simulate_regularized · Samplinglib

Plain-English statement

- Correct each original value/gradient reply using known quadratic data. For every adaptive program and fuel, this preserves the actual regularized execution, including its state, halt/exhaustion outcome and query count.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem simulate_regularized
    {E S X : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
    (next : S → Sum X E) (update : S → E → (ℝ × E) → S)
    {f : E → ℝ} (hd : Differentiable ℝ f) (hc : ConvexOn ℝ univ f)
    {β δ : ℝ≥0} (hL : LipschitzWith β (gradient f)) (u : E) (fuel : ℕ) (s : S) :
    let W := fun x => f x + (δ : ℝ)/2*‖x-u‖^2
    let corrected := fun s x (a : ℝ × E) =>
      update s x (a.1 + (δ : ℝ)/2*‖x-u‖^2, a.2 + (δ : ℝ) • (x-u))
    let base := run next corrected (fun x => (f x, gradient f x)) fuel s
    let target := run next update (fun x => (W x, gradient W x)) fuel s
    base = target ∧ base.2 ≤ fuel := by
  dsimp only
  have hg := (QuadraticRegularizationFirstOrder.curvature_gradient_and_smoothness
    hd hc hL (δ := δ) u).2.2.1
  induction fuel generalizing s with
  | zero => cases h : next s <;> simp [run, h]
  | succ n ih =>
    cases h : next s with
    | inl x => simp [run, h]
    | inr x =>
      have step (upd : S → E → (ℝ × E) → S) (O : E → ℝ × E) :
          run next upd O (n+1) s =
            let tail := run next upd O n (upd s x (O x))
            (tail.1, tail.2+1) := by rw [run, h]
      simp only [step, hg]
      obtain ⟨he, hn⟩ := ih (update s x
        (f x + (δ : ℝ)/2*‖x-u‖^2, gradient f x + (δ : ℝ) • (x-u)))
      refine ⟨?_, Nat.succ_le_succ hn⟩
      simpa only [hg] using congrArg (fun r => (r.1, r.2+1)) he
end AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationOracle

Open AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationOracle.lean:33published source at 0e31a3cda412

Proof architecture

Actual adaptive first-order oracle execution with one original reply per regularized reply.

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