proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxeasy-side

SNIS pointwise core: |a/w − o| ≤ |a − o·w| + 2F·|1 − w|

For all a,w,o,F∈R\htmlData{mx=1}{a, w, o, F \in \htmlData{mx=0}{\mathbb{R}}}, if 0<w0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{w}, ∣aw∣≤F\htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\frac{\htmlData{mx=0}{a}}{\htmlData{mx=1}{w}}} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{F} and ∣o∣≤F\htmlData{mx=1}{\left\lvert \htmlData{mx=0}{o} \right\rvert} \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{F}, then

∣aw−o∣≤∣a−o⋅w∣+2F⋅∣1−w∣\htmlData{mx=5}{\left\lvert \htmlData{mx=4}{\htmlData{mx=2}{\frac{\htmlData{mx=0}{a}}{\htmlData{mx=1}{w}}} - \htmlData{mx=3}{o}} \right\rvert} \mathrel{\htmlData{mx=19}{\leq}} \htmlData{mx=18}{\htmlData{mx=11}{\left\lvert \htmlData{mx=10}{\htmlData{mx=6}{a} - \htmlData{mx=9}{\htmlData{mx=7}{o} \cdot \htmlData{mx=8}{w}}} \right\rvert} + \htmlData{mx=17}{\htmlData{mx=13}{2\htmlData{mx=12}{F}} \cdot \htmlData{mx=16}{\left\lvert \htmlData{mx=15}{1 - \htmlData{mx=14}{w}} \right\rvert}}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
a, w, o, F range over real numbers: the statement must hold for every choice. ((a w o F : ℝ))
<Less than
0 is strictly smaller than w. (0 < w)
|x|Absolute value
The size of a/w without its sign. (|a / w|)
≤At most
|a/w| is less than or equal to F. (|a / w| ≤ F)
a/bDivision
a divided by w. In Lean, dividing by 0 gives 0. (a / w)
-Subtraction
a/w minus o. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (a / w - o)
+Addition
|a − o · w| plus 2 · F · |1 − w|. (|a - o * w| + 2 * F * |1 - w|)
·Multiplication
o times w. (o * w)

From a/w − o = (a − ow) + (a/w − o)(1 − w) and |a/w − o| ≤ 2F. The self-normalized error splits into a mean-zero numerator term and a normalizer term. 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
19 lines of Lean
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