Plain-English statement
- Quadratic regularization of a differentiable convex objective: actual curvature, gradient and smoothness, including zero regularization.
Read the mathematics first, then descend into Lean
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Proof architecture
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Lean statement
theorem curvature_gradient_and_smoothness
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
{f : E → ℝ} (hd : Differentiable ℝ f) (hc : ConvexOn ℝ univ f)
{β δ : ℝ≥0} (hL : LipschitzWith β (gradient f)) (u : E) :
let W := fun x => f x + (δ : ℝ)/2*‖x-u‖^2
Differentiable ℝ W ∧ StrongConvexOn univ (δ : ℝ) W ∧
(∀ x, gradient W x = gradient f x + (δ : ℝ) • (x-u)) ∧
LipschitzWith (β+δ) (gradient W) := by
let W := fun x => f x + (δ : ℝ)/2*‖x-u‖^2
have hq (x : E) : HasFDerivAt (fun z => (δ : ℝ)/2*‖z-u‖^2)
((δ : ℝ) • innerSL ℝ (x-u)) x := by
convert (((hasFDerivAt_id x).sub_const u).norm_sq).const_mul ((δ : ℝ)/2)
using 1 <;> first | rfl | (ext v; simp; ring)
have hWd : Differentiable ℝ W := fun x => ((hd x).hasFDerivAt.add (hq x)).differentiableAt
have hgrad (x : E) : gradient W x = gradient f x + (δ : ℝ) • (x-u) := by
have hfd : fderiv ℝ W x = fderiv ℝ f x + (δ : ℝ) • innerSL ℝ (x-u) :=
((hd x).hasFDerivAt.add (hq x)).fderiv
rw [gradient, hfd, map_add, map_smul]
congr 1
exact congrArg (fun v : E => (δ : ℝ) • v) ((toDual ℝ E).symm_apply_apply (x-u))
have hstrong : StrongConvexOn univ (δ : ℝ) W := by
apply strongConvexOn_iff_convex.mpr
have ha := (hc.add (((-(δ : ℝ)) • (innerSL ℝ u).toLinearMap).convexOn convex_univ)).add_const
((δ : ℝ)/2*‖u‖^2)
convert! ha using 1
ext x
simp [W, norm_sub_sq_real, real_inner_comm]
ring
refine ⟨hWd,hstrong,hgrad,?_⟩
apply LipschitzWith.of_dist_le_mul
intro x y
rw [dist_eq_norm, hgrad, hgrad]
have he : gradient f x + (δ : ℝ) • (x-u) -
(gradient f y + (δ : ℝ) • (y-u)) =
(gradient f x-gradient f y) + (δ : ℝ) • (x-y) := by
simp only [smul_sub]
abel
rw [he]
have hb := hL.dist_le_mul x y
rw [dist_eq_norm, dist_eq_norm] at hb
calc
_ ≤ ‖gradient f x-gradient f y‖ + ‖(δ : ℝ) • (x-y)‖ := norm_add_le _ _
_ ≤ (β : ℝ)*‖x-y‖ + (δ : ℝ)*‖x-y‖ := by
rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg δ.coe_nonneg]
linarith
_ = ((β+δ : ℝ≥0) : ℝ)*dist x y := by rw [dist_eq_norm]; simp; ring
end AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationFirstOrderOpen AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationFirstOrder.lean:21published source at 0e31a3cda412
Proof architecture
Actual regularized curvature and gradient Lipschitz shift at first-order regularity.
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.
- `calc` records an equality or inequality chain matching a paper calculation.
- `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.
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
- Totalized `fderiv` values must not be read as a differentiability theorem.
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.