Plain-English statement
- Weighted final function gap for actual gradient-descent iterates. The coefficient domain is explicit and includes both zero and one.
Read the mathematics first, then descend into Lean
You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.
Proof architecture
Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.
Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.
How to read the exact Lean declaration
Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.
- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.
Source voice and ASTIS voice stay separate
When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.
Lean statement
theorem gradient_descent_weighted_value_bound {f : E → ℝ} {α β h : ℝ}
(hf : ContDiff ℝ 1 f) (hsc : StrongConvexOn univ α f)
(hh : 0 ≤ h) (hstep : β * h ≤ 1) (hcoeff : α * h ≤ 1)
(hu : ∀ x y, f y ≤ f x + inner ℝ (gradient f x) (y - x) + β / 2 * ‖y - x‖ ^ 2)
(x₀ z : E) (N : ℕ) :
2 * h * (∑ k ∈ range N, (1 - α * h) ^ k) *
(f ((fun x => x - h • gradient f x)^[N] x₀) - f z) ≤
(1 - α * h) ^ N * ‖x₀ - z‖ ^ 2 := by
let T : E → E := fun x => x - h • gradient f x
let X : ℕ → E := fun n => T^[n] x₀
let q : ℝ := 1 - α * h
have hq : 0 ≤ q := sub_nonneg.mpr hcoeff
have hX (n : ℕ) : X (n + 1) = T (X n) := Function.iterate_succ_apply' _ _ _
have hm : Antitone (fun n => f (X n)) := by
apply antitone_nat_of_succ_le
intro n
have hd := GradientDescentBasic.gradient_step_descent_of_quadratic_upper_bound hh hstep hu (X n)
rw [hX]
dsimp [T]
have : 0 ≤ h / 2 * ‖gradient f (X n)‖ ^ 2 := by positivity
linarith
have hr (n : ℕ) (_ : 0 ≤ n) :
‖X (n + 1) - z‖ ^ 2 ≤ q * ‖X n - z‖ ^ 2 + (-2 * h * (f (X (n + 1)) - f z)) := by
have he := gradient_step_energy_bound hf hsc hh hstep hu (X n) z
rw [hX]
dsimp [T, q]
linarith
have hg := discrete_gronwall_prod_general (u := fun n => ‖X n - z‖ ^ 2)
(b := fun n => -2 * h * (f (X (n + 1)) - f z)) (c := fun _ => q) hr (fun _ _ => hq) (Nat.zero_le N)
simp only [Finset.prod_const, Nat.card_Ico, Nat.Ico_zero_eq_range, Finset.card_range] at hg
have hs : ∑ k ∈ range N, (-2 * h * (f (X (k + 1)) - f z)) * q ^ (N - (k + 1)) ≤
∑ k ∈ range N, (-2 * h * (f (X N) - f z)) * q ^ (N - (k + 1)) := by
apply sum_le_sum
intro k hk
have hm' := hm (by have := mem_range.mp hk; omega : k + 1 ≤ N)
apply mul_le_mul_of_nonneg_right _ (pow_nonneg hq _)
have hh' : 0 ≤ 2 * h := by positivity
nlinarith
have hsum : (∑ k ∈ range N, q ^ (N - (k + 1))) = ∑ k ∈ range N, q ^ k := by
rw [← sum_range_reflect (fun k => q ^ k) N]
apply sum_congr rfl
intro k _
congr 1
omega
rw [← mul_sum, hsum] at hs
have hu0 : X 0 = x₀ := rfl
rw [hu0] at hg
have huN := sq_nonneg ‖X N - z‖
change 2 * h * (∑ k ∈ range N, q ^ k) * (f (X N) - f z) ≤ q ^ N * ‖x₀ - z‖ ^ 2
nlinarith
end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValueOpen AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentValue.lean:54published source at 0e31a3cda412
Proof architecture
Actual Nth gradient iterate, signed comparator gap and finite geometric weights; normalized rates exercised in Tests.
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.
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.