Provedunbiased-attentionrandomized-algorithmssoftmaxeasy-side

Approximate-score proposal: |s − ŝ| ≤ η ⇒ Σ π²/r ≤ e^{2η} (π = softmax s, r = softmax ŝ)

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{s, t : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{\eta \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=10}{\forall j,\ \htmlData{mx=7}{\left\lvert \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{j})} - \htmlData{mx=5}{\htmlData{mx=3}{t}(\htmlData{mx=4}{j})}} \right\rvert} \mathrel{\htmlData{mx=9}{\leq}} \htmlData{mx=8}{\eta}}$ , then

\begin{aligned} & \htmlData{mx=23}{\sum_{j} \htmlData{mx=22}{\frac{\htmlData{mx=11}{\left(\htmlData{mx=10}{\frac{\htmlData{mx=4}{e^{\htmlData{mx=3}{\htmlData{mx=1}{s}(\htmlData{mx=2}{j})}}}}{\htmlData{mx=9}{\sum_{k} \htmlData{mx=8}{e^{\htmlData{mx=7}{\htmlData{mx=5}{s}(\htmlData{mx=6}{k})}}}}}}\right)^{2}}}{\htmlData{mx=21}{\frac{\htmlData{mx=15}{e^{\htmlData{mx=14}{\htmlData{mx=12}{t}(\htmlData{mx=13}{j})}}}}{\htmlData{mx=20}{\sum_{k} \htmlData{mx=19}{e^{\htmlData{mx=18}{\htmlData{mx=16}{t}(\htmlData{mx=17}{k})}}}}}}}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=26}{e^{\htmlData{mx=25}{2\htmlData{mx=24}{\eta}}}} \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)
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: s, t range over Fin n → ℝ: the statement must hold for every choice. ((s t : Fin n → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, |s j - t j| ≤ η)
|x|Absolute value: The size of s(j) − t(j) without its sign. (|s j - t j|)
≤At most: |s(j) − t(j)| is less than or equal to η. (|s j - t j| ≤ η)
aᵇPower: exp(s(j))/(∑ k, exp(s(k))) raised to the power 2. ((Real.exp (s j) / ∑ k, Real.exp (s k)) …)
-Subtraction: s(j) minus t(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (s j - t j)
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))
RReal numbers: The real numbers: every point on the number line. (ℝ)
∑Summation: Add up exp(s(k)) for every k. (∑ k, Real.exp (s k))
·Multiplication: 2 times η. (2 * η)

theorem ua-approximate-score-proposal {n : ℕ} (s t : Fin n → ℝ) (η : ℝ) (hη : ∀ j, |s j - t j| ≤ η) :
    ∑ j, (Real.exp (s j) / ∑ k, Real.exp (s k)) ^ 2 / (Real.exp (t j) / ∑ k, Real.exp (t k)) ≤ Real.exp (2 * η)

If any cheap stage (low-precision or sketched dot products, a cache, a data structure) yields scores within η of the true ones, importance sampling from softmax(ŝ) has 1 + χ² ≤ e^{2η}: the sample count for constant MSE depends only on η, not on n or the score range. All of the hardness lives in producing the proposal. Part of 'Unbiased randomized attention: cost x variance thresholds', round 5 (approximate-score proposals and the sampling lower bound).

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: 44 lines of Lean

Approximate-score proposal: |s − ŝ| ≤ η ⇒ Σ π²/r ≤ e^{2η} (π = softmax s, r = softmax ŝ)

How it was proved

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