On the Geometry of Regularization in Adversarial Training: High-Dimensional Asymptotics and Generalization Bounds

• URL: http://arxiv.org/abs/2410.16073v1
• Date: Mon, 21 Oct 2024 14:53:12 GMT
• Title: On the Geometry of Regularization in Adversarial Training: High-Dimensional Asymptotics and Generalization Bounds
• Authors: Matteo Vilucchio, Nikolaos Tsilivis, Bruno Loureiro, Julia Kempe,
• Abstract summary: This work investigates the question of how to choose the regularization norm $lVert cdot rVert$ in the context of high-dimensional adversarial training for binary classification. We quantitatively characterize the relationship between perturbation size and the optimal choice of $lVert cdot rVert$, confirming the intuition that, in the data scarce regime, the type of regularization becomes increasingly important for adversarial training as perturbations grow in size.
• Score: 11.30047438005394
• License:
• Abstract: Regularization, whether explicit in terms of a penalty in the loss or implicit in the choice of algorithm, is a cornerstone of modern machine learning. Indeed, controlling the complexity of the model class is particularly important when data is scarce, noisy or contaminated, as it translates a statistical belief on the underlying structure of the data. This work investigates the question of how to choose the regularization norm $\lVert \cdot \rVert$ in the context of high-dimensional adversarial training for binary classification. To this end, we first derive an exact asymptotic description of the robust, regularized empirical risk minimizer for various types of adversarial attacks and regularization norms (including non-$\ell_p$ norms). We complement this analysis with a uniform convergence analysis, deriving bounds on the Rademacher Complexity for this class of problems. Leveraging our theoretical results, we quantitatively characterize the relationship between perturbation size and the optimal choice of $\lVert \cdot \rVert$, confirming the intuition that, in the data scarce regime, the type of regularization becomes increasingly important for adversarial training as perturbations grow in size.

Related papers

• Regularization for Adversarial Robust Learning [18.46110328123008]
  We develop a novel approach to adversarial training that integrates $phi$-divergence regularization into the distributionally robust risk function. This regularization brings a notable improvement in computation compared with the original formulation. We validate our proposed method in supervised learning, reinforcement learning, and contextual learning and showcase its state-of-the-art performance against various adversarial attacks.
  arXiv  Detail & Related papers  (2024-08-19T03:15:41Z)
• Semi-parametric inference based on adaptively collected data [34.56133468275712]
  We construct suitably weighted estimating equations that account for adaptivity in data collection. Our results characterize the degree of "explorability" required for normality to hold. We illustrate our general theory with concrete consequences for various problems, including standard linear bandits and sparse generalized bandits.
  arXiv  Detail & Related papers  (2023-03-05T00:45:32Z)
• Instance-Dependent Generalization Bounds via Optimal Transport [51.71650746285469]
  Existing generalization bounds fail to explain crucial factors that drive the generalization of modern neural networks. We derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the learned prediction function in the data space. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.
  arXiv  Detail & Related papers  (2022-11-02T16:39:42Z)
• Towards Data-Algorithm Dependent Generalization: a Case Study on Overparameterized Linear Regression [19.047997113063147]
  We introduce a notion called data-algorithm compatibility, which considers the generalization behavior of the entire data-dependent training trajectory. We perform a data-dependent trajectory analysis and derive a sufficient condition for compatibility in such a setting.
  arXiv  Detail & Related papers  (2022-02-12T12:42:36Z)
• Benign Overfitting in Adversarially Robust Linear Classification [91.42259226639837]
  "Benign overfitting", where classifiers memorize noisy training data yet still achieve a good generalization performance, has drawn great attention in the machine learning community. We show that benign overfitting indeed occurs in adversarial training, a principled approach to defend against adversarial examples.
  arXiv  Detail & Related papers  (2021-12-31T00:27:31Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Squared $\ell_2$ Norm as Consistency Loss for Leveraging Augmented Data to Learn Robust and Invariant Representations [76.85274970052762]
  Regularizing distance between embeddings/representations of original samples and augmented counterparts is a popular technique for improving robustness of neural networks. In this paper, we explore these various regularization choices, seeking to provide a general understanding of how we should regularize the embeddings. We show that the generic approach we identified (squared $ell$ regularized augmentation) outperforms several recent methods, which are each specially designed for one task.
  arXiv  Detail & Related papers  (2020-11-25T22:40:09Z)
• Precise Statistical Analysis of Classification Accuracies for Adversarial Training [43.25761725062367]
  A variety of recent adversarial training procedures have been proposed to remedy this issue. We derive a precise characterization of the standard and robust accuracy for a class of minimax adversarially trained models.
  arXiv  Detail & Related papers  (2020-10-21T18:00:53Z)
• Failures of model-dependent generalization bounds for least-norm interpolation [39.97534972432276]
  We consider bounds on the generalization performance of the least-norm linear regressor. For a variety of natural joint distributions on training examples, any valid generalization bound must sometimes be very loose.
  arXiv  Detail & Related papers  (2020-10-16T16:30:05Z)
• Good Classifiers are Abundant in the Interpolating Regime [64.72044662855612]
  We develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers. We find that test errors tend to concentrate around a small typical value $varepsilon*$, which deviates substantially from the test error of worst-case interpolating model. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice.
  arXiv  Detail & Related papers  (2020-06-22T21:12:31Z)
• Total Deep Variation: A Stable Regularizer for Inverse Problems [71.90933869570914]
  We introduce the data-driven general-purpose total deep variation regularizer. In its core, a convolutional neural network extracts local features on multiple scales and in successive blocks. We achieve state-of-the-art results for numerous imaging tasks.
  arXiv  Detail & Related papers  (2020-06-15T21:54:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.