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Outliers with Opposing Signals Have an Outsized Effect on Neural Network Optimization
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Identifies paired groups of outlier training points with opposing gradient signals as a cause of progressive sharpening, edge of stability, and oscillation in deep network optimization, supported by experiments, a toy example, and a two-layer linear network analysis.
Problem
Explaining intriguing neural network optimization phenomena such as progressive sharpening, the edge of stability, loss spikes and their connections to grokking, simplicity bias, and SAM, which lack a unified mechanistic cause.
Approach
Heavy-tailed natural data contains paired groups of outliers with large, consistent, opposing gradients; with depth and steepest descent, the network enters a narrow valley balancing these groups, and subsequent sharpening makes their losses oscillate and spike, driving edge-of-stability behavior. Adam's advantages are interpreted through this lens, motivating a simple SplitSGD variant.
Key results
- Paired outlier groups (e.g., sky vs. non-plane in ResNet-18 on CIFAR-10) show loss oscillations with growing amplitude that coincide with spikes in overall training loss under GD.
- SplitSGD (SGD with dampening plus fixed-size signed steps on large-gradient parameters) matches Adam on ResNet-56/110 CIFAR-10 and early GPT-2 training, with only about 10-25% of parameters in the fixed-step regime per step (ResNet).
- Theorem: in a two-layer linear network, sharpness initially decreases and then progressively sharpens after the opposing-signal outliers are fit.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SplitSGD with dampening and thresholded fixed-size signed steps instead of Adam | ResNet-56/110 · CIFAR-10 | test accuracy | No clear effect |
| Use SplitSGD instead of Adam | GPT-2 · OpenWebText, early training | training loss | No clear effect |
| Vary SplitSGD's standard SGD learning rate across seven orders of magnitude | SplitSGD · ResNet-56 on CIFAR-10 | test accuracy: seven orders of magnitude | No clear effect |
| Remove dampening (set τ=0) (ablation) | SplitSGD · ResNet-110 on CIFAR-10 | performance | Mixed |
| Use gradient flow instead of gradient descent, removing oscillation (ablation) | VGG-11-BN · opposing-signal feature norms | feature norm | Hurt |
| Add oscillation through gradient descent rather than approximating gradient flow (ablation) | VGG-11-BN · opposing-signal feature norms | feature norm | Helped |
Surprising findings
- Approximately half of training points increase in loss on each step under both GD and SGD.
- Adam and SplitSGD on early GPT-2 training appear to exhibit exactly the same loss oscillations.
Limitations
- Outlier groups are identified by visual inspection and are not formally defined or automatically clustered.
- Theory is restricted to a two-layer linear network on a simple misspecified regression model, and the oscillation phase is not formally proven.
- SplitSGD is a proof-of-concept only and was evaluated on limited settings, with GPT-2 compared only in the early training stage.
Experimental setup
Datasets: CIFAR-10 OpenWebText Chebyshev polynomial synthetic data Gaussian synthetic data shakespeare character dataset
Benchmarks: CIFAR-10 OpenWebText
Metrics: training loss test accuracy sharpness (top Hessian eigenvalue) feature norm fraction of curvature per layer
Baselines: Adam SGD with momentum SAM gradient descent gradient flow
Builds on: edge of stability progressive sharpening Adam Sharpness-Aware Minimization
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- Understanding and Minimising Outlier Features in Transformer Training (NeurIPS 2024 · similarity 0.87)
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- Understanding Sharpness Dynamics in NN Training with a Minimalist Example: The Effects of Dataset Difficulty, Depth, Stochasticity, and More (ICML 2025 · similarity 0.86)
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