ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Power index functions give polynomial Bregman rates

Example

Let $X, Y$ be real Banach spaces, $D (F) \subseteq X$ , $F : D (F) \to Y$ , and let $R : X \to (- \infty, + \infty]$ be proper and convex. Fix $x^{†} \in D (F) \cap dom R$ , $y = F (x^{†})$ , $ξ^{†} \in \partial R (x^{†})$ , $1 \leq p < \infty$ , $A > 0$ , $0 < μ < 1$ , $0 \leq β < 1$ , and $R_{0} > R (x^{†})$ . Write $D_{R}^{ξ^{†}} (x, x^{†}) : = R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X}$ , and suppose

$- ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X} \leq β D_{R}^{ξ^{†}} (x, x^{†}) + A ∥ F (x) - F (x^{†}) ∥_{Y}^{p μ} (x \in D (F), R (x) \leq R_{0}) .$

For data $y^{δ}$ with $∥ y^{δ} - y ∥_{Y} \leq δ$ , suppose $0 \leq ε_{δ} \leq C_{ε} δ^{p}$ and $x_{α}^{δ, ε} \in D (F)$ satisfies $J_{α}^{δ} (x_{α}^{δ, ε}) \leq \inf_{z \in D (F)} J_{α}^{δ} (z) + ε_{δ}$ , where $J_{α}^{δ} (x) : = p^{- 1} ∥ F (x) - y^{δ} ∥_{Y}^{p} + α R (x)$ . Set $c_{p} = 2^{1 - p} / p$ , $H (q) : = \sup_{s \geq 0} (A s^{μ} - q s)$ , and choose $α (δ) ≍ δ^{p (1 - μ)}$ . Then $H (c_{p} / α) ≍ δ^{p μ}$ and $α H (c_{p} / α) ≍ δ^{p}$ . The fixed-scale conjugate and penalty-index conditions hold. Hence $D_{R}^{ξ^{†}} (x_{α}^{δ, ε}, x^{†}) = O (δ^{p μ})$ and $∥ F (x_{α}^{δ, ε}) - y^{δ} ∥_{Y} = O (δ)$ ; if $D_{R}^{ξ^{†}} (x, x^{†}) \geq c_{R} ∥ x - x^{†} ∥_{X}^{s}$ on $L_{R_{0}}$ for constants $c_{R}, s > 0$ , then $∥ x_{α}^{δ, ε} - x^{†} ∥_{X} = O (δ^{p μ / s})$ .

Example references

Variational source condition hypotheses

Conjugate error bound

Banach VSC rate theorem

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, equation (2.25a)

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equation (1.32a)

Facts & Assumptions

Given: The conditions in the Example hold, with $α (δ) ≍ δ^{p (1 - μ)}$ .

[F1]

Variational source conditions yield nonlinear regularization rates: under the VSC, approximate-minimizer tolerance, and conjugate parameter coupling, the Bregman distance is $O (H (c_{p} / α))$ and the residual is $O (δ)$ .

[F2]

A conjugate bound for residual and Bregman errors: $H (q) = \sup_{s \geq 0} (ψ (s) - q s)$ is the conjugate term in the residual/Bregman bound.

[A1]

$A > 0$ , $0 < μ < 1$ , and $p \geq 1$ are fixed, independent of $δ$ .

Verification

1.1A1algebra

For $q > 0$ , the function $s \mapsto A s^{μ} - q s$ has derivative $A μ s^{μ - 1} - q$ , which decreases from $+ \infty$ to $- q$ and vanishes only at $s_{q} = (A μ / q)^{1 / (1 - μ)}$ . The function is increasing before $s_{q}$ and decreasing after it, with value $0$ at $s = 0$ and limit $- \infty$ as $s \to \infty$ , so this point is its global maximizer. Substitution gives $H (q) = (1 - μ) A^{1 / (1 - μ)} μ^{μ / (1 - μ)} q^{- μ / (1 - μ)}$ .

2.1A1F1F2step 1.1algebra∎

With $q = c_{p} / α$ this gives $H (c_{p} / α) = C_{p, A, μ} α^{μ / (1 - μ)}$ , so the chosen $α$ yields $H (c_{p} / α) ≍ δ^{p μ}$ and $α H (c_{p} / α) ≍ δ^{p}$ . Also $H (c_{p} / (2 α)) = 2^{μ / (1 - μ)} H (c_{p} / α)$ , while $ψ (C_{0} δ^{p}) = A C_{0}^{μ} δ^{p μ} = O (H (c_{p} / α))$ for every fixed $C_{0} > 0$ . Thus the parameter and penalty conditions in [F1] hold, and [F1] gives the stated Bregman and residual rates. Under the additional lower bound with $c_{R}, s > 0$ , division by $c_{R}$ gives $∥ x_{α}^{δ, ε} - x^{†} ∥_{X}^{s} = O (δ^{p μ})$ ; taking the positive $s$ -th root gives the norm rate.

Depends on

Used by

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Sources

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, equation (2.25a) (standard reference, not scraped)
Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equation (1.32a) (standard reference, not scraped)