Despite the wide empirical success of modern machine learning algorithms and models in a multitude of applications, they are known to be highly susceptible to seemingly small indiscernible perturbations to the input data known as \emph{adversarial attacks}. A variety of recent adversarial training procedures have been proposed to remedy this issue. Despite the success of such procedures at increasing accuracy on adversarially perturbed inputs or \emph{robust accuracy}, these techniques often reduce accuracy on natural unperturbed inputs or \emph{standard accuracy}. Complicating matters further, the effect and trend of adversarial training procedures on standard and robust accuracy is rather counter intuitive and radically dependent on a variety of factors including the perceived form of the perturbation during training, size/quality of data, model overparameterization, etc. In this paper we focus on binary classification problems where the data is generated according to the mixture of two Gaussians with general anisotropic covariance matrices and derive a precise characterization of the standard and robust accuracy for a class of minimax adversarially trained models. We consider a general norm-based adversarial model, where the adversary can add perturbations of bounded ellp norm to each input data, for an arbitrary p greater than one. Our comprehensive analysis allows us to theoretically explain several intriguing empirical phenomena and provide a precise understanding of the role of different problem parameters on standard and robust accuracies.
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Precise Tradeoffs in Adversarial Training for Linear Regression
Despite breakthrough performance, modern learning models are known to be highly vulnerable
to small adversarial perturbations in their inputs. While a wide variety of recent adversarial
training methods have been effective at improving robustness to perturbed inputs (robust
accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard
accuracy), leading to a tradeoff between often competing objectives. Complicating matters
further, recent empirical evidence suggest that a variety of other factors (size and quality of
training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this paper
we provide a precise and comprehensive understanding of the role of adversarial training in the
context of linear regression with Gaussian features. In particular, we characterize the fundamental
tradeoff between the accuracies achievable by any algorithm regardless of computational power
or size of the training data. Furthermore, we precisely characterize the standard/robust accuracy
and the corresponding tradeoff achieved by a contemporary mini-max adversarial training
approach in a high-dimensional regime where the number of data points and the parameters of
the model grow in proportion to each other. Our theory for adversarial training algorithms also
facilitates the rigorous study of how a variety of factors (size and quality of training data, model
overparametrization etc.) affect the tradeoff between these two competing accuracies.
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- NSF-PAR ID:
- 10159506
- Date Published:
- Journal Name:
- Conference on Learning Theory
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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