Provedalphabaguettebaguette-zoptimizer

The clipped gradient g·c/max(c, ‖g‖) has norm at most c

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{g : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{c}$ , then

\htmlData{mx=16}{\sqrt{\htmlData{mx=15}{\sum_{i} \htmlData{mx=14}{\left(\htmlData{mx=13}{\htmlData{mx=2}{\htmlData{mx=0}{g}(\htmlData{mx=1}{i})} \cdot \left(\htmlData{mx=12}{\frac{\htmlData{mx=3}{c}}{\htmlData{mx=11}{\max(\htmlData{mx=4}{c}, \htmlData{mx=10}{\sqrt{\htmlData{mx=9}{\sum_{j} \htmlData{mx=8}{\htmlData{mx=7}{\htmlData{mx=5}{g}(\htmlData{mx=6}{j})}^{2}}}}})}}}\right)}\right)^{2}}}}} \mathrel{\htmlData{mx=18}{\leq}} \htmlData{mx=17}{c}

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 n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
<Less than: 0 is strictly smaller than c. (0 < c)
≤At most: √(∑ i, (g(i) · (c/max(c, √(∑ j, g(j)^2))))^2) is less than or equal to c. (Real.sqrt (∑ i, (g i * (c / max c (Real…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: g(i) times c/max(c, √(∑ j, g(j)^2)). (g i * (c / max c (Real.sqrt (∑ j, g j ^…)
a/bDivision: c divided by max(c, √(∑ j, g(j)^2)). In Lean, dividing by 0 gives 0. (c / max c (Real.sqrt (∑ j, g j ^ 2)))
maxMaximum: The larger of c and √(∑ j, g(j)^2). (max c (Real.sqrt (∑ j, g j ^ 2)))
√Square root: The non-negative number whose square is ∑ j, g(j)^2. (Real.sqrt (∑ j, g j ^ 2))
∑Summation: Add up g(j)^2 for every j. (∑ j, g j ^ 2)
aᵇPower: g(j) raised to the power 2. (g j ^ 2)

theorem bz-clipped-norm-bound {n : ℕ} (g : Fin n → ℝ) (c : ℝ) (hc : 0 < c) :
    Real.sqrt (∑ i, (g i * (c / max c (Real.sqrt (∑ j, g j ^ 2)))) ^ 2) ≤ c

Global-norm clipping really enforces ‖g‖₂ ≤ c for every input gradient (c > 0). Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

How it was proved

Proved: 3 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

The clipped gradient g·c/max(c, ‖g‖) has norm at most c

How it was proved

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