exists_gradient_descent_norm_le

Plain-English statement

- Among the first `N` actual gradient iterates, one has small gradient norm. This is a best-iterate guarantee, not a last-iterate or global optimality guarantee.

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_gradient_descent_norm_le {f : E → ℝ} {β h : ℝ} {z : E}
    (hz : IsMinOn f univ z) (hh : 0 < h) (hstep : β * h ≤ 1)
    (hu : ∀ x y, f y ≤ f x + inner ℝ (gradient f x) (y - x) + β / 2 * ‖y - x‖ ^ 2)
    (x₀ : E) {N : ℕ} (hN : 0 < N) :
    ∃ k ∈ Finset.range N,
      ‖gradient f ((fun x => x - h • gradient f x)^[k] x₀)‖ ≤
        Real.sqrt (2 * (f x₀ - f z) / ((N : ℝ) * h)) := by
  let T : E → E := fun x => x - h • gradient f x
  let B : ℝ := 2 * (f x₀ - f z) / ((N : ℝ) * h)
  have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
  have hd := gradient_descent_sum_sq_bound hh.le hstep hu x₀ N
  have hzN : f z ≤ f (T^[N] x₀) := hz (mem_univ _)
  have hsum : (∑ k ∈ Finset.range N, ‖gradient f (T^[k] x₀)‖ ^ 2) ≤
      ∑ _k ∈ Finset.range N, B := by
    have hb : h / 2 * ((N : ℝ) * B) = f x₀ - f z := by
      dsimp [B]
      field_simp
    simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
    apply (mul_le_mul_iff_right₀ (show 0 < h / 2 by positivity)).mp
    change h / 2 * (∑ k ∈ Finset.range N, ‖gradient f (T^[k] x₀)‖ ^ 2) ≤
      h / 2 * ((N : ℝ) * B)
    rw [hb]
    change h / 2 * (∑ k ∈ Finset.range N, ‖gradient f (T^[k] x₀)‖ ^ 2) ≤
      f x₀ - f (T^[N] x₀) at hd
    linarith
  obtain ⟨k, hk, hkle⟩ := Finset.exists_le_of_sum_le ⟨0, Finset.mem_range.mpr hN⟩ hsum
  refine ⟨k, hk, ?_⟩
  exact Real.le_sqrt_of_sq_le hkle

end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity

Open AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentStationarity.lean:49published source at 0e31a3cda412

Proof architecture

Nonconvex best-iterate gradient-norm guarantee from actual updates, supplied global minimum, positive step and nonempty horizon.

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