \left.\right) \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)
Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to Ω. Lean writes f x; papers write f(x). (Fintype Ω)
⟹Implies: If Ω, then ℝ. (Between types, A → B is the type of functions from A to B.) (Ω → ℝ)
x : TVariables: m, D range over ℕ → ℝ: the statement must hold for every choice. ((m D : ℕ → ℝ))
∀For all: The statement after the comma holds for every ω. (∀ ω, 0 ≤ w ω)
∑Summation: Add up w(ω) for every ω. (∑ ω, w ω)
=Equality: ∑ ω, w(ω) and 1 are the same. (∑ ω, w ω = 1)
∧And: Both hold: (∑ l, ∑ ω, p(l) · w(ω) · (Y(0, ω) + (Y(l + 1, ω) − Y(l, ω)…, and (∑ l, ∑ ω, p(l) · w(ω) · (Y(0, ω) + (Y(l + 1, ω) − Y(l, ω)…. ((∑ l ∈ Finset.range L, ∑ ω, p l * w ω *…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
≤At most: 0 is less than or equal to w(ω). (0 ≤ w ω)
<Less than: l is strictly smaller than L. (l < L)
·Multiplication: w(ω) times Y(l, ω). (w ω * Y l ω)
aᵇPower: Y(l + 1, ω) − Y(l, ω) raised to the power 2. ((Y (l + 1) ω - Y l ω) ^ 2)
-Subtraction: Y(l + 1, ω) minus Y(l, ω). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (Y (l + 1) ω - Y l ω)
+Addition: l plus 1. (l + 1)
a/bDivision: Y(l + 1, ω) − Y(l, ω) divided by p(l). In Lean, dividing by 0 gives 0. ((Y (l + 1) ω - Y l ω) / p l)

theorem ua-end-to-end-mlmc {L : ℕ} {Ω : Type} [Fintype Ω] (w : Ω → ℝ) (p : ℕ → ℝ) (Y : ℕ → Ω → ℝ) (m D : ℕ → ℝ) (hw : ∀ ω, 0 ≤ w ω) (hsum : ∑ ω, w ω = 1) (hp : ∀ l, l < L → 0 < p l) (hps : ∑ l ∈ Finset.range L, p l = 1) (hm : ∀ l, l ≤ L → ∑ ω, w ω * Y l ω = m l) (hD : ∀ l, l < L → ∑ ω, w ω * (Y (l + 1) ω - Y l ω) ^ 2 ≤ D l) :
    (∑ l ∈ Finset.range L, ∑ ω, p l * w ω * (Y 0 ω + (Y (l + 1) ω - Y l ω) / p l) = m L) ∧ (∑ l ∈ Finset.range L, ∑ ω, p l * w ω * (Y 0 ω + (Y (l + 1) ω - Y l ω) / p l - m L) ^ 2 ≤ 2 * ∑ ω, w ω * (Y 0 ω - m 0) ^ 2 + 2 * ∑ l ∈ Finset.range L, D l / p l)

Second single-statement composition (external review asked for end-to-end claims): unbiasedness AND the variance bound of the randomized multilevel estimator in one theorem over the exact finite product space (levels × sample draws). With Y_ℓ the SNIS estimates at 2^ℓ draws (D_ℓ ≤ 2(MSE_{ℓ+1}+MSE_ℓ) from ua-level-difference-bound and ua-end-to-end-uniform-snis) and Y_L exact, this is the unbiased easy side of the sharp threshold. Part of 'Unbiased randomized attention: cost x variance thresholds' (end-to-end composition round).

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: 76 lines of Lean

END-TO-END debiasing: single-term MLMC Z = Y₀ + (Y_{ℓ+1} − Y_ℓ)/p_ℓ (level ℓ ~ p independent of the sample draw) is exactly unbiased for m_L and E(Z − m_L)² ≤ 2E(Y₀ − m₀)² + 2Σ_ℓ D_ℓ/p_ℓ

How it was proved

More proved statements