Proximal point algorithm, Douglas-Rachford algorithm and alternating\n projections: a case study
Bibliographic record
Abstract
Many iterative methods for solving optimization or feasibility problems have\nbeen invented, and often convergence of the iterates to some solution is\nproven. Under favourable conditions, one might have additional bounds on the\ndistance of the iterate to the solution leading thus to worst case estimates,\ni.e., how fast the algorithm must converge.\n Exact convergence estimates are typically hard to come by. In this paper, we\nconsider the complementary problem of finding best case estimates, i.e., how\nslow the algorithm has to converge, and we also study exact asymptotic rates of\nconvergence. Our investigation focuses on convex feasibility in the Euclidean\nplane, where one set is the real axis while the other is the epigraph of a\nconvex function. This case study allows us to obtain various convergence rate\nresults. We focus on the popular method of alternating projections and the\nDouglas-Rachford algorithm. These methods are connected to the proximal point\nalgorithm which is also discussed. Our findings suggest that the\nDouglas-Rachford algorithm outperforms the method of alternating projections in\nthe absence of constraint qualifications. Various examples illustrate the\ntheory.\n
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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 | 0.003 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".