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On the Variance of Neural Network Training with respect to Test Sets and Distributions
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Shows that standard CIFAR-10 and ImageNet trainings have large test-set variance but tiny distribution-wise variance, explained by independent errors and class-wise calibration, with a formula predicting variance.
Problem
Repeated runs of neural network training vary substantially in test-set accuracy; it is unclear whether this reflects genuine differences in model quality or finite-sample noise, and how to estimate that variance a priori.
Approach
Decomposes test-set variance into distribution-wise and example-wise terms, gives an unbiased estimator of distribution-wise variance, and proves that test-set variance follows from class-wise calibration, yielding a simple formula (err/2n) for binary tasks.
Key results
- On CIFAR-10 64-epoch ResNet-9 training, distribution-wise standard deviation estimated at 0.033%, versus 0.56% test-set std at 4 epochs and 0.149% observed at 64 epochs.
- The binary formula E[Var(err_S)] ≈ err/2n predicts test-set std across 511 CIFAR-10 binary tasks with R^2 = 0.996, about 70x more accurate in MSE than the binomial assumption.
- BERT-LARGE finetuning on MRPC has distribution-wise std of 2.08% versus 0.21% for BERT-BASE; distribution-shifted ImageNet test sets have distribution-wise std of 0.18-0.26% versus 0.034% on the ImageNet validation set.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train CIFAR-10 ResNet-9 for 64 epochs instead of 4 epochs | CIFAR-10 ResNet-9 training · CIFAR-10 test distribution | distribution-wise error stddev: 0.52\% to 0.033\% | Helped |
| Train for 64 epochs rather than 4 epochs | CIFAR-10 ResNet-9 training · CIFAR-10 test set | test-set accuracy stddev: 0.56\% to 0.149\% | Helped |
| Assume independent errors for 64-epoch training | CIFAR-10 ResNet-9 training · CIFAR-10 test set | accuracy stddev: 0.145\% predicted; 0.149\% empirical | No clear effect |
| Remove data augmentation (ablation) | CIFAR-10 ResNet-9 training · CIFAR-10 | variance between runs: 3.5$\times$ more variance | Hurt |
| Add data augmentation rather than remove it (ablation) | CIFAR-10 ResNet-9 training · CIFAR-10 | variance between runs: 3.5$\times$ more variance without augmentation | Helped |
| Remove 80% of training data (ablation) | CIFAR-10 ResNet-9 training · CIFAR-10 | large-ensemble accuracy: 89.8\% | Hurt |
| Remove data augmentation rather than remove 80% of training data (ablation) | CIFAR-10 ResNet-9 training · CIFAR-10 | large-ensemble accuracy: 89.8\% to 91.2\% | Helped |
| Raise learning rate from 0.5 to 1.0 | CIFAR-10 ResNet-9, 64 epochs · CIFAR-10 test set | test-set accuracy stddev: 0.148\% to 0.168\% | Hurt |
| Increase model scale from BERT_BASE to BERT_LARGE | BERT finetuning · MRPC test distribution | distribution-wise error stddev: 0.21\% to 2.08\% | Hurt |
| Evaluate ImageNet-trained ResNet-18 on shifted distributions instead of ImageNet validation | ResNet-18 ImageNet training · ImageNet-R, ObjectNet, and ImageNet-Sketch | distribution-wise accuracy stddev: 0.034\%; 0.181\%, 0.179\%, and 0.257\% | Hurt |
| Use class-calibration formula instead of the binomial assumption | Binary classification variance prediction · CIFAR-10 binary classification tasks | mean squared distance to empirical variance: 70$\times$ more accurate; $R^2 = 0.996$ | Helped |
Surprising findings
- Lucky random seeds on one test split do not outperform average on a disjoint split when training to convergence (top 1/4 only 0.02% better).
- Variance on distribution-shifted test sets (ImageNet-Sketch, ObjectNet, ImageNet-R) is several times larger than on in-distribution sets, even though standard trainings are otherwise near-zero variance.
Limitations
- Distribution-shift and pathological-instability cases (BERT-LARGE) show large distribution-wise variance, not explained by the theory.
- Binary calibration formula is exact only for binary tasks; general k-way result is only a lower bound.
- Data augmentation mechanism behind variance reduction is not identified; preliminary studies only.
Experimental setup
Datasets: CIFAR-10 ImageNet ImageNet-V2 ImageNet-R ObjectNet ImageNet-Sketch MRPC
Benchmarks: CIFAR-10 ImageNet ImageNet-V2 ImageNet-R ObjectNet ImageNet-Sketch MRPC
Metrics: test-set accuracy standard deviation distribution-wise variance estimate R^2 mean squared error of variance prediction
Baselines: binomial assumption regularized linear model
Builds on: class-wise calibration (Jiang et al.) disagreement rate theory (Jiang et al.) linear mode connectivity
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