Limit Theorems for Stochastic Gradient Descent in High-Dimensional Single-Layer Networks
Bibliographic record
Abstract
This paper studies the high-dimensional scaling limits of online stochastic gradient descent (SGD). Building on the work of Ben Arous, Gheissari, and Jagannath on the effective dynamics of SGD, we study the critical scaling regime of the step size for single-layer networks. Below this regime the effective dynamics are governed by deterministic (ballistic) limits, whereas at the critical scale a correction term emerges that changes the phase diagram. Near the fixed points of these dynamics, we show that the diffusive (SDE) limit of the rescaled correlation is an Ornstein-Uhlenbeck process. More precisely, it is mean-reverting whenever the information exponent is at least three. At information exponent two the drift has no universal sign, and the fixed point may become repelling; we show this explicitly for phase retrieval, where the sign is determined by the step size and the noise level. These results illustrate the limitations of deterministic scaling limits in capturing stochastic fluctuations in high-dimensional learning dynamics.
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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 | 0.030 |
| 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 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 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".