Provedunbiased-attentionrandomized-algorithmssoftmaxproposal

Importance sampling of the attention numerator is unbiased: Σ r_j·(w_j v_j / r_j) = Σ w_j v_j

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{w, v, r : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \mathrel{\htmlData{mx=3}{\neq}} 0}$ , then

\htmlData{mx=15}{\sum_{j} \htmlData{mx=14}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \left(\htmlData{mx=13}{\frac{\htmlData{mx=9}{\htmlData{mx=5}{\htmlData{mx=3}{w}(\htmlData{mx=4}{j})} \cdot \htmlData{mx=8}{\htmlData{mx=6}{v}(\htmlData{mx=7}{j})}}}{\htmlData{mx=12}{\htmlData{mx=10}{r}(\htmlData{mx=11}{j})}}}\right)}} \mathrel{\htmlData{mx=24}{=}} \htmlData{mx=23}{\sum_{j} \htmlData{mx=22}{\htmlData{mx=18}{\htmlData{mx=16}{w}(\htmlData{mx=17}{j})} \cdot \htmlData{mx=21}{\htmlData{mx=19}{v}(\htmlData{mx=20}{j})}}}

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: w, v, r range over Fin n → ℝ: the statement must hold for every choice. ((w v r : Fin n → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, r j ≠ 0)
∑Summation: Add up r(j) · (w(j) · v(j)/r(j)) for every j. (∑ j, r j * (w j * v j / r j))
=Equality: ∑ j, r(j) · (w(j) · v(j)/r(j)) and ∑ j, w(j) · v(j) are the same. (∑ j, r j * (w j * v j / r j) = ∑ j, w j…)
≠Not equal: r(j) and 0 are different. (r j ≠ 0)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: w(j) times v(j). (w j * v j)
a/bDivision: w(j) · v(j) divided by r(j). In Lean, dividing by 0 gives 0. (w j * v j / r j)

theorem ua-importance-sampling-unbiased {n : ℕ} (w v r : Fin n → ℝ) (hr : ∀ j, r j ≠ 0) :
    ∑ j, r j * (w j * v j / r j) = ∑ j, w j * v j

Sampling key j with probability r_j and returning w_j v_j / r_j is unbiased for the unnormalized attention numerator, for any proposal with full support. 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: 11 lines of Lean

Importance sampling of the attention numerator is unbiased: Σ r_j·(w_j v_j / r_j) = Σ w_j v_j

How it was proved

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