PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization
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Bibliographic record
Abstract
In this paper, we propose a novel stochastic gradient estimator-ProbAbilistic Gradient Estimator (PAGE)-for nonconvex optimization. PAGE is easy to implement as it is designed via a small adjustment to vanilla SGD: in each iteration, PAGE uses the vanilla minibatch SGD update with probability p t or reuses the previous gradient with a small adjustment, at a much lower computational cost, with probability 1 - p(t). We give a simple formula for the optimal choice of p(t). Moreover, we prove the first tight lower bound Omega (n + root n/epsilon(2)), for non-convex finite-sum problems, which also leads to a tight lower bound Omega (b + root b/epsilon(2)) for non- convex online problems, where b := min{sigma(2)/epsilon(2), n} . Then, we show that PAGE obtains the optimal convergence results O(n + root n/epsilon(2)) (finite-sum) and O(b + root b/is an element of(2)) (online) matching our lower bounds for both nonconvex finite-sum and online problems. Besides, we also show that for nonconvex functions satisfying the Polyak-Lojasiewicz (PL) condition, PAGE can automatically switch to a faster linear convergence rate O(. log 1/epsilon). Finally, we conduct several deep learning experiments (e.g., LeNet, VGG, ResNet) on real datasets in PyTorch showing that PAGE not only converges much faster than SGD in training but also achieves the higher test accuracy, validating the optimal theoretical results and confirming the practical superiority of PAGE.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.006 |
| Science and technology studies | 0.003 | 0.020 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it