Bibliographic record
Abstract
Abstract. We consider the problem of finding an [Formula: see text]-approximate stationary point of a smooth function on a compact domain of [Formula: see text]. In contrast with dimension-free approaches such as gradient descent, we focus here on the case where [Formula: see text] is finite, and potentially small. This viewpoint was explored in 1993 by Vavasis, who proposed an algorithm which, for any fixed finite dimension [Formula: see text] , improves upon the [Formula: see text] oracle complexity of gradient descent. For example for [Formula: see text], Vavasis’s approach obtains the complexity [Formula: see text]. Moreover, for [Formula: see text] he also proved a lower bound of [Formula: see text] for deterministic algorithms (we extend this result to randomized algorithms). Our main contribution is an algorithm, which we call gradient flow trapping (GFT), and the analysis of its oracle complexity. In dimension [Formula: see text], GFT closes the gap with Vavasis’s lower bound (up to a logarithmic factor), as we show that it has complexity [Formula: see text]. In dimension [Formula: see text], we show a complexity of [Formula: see text], improving upon Vavasis’s [Formula: see text]. In higher dimensions, GFT has the remarkable property of being a logarithmic parallel depth strategy, in stark contrast with the polynomial depth of gradient descent or Vavasis’s algorithm. We augment this result with another algorithm, named cut and flow (CF), which improves upon Vavasis’s algorithm in any fixed dimension.
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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 | 0.009 |
| 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 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".