optimal_gradient_step · Samplinglib

Plain-English statement

- The step `2/(α+β)` gives the sharp uniform curvature-envelope contraction. Its factor minimizes the endpoint max-envelope over every real step. The `α=0` boundary is nonexpansive, and `α=β` is retained.

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 optimal_gradient_step {f : E → ℝ} {α β : ℝ}
    (hf : ContDiff ℝ 2 f) (hsc : StrongConvexOn univ α f)
    (hα : 0 ≤ α) (hβ : 0 < β) (hαβ : α ≤ β)
    (hu : ∀ x y, f y ≤ f x + inner ℝ (gradient f x) (y - x) + β / 2 * ‖y - x‖ ^ 2)
    (x y : E) :
    ‖(y - (2 / (α + β)) • gradient f y) - (x - (2 / (α + β)) • gradient f x)‖ ≤
        ((β - α) / (α + β)) * ‖y - x‖ ∧
      ∀ h : ℝ, (β - α) / (α + β) ≤ max |1 - h * α| |1 - h * β| := by
  have hD : 0 < α + β := add_pos_of_nonneg_of_pos hα hβ
  have hq : 0 ≤ (β - α) / (α + β) := div_nonneg (sub_nonneg.mpr hαβ) hD.le
  have ha : 1 - 2 / (α + β) * α = (β - α) / (α + β) := by field_simp; ring
  have hb : 1 - 2 / (α + β) * β = -((β - α) / (α + β)) := by field_simp; ring
  constructor
  · have hc := gradient_step_endpoint_bound hf hsc (show 0 ≤ 2 / (α + β) by positivity) hu x y
    simpa only [ha, hb, abs_neg, abs_of_nonneg hq, max_self] using hc
  · intro h
    let M := max |1 - h * α| |1 - h * β|
    have ha' : 1 - h * α ≤ M := (le_abs_self _).trans (le_max_left _ _)
    have hb' : -(1 - h * β) ≤ M := (neg_le_abs _).trans (le_max_right _ _)
    apply (div_le_iff₀ hD).mpr
    have h1 := mul_le_mul_of_nonneg_left ha' hβ.le
    have h2 := mul_le_mul_of_nonneg_left hb' hα
    nlinarith

end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep

Open AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentOptimalStep.lean:102published source at 0e31a3cda412

Proof architecture

Balanced step 2/(alpha+beta) gives factor (beta-alpha)/(alpha+beta), minimizing the endpoint envelope over all real steps.

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