A LEVENBERG-MARQUARDT METHOD FOR NONSMOOTH REGULARIZED LEAST SQUARES
Bibliographic record
Abstract
We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squar es term $f(x) = \tfrac{1}{2} \|F(x)\|_2^2$ and a nonsmooth term $h$. Both $f$ and $h$ may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of $h$ using a first-order method such as the proximal gradient method. We establish global convergence to a first-order stationary point of both a trust-region and a regularization variant of the Levenberg-Marquardt method under the assumptions that $F$ and its Jacobian are Lipschitz continuous and $h$ is proper and lower semi-continuous. In the worst case, both methods perform $O(ε^{-2})$ iterations to bring a measure of stationarity below $ε\in (0, 1)$. We report numerical results on three examples: a group-lasso basis-pursuit denoise example, a nonlinear support vector machine, and parameter estimation in neuron firing. For those examples to be implementable, we describe in detail how to evaluate proximal operators for separable $h$ and for the group lasso with trust-region constraint. In all cases, the Levenberg-Marquardt methods perform fewer outer iterations than a proximal-gradient method with adaptive step length and 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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".