Provedalphabaguettebaguette-zv2

AdamW's parameter box holds for ANY learning-rate schedule: |θ(t)| ≤ max(|θ₀|, K/λ)

For all $\htmlData{mx=3}{\theta, u, \eta : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{\mathit{lam}, K \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\mathit{lam}}$ , $\htmlData{mx=4}{\forall t,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\eta}(\htmlData{mx=1}{t})}}$ , $\htmlData{mx=6}{\forall t,\ \htmlData{mx=4}{\htmlData{mx=2}{\htmlData{mx=0}{\eta}(\htmlData{mx=1}{t})} \cdot \htmlData{mx=3}{\mathit{lam}}} \mathrel{\htmlData{mx=5}{\leq}} 1}$ , $\htmlData{mx=6}{\forall t,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{u}(\htmlData{mx=1}{t})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{K}}$ and $\htmlData{mx=23}{\forall t,\ \htmlData{mx=3}{\htmlData{mx=0}{\theta}(\htmlData{mx=2}{\htmlData{mx=1}{t} + 1})} \mathrel{\htmlData{mx=22}{=}} \htmlData{mx=21}{\htmlData{mx=13}{(\htmlData{mx=9}{1 - \htmlData{mx=8}{\htmlData{mx=6}{\htmlData{mx=4}{\eta}(\htmlData{mx=5}{t})} \cdot \htmlData{mx=7}{\mathit{lam}}}}) \cdot \htmlData{mx=12}{\htmlData{mx=10}{\theta}(\htmlData{mx=11}{t})}} - \htmlData{mx=20}{\htmlData{mx=16}{\htmlData{mx=14}{\eta}(\htmlData{mx=15}{t})} \cdot \htmlData{mx=19}{\htmlData{mx=17}{u}(\htmlData{mx=18}{t})}}}}$ , then

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

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

theorem bz2-adamw-box-any-schedule (θ u η : ℕ → ℝ) (lam K : ℝ) (hlam : 0 < lam) (hη : ∀ t, 0 ≤ η t) (hηl : ∀ t, η t * lam ≤ 1) (hu : ∀ t, |u t| ≤ K) (hθ : ∀ t, θ (t + 1) = (1 - η t * lam) * θ t - η t * u t) :
    ∀ t, |θ t| ≤ max |θ 0| (K / lam)

With decoupled decay coupled to the learning rate (θ ← (1 − η_t λ)θ − η_t u_t, 0 ≤ η_t λ ≤ 1) and |u_t| ≤ K, the box radius K/λ does not depend on the schedule η_t at all. Fixes the constant-rate restriction of bz-adamw-uniform-box. This is the sup-bound reading of the EMA view of AdamW (Wang & Aitchison, ICML 2025); the far tighter expected equilibrium is Kosson et al. (ICML 2024). Part of 'Baguette-Z v2' (revision after the hostile review).

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: 26 lines of Lean

AdamW's parameter box holds for ANY learning-rate schedule: |θ(t)| ≤ max(|θ₀|, K/λ)

How it was proved

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