On margin-based generalization prediction in deep neural networks

TL;DR

This work probes margin-based generalization prediction in deep neural networks, showing that margins measured in the input and hidden representations provide inconsistent generalization signals across settings. It reveals that margins interact with how data is corrupted and that naive margin definitions often fail to predict generalization. To address this, the authors introduce constrained margins, which restrict the margin search to directions spanning the data manifold via PCA, and demonstrate that these margins substantially improve predictive power on PGDL and related benchmarks, outperforming several contemporary complexity measures. The study further analyzes margin estimation methods (Taylor vs DeepFool) and finds that marginally more accurate margin estimation does not always improve predictions, underscoring the importance of aligning margins with data geometry. Overall, constrained margins offer a robust, interpretable, input-space-based predictor with potential regularization applications and implications for adversarial robustness and future margin-based theory.

Abstract

Understanding generalization in deep neural networks is an active area of research. A promising avenue of exploration has been that of margin measurements: the shortest distance to the decision boundary for a given sample or that sample's representation internal to the network. Margin-based complexity measures have been shown to be correlated with the generalization ability of deep neural networks in some circumstances but not others. The reasons behind the success or failure of these metrics are currently unclear. In this study, we examine margin-based generalization prediction methods in different settings. We motivate why these metrics sometimes fail to accurately predict generalization and how they can be improved. First, we analyze the relationship between margins measured in the input space and sample noise. We find that different types of sample noise can have a very different effect on the overall margin of a network that has modeled noisy data. Following this, we empirically evaluate how robust margins measured at different representational spaces are at predicting generalization. We find that these metrics have several limitations and that a large margin does not exhibit a strong correlation with empirical risk in many cases. Finally, we introduce a new margin-based measure that incorporates an approximation of the underlying data manifold. It is empirically demonstrated that this measure is generally more predictive of generalization than all other margin-based measures. Furthermore, we find that this measurement also outperforms other contemporary complexity measures on a well-known generalization prediction benchmark. In addition, we analyze the utility and limitations of this approach and find that this metric is well aligned with intuitions expressed in prior work.

Figures (35)

• Figure 3.1: Example of label corruption and Gaussian input corruption for MNIST (top) and CIFAR10 (bottom). Left: Original training sample. Middle: Label corrupted sample. Right: Gaussian input corrupted sample.
• Figure 3.2: Validation error for MNIST models (left) and CIFAR10 models (right). Values are averaged over three random seeds and shaded areas indicate standard deviation.
• Figure 3.3: Mean margins for all MNIST (left) and CIFAR10 (right) models as a function of model capacity.
• Figure 3.4: Margin distributions for MNIST models (left) and CIFAR10 models (right) trained on clean (top), label-corrupted (middle) and input-corrupted (bottom) training sets. Within each plot, from top to bottom, the distributions are ordered by ascending model size. The relevant capacity metric is shown on the right. Green and red distributions are constructed from clean and corrupted samples, respectively. The corrupted sample distributions are also visualized separately in Figure \ref{['fig:noise_margin_distributions_noise_only']}
• Figure 3.5: Corrupt sample margin distributions for MNIST models (left) and CIFAR10 models (right) trained on label-corrupted (top) and input-corrupted (bottom) training sets. Within each plot, from top to bottom, distributions are ordered by ascending model size. The relevant capacity metric is shown on the right.