Deep learning and the geometry of compactness in stability and generalization
Bibliographic record
Abstract
Deep learning models often continue to generalize well even when they have far more parameters than available training examples. This observation naturally leads to two questions: why does training remain stable, and why do the resulting predictors generalize at all? To address these questions, we return to the classical Extreme Value Theorem and interpret modern training as optimization over compact sets in parameter space or function space. Our main results show that continuity together with coercive or Lipschitz based regularization gives existence of minimizers and uniform control of the excess risk, by bounding rare high loss events. We apply this framework to weight decay, gradient penalties, and spectral normalization, and we introduce simple diagnostics that monitor compactness in parameter space, representation space, and function space. Experiments on synthetic examples, standard image data sets (MNIST, CIFAR ten, Tiny ImageNet), and the UCI Adult tabular task are consistent with the theory: mild regularization leads to smoother optimization, reduced variation across random seeds, and better robustness and calibration while preserving accuracy. Taken together, these results highlight compactness as a practical geometric guideline for training stable and reliable deep networks.
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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 | 0.006 | 0.035 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.009 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| 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".