Data-Dependent Complexity of First-Order Methods for Binary Classification
Bibliographic record
Abstract
Large-scale problems in data science are often modeled with optimization, and the optimization model is usually solved with first-order methods that may converge at a sublinear rate. Therefore, it is of interest to terminate the optimization algorithm as soon as the underlying data science task is accomplished. We consider FISTA for solving two binary classification problems: the ellipsoid separation problem (ESP), and the soft-margin support-vector machine (SVM). For the ESP, we cast the dual second-order cone program into a form amenable to FISTA and show that the FISTA residual converges to the infimal displacement vector of the primal-dual hybrid gradient (PDHG) algorithm, that directly encodes a separating hyperplane. We further derive a data-dependent iteration upper bound scaling as $\mathcal{O}(1/δ_{\mathcal{A}}^2)$, where $δ_{\mathcal{A}}$ is the minimal perturbation that destroys separability. For the SVM, we propose a strongly-concave perturbed dual that admits efficient FISTA updates under a linear time projection scheme, and with our parameter choices, the objective has small condition number, enabling rapid convergence. We prove that, under a reasonable data model, early-stopped iterates identify well-classified points and yield a hyperplane that exactly separates them, where the accuracy required of the dual iterate is governed by geometric properties of the data. In particular, the proposed early-stopping criteria diminish the need for hard-to-select tolerance-based stopping conditions. Our numerical experiments on ESP instances derived from MNIST data and on soft-margin SVM benchmarks indicate competitive runtimes and substantial speedups from stopping early.
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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 | 0.008 | 0.041 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".