Provedunbiased-attentionrandomized-algorithmssoftmaxaggregate

Exact-row subsampling is unbiased: (1/n) Σ_i n·o_i = Σ_i o_i

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{o : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ , then

\htmlData{mx=10}{\sum_{i} \htmlData{mx=9}{\htmlData{mx=2}{\frac{1}{\htmlData{mx=1}{\htmlData{mx=0}{n}}}} \cdot (\htmlData{mx=8}{\htmlData{mx=4}{\htmlData{mx=3}{n}} \cdot \htmlData{mx=7}{\htmlData{mx=5}{o}(\htmlData{mx=6}{i})}})}} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\sum_{i} \htmlData{mx=13}{\htmlData{mx=11}{o}(\htmlData{mx=12}{i})}}

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)
<Less than: 0 is strictly smaller than n. (0 < n)
∑Summation: Add up 1/n · (n · o(i)) for every i. (∑ i, (1 / (n : ℝ)) * ((n : ℝ) * o i))
=Equality: ∑ i, 1/n · (n · o(i)) and ∑ i, o(i) are the same. (∑ i, (1 / (n : ℝ)) * ((n : ℝ) * o i) = …)
a/bDivision: 1 divided by n. In Lean, dividing by 0 gives 0. (1 / (n : ℝ))
RReal numbers: The real numbers: every point on the number line. (ℝ)
(x : T)As real numbers: n read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((n : ℝ))
·Multiplication: n times o(i). ((n : ℝ) * o i)

theorem ua-row-subsampling-unbiased {n : ℕ} (o : Fin n → ℝ) (hn : 0 < n) :
    ∑ i, (1 / (n : ℝ)) * ((n : ℝ) * o i) = ∑ i, o i

The estimator n·o_I with I uniform. Part of 'Unbiased randomized attention: cost x variance thresholds', round 15 (aggregate conjecture refuted: exact-row subsampling).

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

Exact-row subsampling is unbiased: (1/n) Σ_i n·o_i = Σ_i o_i

How it was proved

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