A Levenberg–Marquardt Method for Nonsmooth Regularized Least Squares
Bibliographic record
Abstract
Abstract. We develop a Levenberg–Marquardt method for minimizing the sum of a smooth nonlinear least-squares term [Formula: see text] and a nonsmooth term [Formula: see text]. Both [Formula: see text] and [Formula: see text] may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of [Formula: see text] using a first-order method such as the proximal gradient method. We establish global convergence to a first-order stationary point under the assumptions that [Formula: see text] and its Jacobian are Lipschitz continuous and [Formula: see text] is proper and lower semicontinuous. In the worst case, our method performs [Formula: see text] iterations to bring a measure of stationarity below [Formula: see text]. We also derive a trust-region variant that enjoys similar asymptotic worst-case iteration complexity as a special case of the trust-region algorithm of Aravkin, Baraldi, and Orban [ SIAM J. Optim., 32 (2022), pp. 900–929]. We report numerical results on three examples: a group-lasso basis-pursuit denoise example, a nonlinear support vector machine, and parameter estimation in a neuroscience application. To implement those examples, we describe in detail how to evaluate proximal operators for separable [Formula: see text] and for the group lasso with trust-region constraint. In all cases, the Levenberg–Marquardt methods perform fewer outer iterations than either a proximal gradient method with adaptive step length or a quasi-Newton trust-region method, neither of which exploit the least-squares structure of the problem. Our results also highlight the need for more sophisticated subproblem solvers than simple first-order methods.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.006 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".