Uncertainty quantification from ensemble variance scaling laws in deep neural networks
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Bibliographic record
Abstract
Abstract Quantifying the uncertainty from machine learning analyses is critical to their use in the physical sciences. In this work we focus on uncertainty inherited from the initialization distribution of neural networks. We compute the mean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and variance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> of the test loss <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:math> for an ensemble of multi-layer perceptrons with neural tangent kernel initialization in the infinite-width limit, and compare empirically to the results from finite-width networks for three example tasks: MNIST classification, CIFAR classification and calorimeter energy regression. We observe scaling laws as a function of training set size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> for both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , but find that the coefficient of variation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">L</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> becomes independent of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> at both infinite and finite width for sufficiently large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">D</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . This implies that the coefficient of variation of a finite-width network may be approximated by its infinite-width value, and may in principle be calculable using finite-width perturbation theory.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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