Provedunbiased-attentionrandomized-algorithmssoftmaxproposal

Σ π_j²/r_j = 1 + χ²(π‖r) for probability vectors π, r

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

\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\frac{\htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}^{2}}}{\htmlData{mx=6}{\htmlData{mx=4}{r}(\htmlData{mx=5}{j})}}}} \mathrel{\htmlData{mx=23}{=}} \htmlData{mx=22}{1 + \htmlData{mx=21}{\sum_{j} \htmlData{mx=20}{\frac{\htmlData{mx=16}{(\htmlData{mx=15}{\htmlData{mx=11}{\htmlData{mx=9}{\pi}(\htmlData{mx=10}{j})} - \htmlData{mx=14}{\htmlData{mx=12}{r}(\htmlData{mx=13}{j})}})^{2}}}{\htmlData{mx=19}{\htmlData{mx=17}{r}(\htmlData{mx=18}{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: π, r range over Fin n → ℝ: the statement must hold for every choice. ((π r : Fin n → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, r j ≠ 0)
∑Summation: Add up π(j) for every j. (∑ j, π j)
=Equality: ∑ j, π(j) and 1 are the same. (∑ j, π j = 1)
≠Not equal: r(j) and 0 are different. (r j ≠ 0)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
aᵇPower: π(j) raised to the power 2. (π j ^ 2)
a/bDivision: π(j)^2 divided by r(j). In Lean, dividing by 0 gives 0. (π j ^ 2 / r j)
RReal numbers: The real numbers: every point on the number line. (ℝ)
+Addition: 1 plus ∑ j, (π(j) − r(j))^2/r(j). (1 + ∑ j, (π j - r j) ^ 2 / r j)
-Subtraction: π(j) minus r(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (π j - r j)

theorem ua-chi-square-identity {n : ℕ} (π r : Fin n → ℝ) (hr : ∀ j, r j ≠ 0) (hπ : ∑ j, π j = 1) (hrs : ∑ j, r j = 1) :
    ∑ j, π j ^ 2 / r j = 1 + ∑ j, (π j - r j) ^ 2 / r j

The second-moment factor of importance sampling attention row π from proposal r is exactly 1 + χ². All estimator variances below are measured in this unit. 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

Σ π_j²/r_j = 1 + χ²(π‖r) for probability vectors π, r

How it was proved

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