Provedunbiased-attentionrandomized-algorithmssoftmaxproposal

Score range ≤ R ⇒ uniform key sampling has n·Σπ_j² ≤ e^R (χ² ≤ e^R − 1)

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{s : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{R \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=9}{\forall i, j,\ \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{i})} - \htmlData{mx=5}{\htmlData{mx=3}{s}(\htmlData{mx=4}{j})}} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{R}}$ , then

\htmlData{mx=14}{\htmlData{mx=1}{\htmlData{mx=0}{n}} \cdot \htmlData{mx=13}{\sum_{j} \htmlData{mx=12}{\left(\htmlData{mx=11}{\frac{\htmlData{mx=5}{e^{\htmlData{mx=4}{\htmlData{mx=2}{s}(\htmlData{mx=3}{j})}}}}{\htmlData{mx=10}{\sum_{k} \htmlData{mx=9}{e^{\htmlData{mx=8}{\htmlData{mx=6}{s}(\htmlData{mx=7}{k})}}}}}}\right)^{2}}}} \mathrel{\htmlData{mx=17}{\leq}} \htmlData{mx=16}{e^{\htmlData{mx=15}{R}}}

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)
∀For all: The statement after the comma holds for every i, j. (∀ i j, s i - s j ≤ 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 ∑ j, (exp(s(j))/(∑ k, exp(s(k))))^2. ((n : ℝ) * ∑ j, (Real.exp (s j) / ∑ k, R…)
-Subtraction: s(i) minus s(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (s i - s j)
≤At most: s(i) − s(j) is less than or equal to R. (s i - s j ≤ R)
RReal numbers: The real numbers: every point on the number line. (ℝ)
aᵇPower: exp(s(j))/(∑ k, exp(s(k))) raised to the power 2. ((Real.exp (s j) / ∑ k, Real.exp (s k)) …)
eˣExponential function: e raised to the power s(j), where e ≈ 2.718. (Real.exp (s j))
a/bDivision: exp(s(j)) divided by ∑ k, exp(s(k)). In Lean, dividing by 0 gives 0. (Real.exp (s j) / ∑ k, Real.exp (s k))
∑Summation: Add up exp(s(k)) for every k. (∑ k, Real.exp (s k))

theorem ua-uniform-proposal-chi-square-bound {n : ℕ} (s : Fin n → ℝ) (R : ℝ) (hR : ∀ i j, s i - s j ≤ R) :
    (n : ℝ) * ∑ j, (Real.exp (s j) / ∑ k, Real.exp (s k)) ^ 2 ≤ Real.exp R

The easy side of the threshold. If every row's scores lie in an interval of width R, the softmax row π satisfies n Σ π² ≤ e^R. Under Alman–Song bounded entries (|s| ≤ B²), R ≤ 2B², so for B = o(√log n) plain uniform sampling + debiasing has n^{o(1)} work per unit variance per query: subquadratic unbiased attention with a trivial proposal. 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: 37 lines of Lean

Score range ≤ R ⇒ uniform key sampling has n·Σπ_j² ≤ e^R (χ² ≤ e^R − 1)

How it was proved

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