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Koopman-based generalization bound: New aspect for full-rank weights
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Derives a Koopman-operator-based Rademacher complexity bound for deep networks whose determinant factors make it tight for full-rank, well-conditioned (near-orthogonal) weight matrices, and combines it with existing norm/compression bounds.
Problem
Generalization bounds for neural networks mostly rely on low-rank structure or norms of weight matrices, and do not explain why full-rank weight matrices generalize well.
Approach
Represents the network as a product of Koopman operators on RKHS (Sobolev-type) spaces; the change-of-variables in the Koopman operator norm yields determinant factors det(W^*W)^{-1/4}, so the bound shrinks as the condition number of weight matrices decreases and becomes width-independent for orthogonal weights.
Key results
- Bound O(prod ||W_j||^{s_j}/det(W_j^*W_j)^{1/4}) shrinks in proportion to generalization error across 5 runs of a small 3-dimensional regression network.
- Adding a regularizer based on ||W_j|| + 1/det(I+W_j^*W_j) to a four-layer MNIST network trained on n=1000 samples improves test accuracy over unregularized training.
- On AlexNet/CIFAR-10, the condition number of the second layer's weights tends to decrease during training, while layers 3 and 4 show large condition numbers, suggesting the Koopman bound applies to lower layers and norm-based bounds to higher layers.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add regularization based on the Koopman bound | Four-layer dense network · MNIST | test accuracy | Helped |
| Use the Koopman-based bound for full-rank weights with small condition numbers | Neural networks with full-rank weights · Generalization bound | bound tightness | Helped |
| Use orthogonal weight matrices | Neural networks · Generalization bound | width dependence | Helped |
Surprising findings
- The bound is independent of network width when weight matrices are orthogonal.
- Condition numbers of higher layers (3 and 4) of AlexNet grow during training, the opposite of the trend in the second layer.
Limitations
- Excludes non-smooth activation functions and sigmoid/tanh are only handled via weighted Koopman construction.
- Non-injective weights require modifying the network structure (graph-based or weighted Koopman), so the bound applies to a modified network.
- Dependence on activation factor ||K_sigma|| is not clarified; tightness for larger smoothness s is open.
Experimental setup
Datasets: MNIST CIFAR-10 synthetic regression data
Benchmarks: MNIST CIFAR-10
Metrics: generalization error test accuracy condition number of weight matrices bound value
Baselines: Bartlett et al. spectrally-normalized bound Neyshabur et al. (p,q)-norm bound Long and Sedghi bound Arora et al. compression bound Wei and Ma bound Galanti et al. bound
Builds on: Rademacher complexity bounds for kernel methods norm-based generalization bounds (Bartlett et al.) compression-based bounds (Arora et al.) peeling argument
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