logarithmic_accuracy_and_cost

Plain-English statement

- A rounded logarithmic horizon gives actual restart accuracy and explicit cost, including zero rounds; budget absorption and small-error order are conditional.

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 logarithmic_accuracy_and_cost {f : E → ℝ} {α β R ε : ℝ}
    (hf : ContDiff ℝ 1 f) (hsc : StrongConvexOn univ α f)
    (hα : 0 < α) (hβ : 0 < β) (hR : 0 < R) (hε : 0 < ε)
    {z x₀ : E} (hmin : IsMinOn f univ z) (hx₀ : ‖x₀-z‖ ≤ R)
    (A : E → ℝ → ℝ → E × ℕ) (φ : ℝ → ℕ)
    (hA : ∀ y r δ, 0 < r → 0 < δ → ‖y-z‖ ≤ r →
      f (A y r δ).1 - f z ≤ δ ∧ (A y r δ).2 ≤ φ (β*r^2/δ)) :
    let L := Real.log (α*R^2/ε) / Real.log 4
    let N := ⌈L⌉₊
    let r := fun k : ℕ => R / (2:ℝ)^k
    let run : ℕ → E × ℕ := Nat.rec (x₀, 0) (fun k p =>
      let w := A p.1 (r k) (α*(r k)^2/8)
      (w.1, p.2 + w.2))
    let last := A (run N).1 (Real.sqrt (ε/α)) ε
    let C := (run N).2 + last.2
    (N : ℝ) ≤ max 0 L + 1 ∧ (N = 0 ↔ α*R^2 ≤ ε) ∧
      ‖(run N).1-z‖ ≤ r N ∧ f last.1 - f z ≤ ε ∧
      C ≤ N * φ (8*(β/α)) + φ (β/α) ∧
      (C : ℝ) ≤ (max 0 L + 1) * φ (8*(β/α)) + φ (β/α) ∧
      (φ (β/α) ≤ φ (8*(β/α)) →
        (C : ℝ) ≤ (max 0 L + 2) * φ (8*(β/α)) ∧
        (4 ≤ α*R^2/ε → (C : ℝ) ≤ 3*L*φ (8*(β/α)))) := by
  let L := Real.log (α*R^2/ε) / Real.log 4
  let N := ⌈L⌉₊
  have hlog : 0 < Real.log 4 := Real.log_pos (by norm_num)
  have hq : 0 < α*R^2/ε := by positivity
  have hround : (N : ℝ) ≤ max 0 L + 1 := by
    by_cases hL : 0 ≤ L
    · exact (Nat.ceil_lt_add_one hL).le.trans (by rw [max_eq_right hL])
    · have hz : N = 0 := Nat.ceil_eq_zero.mpr (le_of_not_ge hL)
      rw [hz, Nat.cast_zero, max_eq_left (le_of_not_ge hL)]
      norm_num
  have hzero : N = 0 ↔ α*R^2 ≤ ε := by
    change ⌈Real.log (α*R^2/ε) / Real.log 4⌉₊ = 0 ↔ _
    rw [Nat.ceil_eq_zero, div_le_iff₀ hlog, zero_mul]
    rw [← Real.log_one, Real.log_le_log_iff hq (by norm_num)]
    exact (div_le_iff₀ hε).trans (by simp)
  have hpow : α*R^2/ε ≤ (4:ℝ)^N := by
    apply (Real.log_le_log_iff hq (by positivity)).mp
    rw [Real.log_pow]
    exact (div_le_iff₀ hlog).mp (Nat.le_ceil L)
  have hid : ((2:ℝ)^N)^2 = (4:ℝ)^N := by
    rw [← pow_mul, Nat.mul_comm N 2, pow_mul]
    norm_num
  have hN : α*(R/(2:ℝ)^N)^2 ≤ ε := by
    rw [div_pow, ← mul_div_assoc, hid]
    apply (div_le_iff₀ (by positivity : 0 < (4:ℝ)^N)).mpr
    have h := (div_le_iff₀ hε).mp hpow
    nlinarith
  have hb := RestartReduction.radius_accuracy_and_cost hf hsc hα hβ hR hε
    hmin hx₀ A φ hA N hN
  let r := fun k : ℕ => R / (2:ℝ)^k
  let run : ℕ → E × ℕ := Nat.rec (x₀, 0) (fun k p =>
    let w := A p.1 (r k) (α*(r k)^2/8)
    (w.1, p.2 + w.2))
  let last := A (run N).1 (Real.sqrt (ε/α)) ε
  let C := (run N).2 + last.2
  have hcast : (C : ℝ) ≤ (N : ℝ) * φ (8*(β/α)) + φ (β/α) := by
    have hc : C ≤ N * φ (8*(β/α)) + φ (β/α) := hb.2.2.2
    simpa only [Nat.cast_add, Nat.cast_mul] using ((Nat.cast_le (α := ℝ)).mpr hc)
  have hm : (N : ℝ) * φ (8*(β/α)) ≤
      (max 0 L + 1) * φ (8*(β/α)) :=
    mul_le_mul_of_nonneg_right hround (Nat.cast_nonneg _)
  have hreal : (C : ℝ) ≤ (max 0 L + 1) * φ (8*(β/α)) + φ (β/α) := by
    linarith
  refine ⟨hround, hzero, hb.1, hb.2.2.1, hb.2.2.2, hreal, ?_⟩
  intro hφ
  have hp : (φ (β/α) : ℝ) ≤ φ (8*(β/α)) := by exact_mod_cast hφ
  have ha : (C : ℝ) ≤ (max 0 L + 2) * φ (8*(β/α)) := by nlinarith
  refine ⟨ha, ?_⟩
  intro hsmall
  have hL : 1 ≤ L := by
    apply (le_div_iff₀ hlog).mpr
    simpa using (Real.log_le_log (by norm_num : (0:ℝ)<4) hsmall)
  rw [max_eq_right (by linarith : 0 ≤ L)] at ha
  have hfactor : L+2 ≤ 3*L := by linarith
  exact ha.trans (mul_le_mul_of_nonneg_right hfactor (Nat.cast_nonneg _))

end AutoSamplingTheory.TechnicalLemmas.Analysis.RestartLogComplexity

Open AutoSamplingTheory/TechnicalLemmas/Analysis/RestartLogComplexity.lean:22published source at 0e31a3cda412

Proof architecture

Actual logarithmic restart horizon, final accuracy and explicit certified call-cost bounds.

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

No additional hidden-contract keyword was inferred; the exact Lean hypotheses remain controlling.