Provedunbiased-attentionrandomized-algorithmssoftmaxdebiasing

The bound is attained by p_l ∝ √(V_l/c_l): (Σ p c)(Σ V/p) = (Σ s)², s_l² = c_l V_l

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{c, V, s : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{\kappa \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{c}(\htmlData{mx=1}{l})}}$ , $\htmlData{mx=4}{\forall l,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{l})}}$ , $\htmlData{mx=12}{\forall l,\ \htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{l})}^{2}} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=4}{c}(\htmlData{mx=5}{l})} \cdot \htmlData{mx=9}{\htmlData{mx=7}{V}(\htmlData{mx=8}{l})}}}$ and $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\kappa}$ , then

\begin{aligned} & \htmlData{mx=29}{\left(\htmlData{mx=14}{\sum_{l} \htmlData{mx=13}{\htmlData{mx=9}{\frac{\htmlData{mx=5}{\htmlData{mx=1}{\kappa} \cdot \htmlData{mx=4}{\htmlData{mx=2}{s}(\htmlData{mx=3}{l})}}}{\htmlData{mx=8}{\htmlData{mx=6}{c}(\htmlData{mx=7}{l})}}} \cdot \htmlData{mx=12}{\htmlData{mx=10}{c}(\htmlData{mx=11}{l})}}}\right) \cdot \left(\htmlData{mx=28}{\sum_{l} \htmlData{mx=27}{\frac{\htmlData{mx=17}{\htmlData{mx=15}{V}(\htmlData{mx=16}{l})}}{\htmlData{mx=26}{\frac{\htmlData{mx=22}{\htmlData{mx=18}{\kappa} \cdot \htmlData{mx=21}{\htmlData{mx=19}{s}(\htmlData{mx=20}{l})}}}{\htmlData{mx=25}{\htmlData{mx=23}{c}(\htmlData{mx=24}{l})}}}}}}\right)} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=34}{\left(\htmlData{mx=33}{\sum_{l} \htmlData{mx=32}{\htmlData{mx=30}{s}(\htmlData{mx=31}{l})}}\right)^{2}} \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 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, s range over Fin L → ℝ: the statement must hold for every choice. ((c V s : Fin L → ℝ))
∀For all: The statement after the comma holds for every l. (∀ l, 0 < c l)
∑Summation: Add up κ · s(l)/c(l) · c(l) for every l. (∑ l, (κ * s l / c l) * c l)
<Less than: 0 is strictly smaller than c(l). (0 < c l)
aᵇPower: s(l) raised to the power 2. (s l ^ 2)
=Equality: s(l)^2 and c(l) · V(l) are the same. (s l ^ 2 = c l * V l)
a/bDivision: κ · s(l) divided by c(l). In Lean, dividing by 0 gives 0. (κ * s l / c l)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: c(l) times V(l). (c l * V l)

theorem ua-optimal-allocation-attains {L : ℕ} (c V s : Fin L → ℝ) (κ : ℝ) (hc : ∀ l, 0 < c l) (hs : ∀ l, 0 < s l) (hsq : ∀ l, s l ^ 2 = c l * V l) (hκ : 0 < κ) :
    (∑ l, (κ * s l / c l) * c l) * (∑ l, V l / (κ * s l / c l)) = (∑ l, s l) ^ 2

Taking p_l = κ·s_l/c_l with s_l = √(c_l V_l) (κ normalizes) gives equality in the Cauchy–Schwarz bound, for every κ > 0. 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: 17 lines of Lean

The bound is attained by p_l ∝ √(V_l/c_l): (Σ p c)(Σ V/p) = (Σ s)², s_l² = c_l V_l

How it was proved

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