When does preconditioning help or hurt generalization
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
While second order optimizers such as natural gradient descent (NGD) often speed up optimization, their effect on generalization has been called into question. This work presents a more nuanced view on how the implicit bias of optimizers affects the comparison of generalization properties. We provide an exact asymptotic bias-variance decomposition of the generalization error of preconditioned ridgeless regression in the overparameterized regime, and consider the inverse population Fisher information matrix (used in NGD) as a particular example. We determine the optimal preconditioner P for both the bias and variance, and find that the relative generalization performance of different optimizers depends on label noise and “shape” of the signal (true parameters): when the labels are noisy, the model is misspecified, or the signal is misaligned with the features, NGD can achieve lower risk; conversely, GD generalizes better under clean labels, a well-specified model, or aligned signal. Based on this analysis, we discuss several approaches to manage the bias-variance tradeoff, and the potential benefit of interpolating between first- and second-order updates. We then extend our analysis to regression in the reproducing kernel Hilbert space and demonstrate that preconditioning can lead to more efficient decrease in the population risk. Lastly, we empirically compare the generalization error of first- and second-order optimizers in neural network experiments, and observe robust trends matching our theoretical analysis.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.000 |
| 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