Provedunbiased-attentionrandomized-algorithmssoftmaxdebiasing

Cost × variance lower bound: (Σ√(c_l V_l))² ≤ (Σ p_l c_l)(Σ V_l/p_l)

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{c, V, p : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{c}(\htmlData{mx=1}{l})}}$ , $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{V}(\htmlData{mx=1}{l})}}$ and $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{l})}}$ , then

\htmlData{mx=9}{\left(\htmlData{mx=8}{\sum_{l} \htmlData{mx=7}{\sqrt{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{c}(\htmlData{mx=1}{l})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{V}(\htmlData{mx=4}{l})}}}}}\right)^{2}} \mathrel{\htmlData{mx=27}{\leq}} \htmlData{mx=26}{\left(\htmlData{mx=17}{\sum_{l} \htmlData{mx=16}{\htmlData{mx=12}{\htmlData{mx=10}{p}(\htmlData{mx=11}{l})} \cdot \htmlData{mx=15}{\htmlData{mx=13}{c}(\htmlData{mx=14}{l})}}}\right) \cdot \left(\htmlData{mx=25}{\sum_{l} \htmlData{mx=24}{\frac{\htmlData{mx=20}{\htmlData{mx=18}{V}(\htmlData{mx=19}{l})}}{\htmlData{mx=23}{\htmlData{mx=21}{p}(\htmlData{mx=22}{l})}}}}\right)}

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 L. Lean writes f x; papers write f(x). (Fin L)
⟹Implies: If Fin(L), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin L → ℝ)
x : TVariables: c, V, p range over Fin L → ℝ: the statement must hold for every choice. ((c V p : Fin L → ℝ))
∀For all: The statement after the comma holds for every l. (∀ l, 0 ≤ c l)
aᵇPower: ∑ l, √(c(l) · V(l)) raised to the power 2. ((∑ l, Real.sqrt (c l * V l)) ^ 2)
∑Summation: Add up √(c(l) · V(l)) for every l. (∑ l, Real.sqrt (c l * V l))
≤At most: 0 is less than or equal to c(l). (0 ≤ c l)
<Less than: 0 is strictly smaller than p(l). (0 < p l)
√Square root: The non-negative number whose square is c(l) · V(l). (Real.sqrt (c l * V l))
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: c(l) times V(l). (c l * V l)
a/bDivision: V(l) divided by p(l). In Lean, dividing by 0 gives 0. (V l / p l)

theorem ua-cost-variance-cauchy-schwarz {L : ℕ} (c V p : Fin L → ℝ) (hc : ∀ l, 0 ≤ c l) (hV : ∀ l, 0 ≤ V l) (hp : ∀ l, 0 < p l) :
    (∑ l, Real.sqrt (c l * V l)) ^ 2 ≤ (∑ l, p l * c l) * (∑ l, V l / p l)

For any level law p > 0, expected cost Σ p c times variance bound Σ V/p is at least (Σ √(cV))² (Cauchy–Schwarz). This is the work-normalized variance floor of randomized MLMC. 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: 23 lines of Lean

Cost × variance lower bound: (Σ√(c_l V_l))² ≤ (Σ p_l c_l)(Σ V_l/p_l)

How it was proved

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