Ramchandran Muthukumar

Ramchandran Muthukumar, Jeremias Sulam

Deep artificial neural networks achieve surprising generalization abilities that remain poorly understood. 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.

Ramchandran Muthukumar, Ambar Pal, Jeremias Sulam et al.

State-of-the-art machine learning systems are vulnerable to small perturbations to their input, where ``small'' is defined according to a threat model that assigns a positive threat to each perturbation. Most prior works define a task-agnostic, isotropic, and global threat, like the $\ell_p$ norm, where the magnitude of the perturbation fully determines the degree of the threat and neither the direction of the attack nor its position in space matter. However, common corruptions in computer vision, such as blur, compression, or occlusions, are not well captured by such threat models. This paper proposes a novel threat model called \texttt{Projected Displacement} (PD) to study robustness beyond existing isotropic and global threat models. The proposed threat model measures the threat of a perturbation via its alignment with \textit{unsafe directions}, defined as directions in the input space along which a perturbation of sufficient magnitude changes the ground truth class label. Unsafe directions are identified locally for each input based on observed training data. In this way, the PD threat model exhibits anisotropy and locality. Experiments on Imagenet-1k data indicate that, for any input, the set of perturbations with small PD threat includes \textit{safe} perturbations of large $\ell_p$ norm that preserve the true label, such as noise, blur and compression, while simultaneously excluding \textit{unsafe} perturbations that alter the true label. Unlike perceptual threat models based on embeddings of large-vision models, the PD threat model can be readily computed for arbitrary classification tasks without pre-training or finetuning. Further additional task annotation such as sensitivity to image regions or concept hierarchies can be easily integrated into the assessment of threat and thus the PD threat model presents practitioners with a flexible, task-driven threat specification.

Ramchandran Muthukumar, Jeremias Sulam

This work studies the adversarial robustness of parametric functions composed of a linear predictor and a non-linear representation map. % that satisfies certain stability condition. Our analysis relies on \emph{sparse local Lipschitzness} (SLL), an extension of local Lipschitz continuity that better captures the stability and reduced effective dimensionality of predictors upon local perturbations. SLL functions preserve a certain degree of structure, given by the sparsity pattern in the representation map, and include several popular hypothesis classes, such as piece-wise linear models, Lasso and its variants, and deep feed-forward \relu networks. % are sparse local Lipschitz. We provide a tighter robustness certificate on the minimal energy of an adversarial example, as well as tighter data-dependent non-uniform bounds 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.

Jeremias Sulam, Ramchandran Muthukumar, Raman Arora

Several recent results provide theoretical insights into the phenomena of adversarial examples. Existing results, however, are often limited due to a gap between the simplicity of the models studied and the complexity of those deployed in practice. In this work, we strike a better balance by considering a model that involves learning a representation while at the same time giving a precise generalization bound and a robustness certificate. We focus on the hypothesis class obtained by combining a sparsity-promoting encoder coupled with a linear classifier, and show an interesting interplay between the expressivity and stability of the (supervised) representation map and a notion of margin in the feature space. We bound the robust risk (to $\ell_2$-bounded perturbations) of hypotheses parameterized by dictionaries that achieve a mild encoder gap on training data. Furthermore, we provide a robustness certificate for end-to-end classification. We demonstrate the applicability of our analysis by computing certified accuracy on real data, and compare with other alternatives for certified robustness.