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Logarithmic index functions give log-rate Bregman convergence

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. Choose $x^{†} \in D (F) \cap dom R$ , set $y = F (x^{†})$ , and choose $ξ^{†} \in \partial R (x^{†})$ . Fix $1 \leq p < \infty$ , $ν > 0$ , $0 < t_{0} < e^{- (ν + 1)}$ , $0 \leq β < 1$ , and $R_{0} > R (x^{†})$ . Define $ψ (0) = 0$ , $ψ (t) = (- \log t)^{- ν}$ for $0 < t < t_{0}$ , and $ψ (t) = (- \log t_{0})^{- ν}$ for $t \geq t_{0}$ . Suppose

$- ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X} \leq β D_{R}^{ξ^{†}} (x, x^{†}) + ψ ⁣ (∥ F (x) - F (x^{†}) ∥_{Y}^{p}) (x \in L_{R_{0}} : = {x \in D (F) : R (x) \leq R_{0}}),$

where $D_{R}^{ξ^{†}} (x, x^{†}) : = R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩_{X^{*}, X}$ . For data $y^{δ} \in Y$ with $∥ y^{δ} - y ∥_{Y} \leq δ$ , let $0 \leq ε_{δ} \leq C_{ε} δ^{p}$ and suppose $x_{α}^{δ, ε} \in D (F)$ satisfies

$J_{α}^{δ} (x_{α}^{δ, ε}) \leq \inf_{z \in D (F)} J_{α}^{δ} (z) + ε_{δ}, J_{α}^{δ} (x) : = p^{- 1} ∥ F (x) - y^{δ} ∥_{Y}^{p} + α R (x) .$

Choose $α (δ) = δ^{p} (\log (1 / δ))^{ν}$ for sufficiently small $0 < δ < 1$ , and write $c_{p} = 2^{1 - p} / p$ . For $H (q) : = \sup_{s \geq 0} (ψ (s) - q s)$ , one has $H (q) \sim (\log q)^{- ν}$ as $q \to \infty$ , so $ρ_{δ} : = H (c_{p} / α (δ)) ≍ (\log (1 / δ))^{- ν}$ and $α (δ) ρ_{δ} ≍ δ^{p}$ . The variational source rate theorem therefore gives

$D_{R}^{ξ^{†}} (x_{α}^{δ, ε}, x^{†}) = O ((\log (1 / δ))^{- ν}), ∥ F (x_{α}^{δ, ε}) - y^{δ} ∥_{Y} = O (δ), ∣ R (x_{α}^{δ, ε}) - R (x^{†}) ∣ = O ((\log (1 / δ))^{- ν}) .$

If also $D_{R}^{ξ^{†}} (x, x^{†}) \geq c_{R} ∥ x - x^{†} ∥_{X}^{s}$ on $L_{R_{0}}$ for constants $c_{R}, s > 0$ , then $∥ x_{α}^{δ, ε} - x^{†} ∥_{X} = O ((\log (1 / δ))^{- ν / s})$ . The parameter coupling here is derived from $α H (c_{p} / α) ≍ δ^{p}$ .

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 (2.25b)

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equations (1.31)–(1.32b)

Facts & Assumptions

Given: The conditions in the Example hold for sufficiently small $0 < δ < 1$ .

[F1]

Variational source conditions on a regularizer sublevel: the VSC uses a continuous, nondecreasing, concave index function vanishing at zero and a coefficient $0 \leq β < 1$ on the stated regularizer sublevel.

[F2]

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

[F3]

Variational source conditions yield nonlinear regularization rates: if $α H (c_{p} / α) ≍ δ^{p}$ and $H (c_{p} / (2 α)) = O (H (c_{p} / α))$ , then the theorem yields Bregman error $O (H (c_{p} / α))$ and noisy-data residual $O (δ)$ ; it also yields the penalty rate when $ψ (C_{0} δ^{p}) = O (H (c_{p} / α))$ .

[A1]

$ν > 0$ and $0 < t_{0} < e^{- (ν + 1)}$ are fixed, and $α (δ) = δ^{p} (\log (1 / δ))^{ν}$ .

Verification

1.1A1F1algebra

For $0 < t < t_{0}$ , put $L = - \log t$ . Differentiation gives $ψ^{'} (t) = ν / (t L^{ν + 1})$ and $ψ^{''} (t) = ν ((ν + 1) - L) / (t^{2} L^{ν + 2}) < 0$ , since $L \geq - \log t_{0} > ν + 1$ . Thus $ψ^{'}$ decreases on $(0, t_{0})$ and has positive left limit at $t_{0}$ ; after $t_{0}$ the function is constant, so its derivative drops to zero and remains nonincreasing. Also $ψ (t) \to 0$ as $t ↓ 0$ . Therefore its continuous extension at zero is nondecreasing, concave on $[0, \infty)$ , vanishes at zero and is positive for $t > 0$ , as required by [F1].

2.1A1step 1.1algebra

For $q > ψ^{'} (t_{0} -)$ , the derivative of $s \mapsto ψ (s) - q s$ decreases from $+ \infty$ to a negative value on $(0, t_{0})$ and is $- q < 0$ for $s > t_{0}$ . Hence the unique global maximizer is $s_{q} \in (0, t_{0})$ satisfying $q = ν / (s_{q} L_{q}^{ν + 1})$ , where $L_{q} = - \log s_{q}$ . This equation gives $\log q = L_{q} - (ν + 1) \log L_{q} + \log ν$ , so $L_{q} \to \infty$ and $L_{q} / \log q \to 1$ . At the maximizer, $q s_{q} = ν L_{q}^{- ν - 1}$ ; consequently $H (q) = L_{q}^{- ν} (1 - ν / L_{q}) \sim (\log q)^{- ν}$ .

3.1A1F2F3step 2.1algebra∎

Set $ℓ_{δ} = \log (1 / δ)$ . Then $α (δ) \to 0$ and $q_{δ} = c_{p} / α (δ) \to \infty$ , while $\log q_{δ} = p ℓ_{δ} - ν \log ℓ_{δ} + \log c_{p} \sim p ℓ_{δ}$ . By step 2.1, $ρ_{δ} = H (q_{δ}) ≍ ℓ_{δ}^{- ν}$ and $α ρ_{δ} ≍ δ^{p}$ . Replacing $q_{δ}$ by $q_{δ} / 2$ changes its logarithm by only $- \log 2$ , so $H (c_{p} / (2 α)) ≍ ρ_{δ}$ . For any fixed $C_{0} > 0$ , eventually $C_{0} δ^{p} < t_{0}$ , and $ψ (C_{0} δ^{p}) = (- \log (C_{0} δ^{p}))^{- ν} ≍ ℓ_{δ}^{- ν} ≍ ρ_{δ}$ . Thus all coupling and penalty-index hypotheses in [F3] hold; applying [F3] gives the three asserted rates. If the additional Bregman lower bound holds, its combination with the Bregman rate gives $∥ x_{α}^{δ, ε} - x^{†} ∥_{X}^{s} = O (ℓ_{δ}^{- ν})$ , and taking the positive $s$ -th root proves the stated norm rate.

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Sources

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, equation (2.25b) (standard reference, not scraped)
Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, equations (1.31)–(1.32b) (standard reference, not scraped)