proofs
Provedunbiased-gradientstrainingfirst-principles

Level-error propagation through a Lipschitz stage: E‖X₂ − f(x₁)‖² ≤ 2·E‖X₂ − f(X₁)‖² + 2·Lip²·E‖X₁ − x₁‖²

For all Ω∈Type⁡\htmlData{mx=1}{\Omega \in \htmlData{mx=0}{\operatorname{Type}}}, d1,d2∈N\htmlData{mx=1}{d_{1}, d_{2} \in \htmlData{mx=0}{\mathbb{N}}}, w:Ω→R\htmlData{mx=3}{w : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}, X1:Ω→Fin⁡ d1→R\htmlData{mx=7}{X_{1} : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{d_{1}}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\mathbb{R}}}, x1:Fin⁡ d1→R\htmlData{mx=5}{x_{1} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d_{1}}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}, X2:Ω→Fin⁡ d2→R\htmlData{mx=7}{X_{2} : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{d_{2}}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\mathbb{R}}}, f:(Fin⁡ d1→R)→Fin⁡ d2→R\htmlData{mx=11}{f : (\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d_{1}}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}) \mathrel{\htmlData{mx=10}{\to}} \htmlData{mx=7}{\htmlData{mx=5}{\operatorname{Fin}}\,\htmlData{mx=6}{d_{2}}} \mathrel{\htmlData{mx=9}{\to}} \htmlData{mx=8}{\mathbb{R}}} and Lip,e1,e2∈R\htmlData{mx=1}{\mathit{Lip}, e_{1}, e_{2} \in \htmlData{mx=0}{\mathbb{R}}}, if ∀ω, 0≤w(ω)\htmlData{mx=4}{\forall \omega,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})}}, ∀x,y, ∑i(f(x,i)−f(y,i))2≤Lip2⋅∑i(x(i)−y(i))2\htmlData{mx=24}{\forall x, y,\ \htmlData{mx=10}{\sum_{i} \htmlData{mx=9}{(\htmlData{mx=8}{\htmlData{mx=3}{\htmlData{mx=0}{f}(\htmlData{mx=1}{x}, \htmlData{mx=2}{i})} - \htmlData{mx=7}{\htmlData{mx=4}{f}(\htmlData{mx=5}{y}, \htmlData{mx=6}{i})}})^{2}}} \mathrel{\htmlData{mx=23}{\leq}} \htmlData{mx=22}{\htmlData{mx=12}{\htmlData{mx=11}{\mathit{Lip}}^{2}} \cdot \htmlData{mx=21}{\sum_{i} \htmlData{mx=20}{(\htmlData{mx=19}{\htmlData{mx=15}{\htmlData{mx=13}{x}(\htmlData{mx=14}{i})} - \htmlData{mx=18}{\htmlData{mx=16}{y}(\htmlData{mx=17}{i})}})^{2}}}}}, ∑ωw(ω)⋅∑i(X1(ω,i)−x1(i))2≤e1\htmlData{mx=14}{\sum_{\omega} \htmlData{mx=13}{\htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})} \cdot \htmlData{mx=12}{\sum_{i} \htmlData{mx=11}{(\htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=3}{X_{1}}(\htmlData{mx=4}{\omega}, \htmlData{mx=5}{i})} - \htmlData{mx=9}{\htmlData{mx=7}{x_{1}}(\htmlData{mx=8}{i})}})^{2}}}}} \mathrel{\htmlData{mx=16}{\leq}} \htmlData{mx=15}{e_{1}} and ∑ωw(ω)⋅∑i(X2(ω,i)−f(X1(ω),i))2≤e2\htmlData{mx=17}{\sum_{\omega} \htmlData{mx=16}{\htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})} \cdot \htmlData{mx=15}{\sum_{i} \htmlData{mx=14}{(\htmlData{mx=13}{\htmlData{mx=6}{\htmlData{mx=3}{X_{2}}(\htmlData{mx=4}{\omega}, \htmlData{mx=5}{i})} - \htmlData{mx=12}{\htmlData{mx=7}{f}(\htmlData{mx=10}{\htmlData{mx=8}{X_{1}}(\htmlData{mx=9}{\omega})}, \htmlData{mx=11}{i})}})^{2}}}}} \mathrel{\htmlData{mx=19}{\leq}} \htmlData{mx=18}{e_{2}} (given Fintype⁡ Ω\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega}}), then

∑ωw(ω)⋅∑i(X2(ω,i)−f(x1,i))2≤2e2+2⋅Lip2⋅e1\begin{aligned} & \htmlData{mx=16}{\sum_{\omega} \htmlData{mx=15}{\htmlData{mx=3}{\htmlData{mx=1}{w}(\htmlData{mx=2}{\omega})} \cdot \htmlData{mx=14}{\sum_{i} \htmlData{mx=13}{(\htmlData{mx=12}{\htmlData{mx=7}{\htmlData{mx=4}{X_{2}}(\htmlData{mx=5}{\omega}, \htmlData{mx=6}{i})} - \htmlData{mx=11}{\htmlData{mx=8}{f}(\htmlData{mx=9}{x_{1}}, \htmlData{mx=10}{i})}})^{2}}}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=24}{\htmlData{mx=18}{2\htmlData{mx=17}{e_{2}}} + \htmlData{mx=23}{\htmlData{mx=21}{2 \cdot \htmlData{mx=20}{\htmlData{mx=19}{\mathit{Lip}}^{2}}} \cdot \htmlData{mx=22}{e_{1}}}} \end{aligned}
What the notation means · or point at any symbol
Type
Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype
Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application
Fintype applied to Ω. Lean writes f x; papers write f(x). (Fintype Ω)
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables
d1, d2 range over natural numbers: the statement must hold for every choice. ((d1 d2 : ℕ))
⟹Implies
If Ω, then ℝ. (Between types, A → B is the type of functions from A to B.) (Ω → ℝ)
∀For all
The statement after the comma holds for every ω. (∀ ω, 0 ≤ w ω)
RReal numbers
The real numbers: every point on the number line. (ℝ)
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)
≤At most
0 is less than or equal to w(ω). (0 ≤ w ω)
∑Summation
Add up (f(x, i) − f(y, i))^2 for every i. (∑ i, (f x i - f y i) ^ 2)
aᵇPower
f(x, i) − f(y, i) raised to the power 2. ((f x i - f y i) ^ 2)
-Subtraction
f(x, i) minus f(y, i). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (f x i - f y i)
·Multiplication
Lip^2 times ∑ i, (x(i) − y(i))^2. (Lip ^ 2 * ∑ i, (x i - y i) ^ 2)
+Addition
2 · e2 plus 2 · Lip^2 · e1. (2 * e2 + 2 * Lip ^ 2 * e1)

The induction step for the whole-network level bound needed by the Lipschitz MLMC theorem: a stage's error against the dense value is at most twice its own sampling error plus twice the Lipschitz-propagated error of its input. Iterating over D stages gives a level constant exponential in the depth-wise product of Lipschitz constants, which is the honest variance price of exactly unbiased training. First-principles reformulation, 2026-10-05.

How it was proved

Proved
5 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
24 lines of Lean
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