Alternating Subgradient Methods for Convex-Concave Saddle-Point Problems
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Bibliographic record
Abstract
We propose an alternating subgradient method with non-constant step sizes for solving convex-concave saddle-point problems associated with general convex-concave functions. We assume that the sequence of our step sizes is not summable but square summable. Then under the popular assumption of uniformly bounded subgradients, we prove that a sequence of convex combinations of function values over our iterates converges to the value of the function at a saddle-point. Additionally, based on our result regarding the boundedness of the sequence of our iterates, we show that a sequence of the function evaluated at convex combinations of our iterates also converges to the value of the function over a saddle-point. We implement our algorithms in examples of a linear program in inequality form, a least-squares problem with $\ell_{1}$ regularization, a matrix game, and a robust Markowitz portfolio construction problem. To accelerate convergence, we reorder the sequence of step sizes in descending order, which turned out to work very-well in our examples. Our convergence results are confirmed by our numerical experiments. Moreover, we also numerically compare our iterate scheme with iterates schemes associated with constant step sizes. Our numerical results support our choice of step sizes. Additionally, we observe the convergence of the sequence of function values over our iterates in multiple experiments, which currently lacks theoretical support.
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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 | 0.008 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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