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arxiv: 1907.03132 · v1 · pith:74UGH5IZnew · submitted 2019-07-06 · 🧮 math.OC · cs.NA· math.NA

On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

Bangti Jin , Zehui Zhou , Jun Zou This is my paper

Pith reviewed 2026-05-25 01:43 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords stochastic gradient descentnonlinear ill-posed problemsregularizationconvergence ratestangential cone conditionLandweber iterationinverse problemsHilbert spaces
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The pith

Stochastic gradient descent regularizes nonlinear ill-posed inverse problems when stopped by a priori rules under the tangential cone condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a randomized version of the Landweber iteration, which draws one equation at random per step to estimate the gradient, acts as a regularizer for nonlinear ill-posed problems in Hilbert spaces. The analysis proves that the iterates remain stable and converge to a solution of the noise-free problem when an a priori stopping rule is applied. Under additional sourcewise and range invariance conditions, explicit convergence rates are derived. Because each iteration uses only a single equation, the method avoids computing the full gradient and therefore scales to large systems.

Core claim

Under the canonical tangential cone condition, the stochastic gradient descent iteration satisfies the regularizing property for a priori stopping rules; convergence rates then follow from suitable sourcewise and range invariance conditions.

What carries the argument

The stochastic gradient descent iteration on the nonlinear system, which forms an unbiased gradient estimate by randomly selecting one equation at each step.

Load-bearing premise

The nonlinear forward operator satisfies the tangential cone condition.

What would settle it

A concrete nonlinear operator satisfying the tangential cone condition for which the stochastic iterates diverge or fail to approach the true solution under the stated a priori stopping rule.

read the original abstract

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method randomly chooses one equation from the nonlinear system to obtain an unbiased stochastic estimate of the gradient, and then performs a descent step with the estimated gradient. It is a randomized version of the classical Landweber method for nonlinear inverse problems, and it is highly scalable to the problem size and holds significant potentials for solving large-scale inverse problems. Under the canonical tangential cone condition, we prove the regularizing property for a priori stopping rules, and then establish the convergence rates under suitable sourcewise condition and range invariance condition.

Editorial analysis

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Referee Report

0 major / 4 minor

Summary. The manuscript analyzes a stochastic gradient descent method (randomized Landweber iteration) for nonlinear ill-posed inverse problems in Hilbert spaces. At each step a single equation is chosen uniformly to form an unbiased estimator of the gradient; the iteration is stopped by an a priori rule. Under the tangential cone condition the method is shown to be regularizing; under additional sourcewise and range-invariance conditions convergence rates are derived that match the deterministic case.

Significance. If the proofs hold, the result supplies the first rigorous regularization theory for a scalable, unbiased stochastic variant of the classical Landweber method. The fact that the same structural hypotheses (tangential cone, source condition, range invariance) suffice, with the stochastic gradient entering only through its unbiasedness, is a clean and useful extension. The work directly addresses the need for theoretically justified iterative solvers on large-scale nonlinear inverse problems.

minor comments (4)
  1. The statement of the tangential cone condition (presumably §2) should explicitly record the constant δ and the radius r in which it holds; these parameters appear in the stopping-rule analysis but are not carried through the rate statements.
  2. Notation for the stochastic index selection (uniform over the m equations) is introduced only in the abstract and the introduction; a dedicated paragraph in §2 would improve readability.
  3. The range-invariance condition is used to obtain the rate but is not compared with the weaker conditions that suffice for the deterministic Landweber method; a short remark would clarify the trade-off.
  4. Several displayed equations in the rate proof contain the same generic constant C without distinguishing its dependence on the source parameter; re-labeling would avoid confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives regularizing properties and convergence rates for stochastic Landweber iteration directly from the tangential cone condition (plus sourcewise and range-invariance assumptions) that are stated as external structural hypotheses on the nonlinear forward operator. The argument treats the stochastic gradient as an unbiased estimator whose expectation recovers the deterministic descent direction, with the same nonlinearity control; this is a standard reduction and does not presuppose the target rates or reduce any claimed result to a fitted parameter or self-citation chain. No load-bearing step is shown to be equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claims rest on three domain-standard assumptions in nonlinear inverse-problem theory; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • domain assumption Tangential cone condition on the nonlinear operator
    Invoked to prove the regularizing property for a priori stopping rules
  • domain assumption Sourcewise condition
    Required to obtain convergence rates
  • domain assumption Range invariance condition
    Required to obtain convergence rates

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