exists_gap_le · Samplinglib

Plain-English statement

- A positive finite run contains an iterate satisfying the normalized Optlib subgradient bound, relative to any comparator (not necessarily a minimizer).

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 exists_gap_le {f : E → ℝ} {G : ℝ≥0} {x g : ℕ → E} {a : ℕ → ℝ}
    (hf : LipschitzWith G f) (ha : ∀ k, 0 < a k)
    (hs : ∀ k, SupportsAt f (x k) (g k))
    (hx : ∀ k, x (k + 1) = x k - a k • g k) (z : E) {N : ℕ} (hN : 0 < N) :
    ∃ i < N, f (x i) - f z ≤
      (‖x 0 - z‖ ^ 2 + (G : ℝ) ^ 2 * (∑ k ∈ Finset.range N, (a k) ^ 2)) /
        (2 * ∑ k ∈ Finset.range N, a k) := by
  have hne : (Finset.range N).Nonempty := ⟨0, Finset.mem_range.mpr hN⟩
  obtain ⟨i, hi, hmin⟩ := Finset.exists_min_image (Finset.range N) (fun k => f (x k)) hne
  refine ⟨i, Finset.mem_range.mp hi, ?_⟩
  have hsumpos : 0 < ∑ k ∈ Finset.range N, a k :=
    Finset.sum_pos (fun k _ => ha k) hne
  have hlower : (∑ k ∈ Finset.range N, a k) * (f (x i) - f z) ≤
      ∑ k ∈ Finset.range N, a k * (f (x k) - f z) := by
    rw [Finset.sum_mul]
    exact Finset.sum_le_sum fun k hk =>
      mul_le_mul_of_nonneg_left (sub_le_sub_right (hmin k hk) _) (ha k).le
  have he := sum_weighted_gap_le hf (fun k => (ha k).le) hs hx z N
  apply (le_div_iff₀ (mul_pos (by norm_num : (0 : ℝ) < 2) hsumpos)).mpr
  nlinarith [sq_nonneg ‖x N - z‖]

end AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod

Open AutoSamplingTheory/TechnicalLemmas/Analysis/SubgradientMethod.lean:90published source at ed709a7f320d

Proof architecture

Finite best-iterate guarantee for a supplied nonsmooth subgradient run and arbitrary comparator

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