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Elliptic Loss Regularization
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Proposes elliptic loss regularization, which constrains the loss over the input-label domain to satisfy an elliptic PDE with boundary data from training losses, implemented via Brownian bridge sampling between training points, improving robustness to group imbalance and distribution shift.
Problem
Regularizing neural network losses over regions of the data space away from training points, to anticipate loss behavior under distribution shift and group/class imbalance.
Approach
Impose that the loss landscape over the data-label domain satisfies an elliptic PDE with boundary values given by training losses; solve it approximately by sampling Brownian bridges between pairs of training points (Feynman-Kac view), with optional importance weighting drift from the loss gradient (Girsanov).
Key results
- Elliptic training gets 70.28% top-1 on Tiny-ImageNet with PreActResNet-50 vs 55.91% ERM and 67.22% mixupE.
- On WaterBirds, Elliptic + IW reaches 92.0% average and 84.1% worst-group accuracy, improving over one-stage baselines but below two-stage UMIX worst-group (90.0%).
- Elliptic + IW reaches 77.9% average accuracy on Camelyon17 and 91.3% average on CelebA, the best one-stage results reported.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use elliptic regularization instead of mixupE | PreActRN50 · Tiny-ImageNet 200 | Top-1 accuracy: 67.22$_{\pm0.4}$ -> 70.28$_{\pm0.1}$ | Helped |
| Use elliptic regularization instead of mixupE | PreActRN50 · CIFAR-10 | classification error: 3.53$_{\pm0.05}$ -> 4.15$_{\pm0.43}$ | Hurt |
| Use elliptic regularization instead of ERM | PreActRN50 · CIFAR-100 | classification error: 24.68$_{\pm0.4}$ -> 18.95$_{\pm0.3}$ | Helped |
| Use elliptic regularization instead of C-mixup | Regression model · Airfoil | RMSE: 2.88$_{\pm0.187}$ -> 2.85$_{\pm0.174}$ | Helped |
| Use elliptic regularization instead of C-mixup | Regression model · NO2 | RMSE: 0.524$_{\pm0.006}$ -> 0.523$_{\pm0.008}$ | No clear effect |
| Use elliptic regularization with importance weighting instead of CVaR-DRO | Robust classifier · Waterbirds | worst-group accuracy: 77.2$_{\pm2.2}$ -> 84.1$_{\pm{1.1}}$ | Helped |
| Use elliptic regularization instead of C-mixup | Regression model · SkillCraft | average RMSE: 6.27$_{\pm0.537}$ -> 5.97$_{\pm0.283}$ | Helped |
| Use elliptic regularization instead of ERM | Med-MNIST Breast classifier · Breast | worst-class accuracy: 32.4$_{\pm13.8}$ -> 67.6$_{\pm4.4}$ | Helped |
| Add importance-weighting drift (ξ=1) rather than removing it (ablation) | Elliptic classifier · Waterbirds | worst-group accuracy: 81.7$_{\pm{2.6}}$ -> 84.1$_{\pm{1.1}}$ | Helped |
| Increase drift strength from ξ=1 to ξ=2 (ablation) | Elliptic classifier · Waterbirds | worst-group accuracy: 84.1$_{\pm{1.1}}$ -> 81.2$_{\pm{2.9}}$ | Hurt |
| Increase drift strength from ξ=1 to ξ=10 (ablation) | Elliptic classifier · Waterbirds | worst-group accuracy: 84.1$_{\pm{1.1}}$ -> 82.5$_{\pm{0.6}}$ | Hurt |
| Increase diffusion coefficient from σ=0.1 to σ=1 (ablation) | Elliptic classifier · Waterbirds | worst-group accuracy: 84.1$_{\pm{1.1}}$ -> 75.6$_{\pm{0.5}}$ | Hurt |
Surprising findings
- Elliptic training without importance weighting already matches or beats mixupE on several CIFAR and Tiny-ImageNet settings, and the paper suggests benefits are greater with more classes.
- Adding 50% label noise, elliptic training still keeps high MedMNIST accuracy, and pairwise distance computation has minor effect on performance.
Limitations
- Without Brownian bridge sampling, hitting times of the boundary may be infinite, so the exact PDE is expensive to solve.
- Computation time is greater than ERM due to stochastic path sampling.
- Theoretical guarantees rely on discretized approximations with numerical error and assumptions such as a two-layer ReLU network and Lipschitz f.
Experimental setup
Datasets: CIFAR-10 CIFAR-100 Tiny-ImageNet 200 WaterBirds CelebA Camelyon17 SkillCraft Crime RCF-MNIST Airfoil Self-Noise NO2 MedMNIST (Breast, Blood, Path, OrganC)
Benchmarks: CIFAR-10 CIFAR-100 Tiny-ImageNet 200 Airfoil NO2 WaterBirds CelebA Camelyon17 SkillCraft Crime RCF-MNIST MedMNIST
Metrics: classification error top-1 accuracy top-5 accuracy RMSE average accuracy worst-group accuracy worst-class accuracy
Baselines: ERM mixup mixupE C-mixup Focal Loss CVaR-DRO chi2-DRO CVaR-DORO chi2-DORO JTT UMIX
Builds on: mixup mixupE c-mixup Feynman-Kac formula Girsanov theorem
Closest papers by meaning
- Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernel (NeurIPS 2019 · similarity 0.86)
- Characterizing the Edge of Stability in Variational Training Without Priors (NeurIPS 2026 · similarity 0.86)
- Implicit Gradient Regularization (ICLR 2021 · similarity 0.86)
- Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss (ICML 2024 · similarity 0.85)
- Neglected Hessian component explains mysteries in sharpness regularization (NeurIPS 2024 · similarity 0.85)
- Cauchy-Schwarz Regularizers (ICLR 2025 · similarity 0.84)
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