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This work studies the adversarial robustness of parametric functions composed of a linear predic-tor and a nonlinear representation map. Our analysis relies on sparse local Lipschitzness (SLL),an extension of local Lipschitz continuity that better captures the stability and reduced effectivedimensionality of predictors upon local perturbations. SLL functions preserve a certain degree ofstructure, given by the sparsity pattern in the representation map, and include several popular hy-pothesis classes, such as piecewise linear models, Lasso and its variants, and deep feedforward ReLUnetworks. Compared with traditional Lipschitz analysis, we provide a tighter robustness certificateon the minimal energy of an adversarial example, as well as tighter data-dependent nonuniformbounds on the robust generalization error of these predictors. We instantiate these results for the case of deep neural networks and provide numerical evidence that supports our results, shedding new insights into natural regularization strategies to increase the robustness of these models.
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Sparsity-aware generalization theory for deep neural networks
Deep artificial neural networks achieve surprising generalization abilities that remain poorly under- stood. In this paper, we present a new approach to analyzing generalization for deep feed-forward ReLU networks that takes advantage of the degree of sparsity that is achieved in the hidden layer activations. By developing a framework that accounts for this reduced effective model size for each input sample, we are able to show fundamental trade-offs between sparsity and generalization. Importantly, our results make no strong assumptions about the degree of sparsity achieved by the model, and it improves over recent norm-based approaches. We illustrate our results numerically, demonstrating non-vacuous bounds when coupled with data-dependent priors in specific settings, even in over-parametrized models.
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- Award ID(s):
- 2007649
- PAR ID:
- 10548035
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Format(s):
- Medium: X
- Location:
- 36th Annual Conference on Learning Theory
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
