Research map / Optimization / Implicit bias and generalization
The Implicit Bias of Gradient Descent toward Collaboration between Layers: A Dynamic Analysis of Multilayer Perceptions
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Proposes co-correlation, a Dirichlet-energy-based measure of inter-layer collaboration, and proves under mild assumptions that gradient descent increases it in linear and two-layer MLPs, implying decreasing adversarial robustness.
Problem
Whether gradient descent's implicit bias pushes neural network layers to collaborate in ways that reduce adversarial robustness, and how network width affects this.
Approach
Decomposes the Dirichlet energy of a composed network into per-layer energies and a co-correlation term measuring alignment between adjacent layers; proves co-correlation grows during gradient descent for linear and MLP models under a weight-initialization scale q>0, with growth speed inversely related to weight operator norm.
Key results
- Co-correlation increases during training for linear and shallow ReLU MLPs at q=0.25, with faster growth at larger widths.
- Lower Dirichlet energy S(f) corresponds to higher L2 AutoAttack robust accuracy across MLPs; wider networks are more robust.
- Width-8192 MLPs with q=-0.15 reach accuracy 0.93; raising co-correlation from 0.25 to 0.83 lowers accuracy to 0.68.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase co-correlation from 0.25 to 0.83 by altering weight initialization (ablation) | MLP ReLU · MNIST test set | accuracy: 0.93 to 0.68 | Hurt |
| Train with q > 0 rather than q < 0 (ablation) | Linear and MLP models · MNIST | co-correlation | Mixed |
| Increase network width (ablation) | MLP ReLU · MNIST | co-correlation | Helped |
| Increase network width (ablation) | MLP ReLU · MNIST test set | robust accuracy | Helped |
| Increase co-correlation through gradient descent | Two-layer MLPs · MNIST | co-correlation | Helped |
| Increase co-correlation through gradient descent | Two-layer MLPs · MNIST, L2 Auto-attack with ϵ = 0.5 | robust accuracy | Hurt |
| Train with Adam without specific weight initialization considerations (ablation) | ResNet50 and WRN50 · CIFAR10 | co-correlation | Hurt |
| Strengthen co-correlation during training | Narrow two-layer MLPs · MNIST | performance | Helped |
| Strengthen co-correlation during training | Wide two-layer MLPs · MNIST | performance | No clear effect |
Surprising findings
- Narrow networks improve by strengthening co-correlation, which the authors call intrinsically brittle.
- Co-correlation also rises for ResNet50 and WRN50 trained with Adam without special initialization.
Limitations
- Theory limited to linear and two-layer MLPs.
- Experiments on small MNIST MLPs and CIFAR10 ResNets only.
Experimental setup
Datasets: MNIST CIFAR10
Benchmarks: MNIST CIFAR10
Metrics: robust accuracy under L2 AutoAttack (eps=0.5) co-correlation Dirichlet energy test accuracy
Baselines: Linear model Frei et al. two-layer ReLU analysis
Builds on: Frei et al. implicit bias and adversarial robustness Dohmatob and Bietti Dirichlet energy robustness Lyu and Li max-margin implicit bias
Closest papers by meaning
- The Double-Edged Sword of Implicit Bias: Generalization vs. Robustness in ReLU Networks (NeurIPS 2023 · similarity 0.91)
- Can Implicit Bias Imply Adversarial Robustness? (ICML 2024 · similarity 0.90)
- Feature Averaging: An Implicit Bias of Gradient Descent Leading to Non-Robustness in Neural Networks (ICLR 2025 · similarity 0.90)
- Adversarial Training Can Provably Improve Robustness: Theoretical Analysis of Feature Learning Process Under Structured Data (ICLR 2025 · similarity 0.88)
- The Price of Implicit Bias in Adversarially Robust Generalization (NeurIPS 2024 · similarity 0.87)
- Layer-Aware Analysis of Catastrophic Overfitting: Revealing the Pseudo-Robust Shortcut Dependency (ICML 2024 · similarity 0.86)
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