Provedunbiased-attentionend-to-endsoftmaxeasy-side

END-TO-END easy side: score range ≤ R and |v| ≤ F ⇒ uniform-proposal SNIS from N i.i.d. keys has MSE ≤ 16 F² e^R / N

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

\begin{aligned} & \htmlData{mx=74}{\sum_{f : \htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{N}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\htmlData{mx=4}{\operatorname{Fin}}\,\htmlData{mx=5}{n}}} \htmlData{mx=73}{\left(\htmlData{mx=14}{\prod_{\mathit{\_k} : \htmlData{mx=10}{\htmlData{mx=8}{\operatorname{Fin}}\,\htmlData{mx=9}{N}}} \htmlData{mx=13}{\frac{1}{\htmlData{mx=12}{\htmlData{mx=11}{n}}}}}\right) \cdot \htmlData{mx=72}{\left(\htmlData{mx=71}{\htmlData{mx=55}{\frac{\htmlData{mx=37}{\sum_{i} \htmlData{mx=36}{\htmlData{mx=30}{\frac{\htmlData{mx=26}{\frac{\htmlData{mx=20}{e^{\htmlData{mx=19}{\htmlData{mx=15}{s}(\htmlData{mx=18}{\htmlData{mx=16}{f}(\htmlData{mx=17}{i})})}}}}{\htmlData{mx=25}{\sum_{k} \htmlData{mx=24}{e^{\htmlData{mx=23}{\htmlData{mx=21}{s}(\htmlData{mx=22}{k})}}}}}}}{\htmlData{mx=29}{\frac{1}{\htmlData{mx=28}{\htmlData{mx=27}{n}}}}}} \cdot \htmlData{mx=35}{\htmlData{mx=31}{v}(\htmlData{mx=34}{\htmlData{mx=32}{f}(\htmlData{mx=33}{i})})}}}}{\htmlData{mx=54}{\sum_{i} \htmlData{mx=53}{\frac{\htmlData{mx=49}{\frac{\htmlData{mx=43}{e^{\htmlData{mx=42}{\htmlData{mx=38}{s}(\htmlData{mx=41}{\htmlData{mx=39}{f}(\htmlData{mx=40}{i})})}}}}{\htmlData{mx=48}{\sum_{k} \htmlData{mx=47}{e^{\htmlData{mx=46}{\htmlData{mx=44}{s}(\htmlData{mx=45}{k})}}}}}}}{\htmlData{mx=52}{\frac{1}{\htmlData{mx=51}{\htmlData{mx=50}{n}}}}}}}}} - \htmlData{mx=70}{\sum_{j} \htmlData{mx=69}{\htmlData{mx=65}{\frac{\htmlData{mx=59}{e^{\htmlData{mx=58}{\htmlData{mx=56}{s}(\htmlData{mx=57}{j})}}}}{\htmlData{mx=64}{\sum_{k} \htmlData{mx=63}{e^{\htmlData{mx=62}{\htmlData{mx=60}{s}(\htmlData{mx=61}{k})}}}}}} \cdot \htmlData{mx=68}{\htmlData{mx=66}{v}(\htmlData{mx=67}{j})}}}}\right)^{2}}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=82}{\frac{\htmlData{mx=80}{\htmlData{mx=77}{16 \cdot \htmlData{mx=76}{\htmlData{mx=75}{F}^{2}}} \cdot \htmlData{mx=79}{e^{\htmlData{mx=78}{R}}}}}{\htmlData{mx=81}{N}}} \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)
x ∈ TVariables: n, N range over natural numbers: the statement must hold for every choice. ((n N : ℕ))
<Less than: 0 is strictly smaller than N. (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, v range over Fin n → ℝ: the statement must hold for every choice. ((s v : Fin n → ℝ))
∀For all: The statement after the comma holds for every i, j. (∀ i j, s i - s j ≤ R)
|x|Absolute value: The size of v(j) without its sign. (|v j|)
-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)
∏Product: Multiply together 1/n for every _k in Fin(N). (∏ _k : Fin N, (1 / (n : ℝ)))
a/bDivision: 1 divided by n. In Lean, dividing by 0 gives 0. (1 / (n : 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))
aᵇPower: (∑ i, exp(s(f(i)))/(∑ k, exp(s(k)))/(1/n) · v(f(i)))/(∑ i,… raised to the power 2. (((∑ i, (Real.exp (s (f i)) / ∑ k, Real.…)
·Multiplication: exp(s(f(i)))/(∑ k, exp(s(k)))/(1/n) times v(f(i)). ((Real.exp (s (f i)) / ∑ k, Real.exp (s …)
eˣExponential function: e raised to the power s(f(i)), where e ≈ 2.718. (Real.exp (s (f i)))
∑Summation: Add up exp(s(k)) for every k. (∑ k, Real.exp (s k))

theorem ua-end-to-end-uniform-snis {n N : ℕ} (hN : 0 < N) (hn : 0 < n) (s v : Fin n → ℝ) (F R : ℝ) (hR : ∀ i j, s i - s j ≤ R) (hv : ∀ j, |v j| ≤ F) :
    ∑ f : Fin N → Fin n, (∏ _k : Fin N, (1 / (n : ℝ))) * ((∑ i, (Real.exp (s (f i)) / ∑ k, Real.exp (s k)) / (1 / (n : ℝ)) * v (f i)) / (∑ i, (Real.exp (s (f i)) / ∑ k, Real.exp (s k)) / (1 / (n : ℝ))) - ∑ j, (Real.exp (s j) / ∑ k, Real.exp (s k)) * v j) ^ 2 ≤ 16 * F ^ 2 * Real.exp R / N

The first single-statement composition of the easy side (answering the external review's '0/9 claims formalized end-to-end'): SNIS MSE bound over the exact finite product space composed with the uniform-proposal χ² bound. With N = 16 F² e^R / ε this is constant-MSE attention at n^{1+α} work for range α ln n. The constant 16 is likely 4× loose (Agapiou et al. 2017). Part of 'Unbiased randomized attention: cost x variance thresholds' (end-to-end composition round).

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: 200 lines of Lean

END-TO-END easy side: score range ≤ R and |v| ≤ F ⇒ uniform-proposal SNIS from N i.i.d. keys has MSE ≤ 16 F² e^R / N

How it was proved

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