Noise premium makes the prediction error of adversarial training converge strictly more slowly than the nonparametric minimax benchmark even after balancing approximation and estimation errors, so the proposed noise-debiased adversarial training is proposed.
Abstract
Adversarial training can improve the robustness of predictive models to bounded perturbations, often at the cost of statistical efficiency. We study this trade-off in kernel regression over a reproducing kernel Hilbert space (RKHS). It is shown that, under squared loss, adversarial training in RKHS introduces a term involving the product of the function norm with the mean absolute value of the response noise, which we call the \textit{noise premium}. Our analysis shows that the noise premium makes the prediction error of adversarial training converge strictly more slowly than the nonparametric minimax benchmark even after balancing approximation and estimation errors. Moreover, for a fixed perturbation budget, once the budget exceeds a certain threshold, the solution to adversarial training collapses to the zero function. To mitigate these effects of the noise premium, we propose noise-debiased adversarial training. The resulting noise-debiased estimator can attain the minimax optimal rate up to a logarithmic factor for the prediction error, raises the collapse threshold, and admits an explicit bound on the increase in adversarial loss. Numerical experiments on synthetic and real data support the theoretical findings and validate the effectiveness of the proposed noise-debiased method.
It is proved that VC classes are adversarially robustly learnable with sample complexity linear in the VC dimension $d$, providing an exponential improvement over the previous upper bound of Montasser, Hanneke, and Srebro (2019).
A consistency theory for robust neural classifiers based on the S-divergence family that requires no identifiability assumption is developed and it is proved that empirical S-divergence minimizers converge to the population-optimal equivalence class under mild regularity conditions.
Subhabrata Majumdar, Anand Deo, Partha Pratim Saha et al.· 0 citations
This work concerns adversarial training against the small-norm adversarial examples that arise from the inherent input instability of a trained deep neural network. Examples in this class are small as measured in the relative $\ell^2$-norm, and therefore lie in the neighborhood of the input on which the model acts approximately linearly, the regime in which the perturbation remains imperceptible. We first show that such examples can be computed directly from the trained network parameters, without input gradient iterations, by means of a linearization called the low-rank Householder expansion (LRHE). The expansion describes the composed affine map rather than any individual layer, and the directions it identifies are read from the activation pattern already available in the forward pass. We then propose a simple adversarial training scheme built on this construction. No differentiation with respect to the input is performed at any point: training requires only additional forward evaluations, with weight parameters updated by the standard backward pass, and the inner maximization of the usual min-max formulation is eliminated entirely. That such a regularizer exists is our main finding: the methods that dispense with the inner search all obtain their local geometry by differentiating with respect to the input, and we show this is not necessary. The regularizer costs the equivalent of $2.8$ PGD steps per epoch, an $8.7\times$ reduction relative to 40-step adversarial training on MNIST and below the cost of 3-step training. The resulting models match three-step PGD adversarial training for relative $\ell^2$ budgets $\varepsilon \le 0.02$ and 40-step training for $\varepsilon \le 0.012$, falling away beyond, consistent with the locality of the expansion.
This work exhibits a learnable multiclass problem that becomes altogether unlearnable under a monotone adversary, and shows an analogous result for partial binary concept classes, and demonstrates that monotone adversaries are frighteningly more powerful in each of these settings.
Julian Asilis, S. Dughmi, Chirag Pabbaraju· 0 citations
Adversarial training has emerged as a leading strategy for enhancing the robustness of machine learning models against adversarial attacks. Its effectiveness often wanes when faced with unseen adversarial examples, resulting in suboptimal robust generalization. To address this issue, we introduce a novel energy-based optimization strategy to improve the robust generalization by incorporating the principles of energy-based models. Our framework models the energy of natural and adversarial examples, where natural samples are assigned to lower energy and adversarial samples to higher energy. During the inference phase, the influence of adversarial perturbation can be alleviated by energy minimization. Theoretically, we show that the proposed energy-based optimization strategy yields a tighter robust-generalization bound through an explicit energy-discrepancy term; this analysis provides an explanatory bound and should not be interpreted as certified robustness. Empirically, a series of evaluations provide evidence for the efficacy of the proposed methodology under the specified threat models and evaluation protocols, showing strong and competitive performance across three extensively utilized datasets. Specifically, EM-AT achieves 77.71% standard-AA robustness on CIFAR-10 and remains highly competitive under comparable lightweight settings. The source codes are available at https://github.com/LitterQ/EM-AT.