Provedalphabaguettebaguette-zoptimizer

AdamW with decoupled decay keeps every parameter in a fixed box forever

For all $\htmlData{mx=3}{\theta, u : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{a, \eta, K \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{a}$ , $\htmlData{mx=0}{a} \mathrel{\htmlData{mx=1}{\leq}} 1$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\eta}$ , $\htmlData{mx=6}{\forall t,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{u}(\htmlData{mx=1}{t})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{K}}$ and $\htmlData{mx=17}{\forall t,\ \htmlData{mx=3}{\htmlData{mx=0}{\theta}(\htmlData{mx=2}{\htmlData{mx=1}{t} + 1})} \mathrel{\htmlData{mx=16}{=}} \htmlData{mx=15}{\htmlData{mx=9}{(\htmlData{mx=5}{1 - \htmlData{mx=4}{a}}) \cdot \htmlData{mx=8}{\htmlData{mx=6}{\theta}(\htmlData{mx=7}{t})}} - \htmlData{mx=14}{\htmlData{mx=10}{\eta} \cdot \htmlData{mx=13}{\htmlData{mx=11}{u}(\htmlData{mx=12}{t})}}}}$ , then

\htmlData{mx=14}{\forall t,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{\theta}(\htmlData{mx=1}{t})} \right\rvert} \mathrel{\htmlData{mx=13}{\leq}} \htmlData{mx=12}{\max(\htmlData{mx=6}{\left\lvert \htmlData{mx=5}{\htmlData{mx=4}{\theta}(0)} \right\rvert}, \htmlData{mx=11}{\frac{\htmlData{mx=9}{\htmlData{mx=7}{\eta} \cdot \htmlData{mx=8}{K}}}{\htmlData{mx=10}{a}}})}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
⟹Implies: If ℕ, then ℝ. (Between types, A → B is the type of functions from A to B.) (ℕ → ℝ)
x : TVariables: θ, u range over ℕ → ℝ: the statement must hold for every choice. ((θ u : ℕ → ℝ))
x ∈ TVariables: a, η, K range over real numbers: the statement must hold for every choice. ((a η K : ℝ))
<Less than: 0 is strictly smaller than a. (0 < a)
≤At most: a is less than or equal to 1. (a ≤ 1)
∀For all: The statement after the comma holds for every t. (∀ t, |u t| ≤ K)
RReal numbers: The real numbers: every point on the number line. (ℝ)
|x|Absolute value: The size of u(t) without its sign. (|u t|)
=Equality: θ(t + 1) and (1 − a) · θ(t) − η · u(t) are the same. (θ (t + 1) = (1 - a) * θ t - η * u t)
f(x)Function application: u applied to t. Lean writes f x; papers write f(x). (u t)
+Addition: t plus 1. (t + 1)
maxMaximum: The larger of |θ(0)| and η · K/a. (max |θ 0| (η * K / a))
·Multiplication: 1 − a times θ(t). ((1 - a) * θ t)
-Subtraction: 1 minus a. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 - a)
a/bDivision: η · K divided by a. In Lean, dividing by 0 gives 0. (η * K / a)

theorem bz-adamw-uniform-box (θ u : ℕ → ℝ) (a η K : ℝ) (ha0 : 0 < a) (ha1 : a ≤ 1) (hη : 0 ≤ η) (hu : ∀ t, |u t| ≤ K) (hθ : ∀ t, θ (t + 1) = (1 - a) * θ t - η * u t) :
    ∀ t, |θ t| ≤ max |θ 0| (η * K / a)

θ(t+1) = (1−a)θ(t) − η·u(t) with 0 < a ≤ 1 (a = lr·decay) and |u| ≤ K gives |θ(t)| ≤ max(|θ(0)|, ηK/a) for ALL t. The master-weight bank therefore has a compile-time integer width valid for an unbounded training run. A short search did not find this uniform-in-time box stated in the literature. 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: 25 lines of Lean

AdamW with decoupled decay keeps every parameter in a fixed box forever

How it was proved

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