proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxeasy-side

Sample count: N ≥ 16F²ρ/ε ⇒ the SNIS MSE bound is ≤ ε

For all F,ρ,ε∈R\htmlData{mx=1}{F, \rho, \varepsilon \in \htmlData{mx=0}{\mathbb{R}}} and N∈N\htmlData{mx=1}{N \in \htmlData{mx=0}{\mathbb{N}}}, if 0<ε0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\varepsilon}, 0≤ρ0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\rho}, 16⋅F2⋅ρε≤N\htmlData{mx=6}{\frac{\htmlData{mx=4}{\htmlData{mx=2}{16 \cdot \htmlData{mx=1}{\htmlData{mx=0}{F}^{2}}} \cdot \htmlData{mx=3}{\rho}}}{\htmlData{mx=5}{\varepsilon}}} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{N} and 0<N0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{N}, then

16⋅F2⋅ρN≤ε\htmlData{mx=6}{\frac{\htmlData{mx=4}{\htmlData{mx=2}{16 \cdot \htmlData{mx=1}{\htmlData{mx=0}{F}^{2}}} \cdot \htmlData{mx=3}{\rho}}}{\htmlData{mx=5}{N}}} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\varepsilon}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
F, ρ, ε range over real numbers: the statement must hold for every choice. ((F ρ ε : ℝ))
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
<Less than
0 is strictly smaller than ε. (0 < ε)
≤At most
0 is less than or equal to ρ. (0 ≤ ρ)
·Multiplication
16 times F^2. (16 * F ^ 2)
a/bDivision
16 · F^2 · ρ divided by ε. In Lean, dividing by 0 gives 0. (16 * F ^ 2 * ρ / ε)
aᵇPower
F raised to the power 2. (F ^ 2)

Arithmetic corollary of the SNIS bound with ρ = 1 + χ²; under the uniform proposal ρ ≤ e^R. Part of 'Unbiased randomized attention: cost x variance thresholds' (open theory problem), round 2.

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
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