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Learning with a Strong Adversary
Ruitong Huang, Bing Xu, Dale Schuurmans, Csaba Szepesvari
TL;DR
Neural networks can be vulnerable to small, human-indistinguishable perturbations despite high accuracy. The paper proposes learning with a strong adversary, using adversarial-example generation within a min-max training procedure, and reports substantially improved robustness while maintaining high normal accuracy.
Problem
Neural networks may misclassify inputs after small, human-indistinguishable perturbations, motivating methods that improve classifier robustness.
Method
Learning with a strong adversary formulates supervised training as a min-max problem and uses adversarial-example finding as an intermediate step.
Results
The experiments report significantly better robustness while maintaining high normal accuracy compared with literature benchmarks.
Takeaways & Limitations
The proposed training procedure produces classifiers that are more robust to stronger data perturbations while retaining high classification accuracy.
Takeaways & Limitations
The paper postpones analysis of the min-max formulation’s regularization effect, and representation-layer perturbations can cause perturbation explosion that remains open for future investigation.
Abstract
from arXiv · showhide
The robustness of neural networks to intended perturbations has recently attracted significant attention. In this paper, we propose a new method, \emph{learning with a strong adversary}, that learns robust classifiers from supervised data. The proposed method takes finding adversarial examples as an intermediate step. A new and simple way of finding adversarial examples is presented and experimentally shown to be efficient. Experimental results demonstrate that resulting learning method greatly improves the robustness of the classification models produced.
1 INTRODUCTION
The paper addresses the lack of robustness in otherwise accurate neural networks by proposing learning with a strong adversary, a supervised training method designed to produce robust classifiers under stronger perturbations.
- Motivation: Deep neural networks can be highly accurate yet vulnerable to human-indistinguishable perturbations that cause misclassification.The same adversarial examples can fool diverse models trained with different architectures and hyperparameters.
- Motivation: Adversarial perturbations have attracted increasing attention as a phenomenon affecting deep learning classifiers.Prior work has explored explanations and methods for addressing this phenomenon.
- Approach: The paper proposes learning with a strong adversary, which formulates training as a min-max problem against perturbations that maximize classification error.The learner minimizes misclassification error while an adversary perturbs each data point.
- Approach: The method requires an efficient procedure for finding adversarial examples as an intermediate step in solving the min-max training problem.The paper presents a new procedure for finding such examples before developing the full training method.
- Evaluation: The paper evaluates the proposed method experimentally on MNIST and CIFAR-10.The evaluation covers both adversarial-example generation and learning with a stronger adversary.
2 FINDING ADVERSARIAL EXAMPLES
The paper finds adversarial examples by approximating the network output locally and selecting the smallest perturbation predicted to induce a wrong label. It derives norm-specific perturbations and packages the procedure into an algorithm.
- Problem formulation: The adversarial-example problem seeks a minimum-norm perturbation r that changes a correctly classified input’s prediction.For an input x with true label y, the target is N(x + r) ≠ y.
- Linear approximation: The method linearizes the network output as ˆg(x + r) = g(x) + Hr, where H is the Jacobian matrix.This local approximation makes perturbation construction tractable.
- Class-wise perturbations: For each incorrect class j, the method computes a perturbation satisfying a necessary linearized condition for changing the prediction to j.The optimal solution to the resulting objective is supplied by Proposition 1.
- Scope: The paper postpones analysis of the regularization effect of the min-max formulation to the appendix.The authors state that this analysis is not closely related to the paper’s main content.
- Norm-specific solutions: Under the L2 norm, the proposed perturbation is r*(j) = (α_y − α_j) / ||H_j − H_y||_2^2 (H_j − H_y).The paper also gives corresponding forms for L∞ and L1 norms.
- Algorithm: The final adversarial perturbation is the candidate with the smallest norm among the class-wise perturbations.The procedure computes candidates for all classes and returns the minimum-norm candidate.
3 TOWARD ROBUST NEURAL NETWORKS
The paper formulates robust neural-network training as a min-max problem against adversarial perturbations and develops practical procedures for generating those perturbations and optimizing the resulting objective. Experiments report improved robustness while maintaining high classification accuracy.
- Learning with an Adversary: The objective minimizes classification loss against perturbations chosen to maximize loss within a bounded norm constraint.The perturbation magnitude is controlled by hyperparameter c, and the loss may be a differentiable surrogate for misclassification.
- Learning with an Adversary: The min-max formulation trains classifiers to be robust against adversarial examples rather than adding a separate heuristic robustness objective.The learning procedure alternates between adversarial perturbation and model optimization.
- Experiments: Experiments report that the approach achieves the best robustness while maintaining high classification accuracy.On MNIST, network accuracy decreases as perturbation magnitude increases; Adv Alpha is slightly more effective than Adv Loss, and both outperform Adv Loss Sign.
- Optimization: The derivative of the maximized loss can be computed using the maximizing perturbation under almost-everywhere differentiability and uniform Lipschitz continuity.This result supports stochastic-gradient optimization of the robust objective.
- Optimization: The training algorithm computes an adversarial perturbation for each batch example, forms a pseudo-sample, and updates the network using forward-backward propagation.The procedure can perturb raw inputs or learned representations in a split-network interpretation.
- Computing the Perturbation: Two perturbation methods are proposed, including a linearized method whose solution is determined by the Jacobian and the chosen norm.For the L1 norm, the perturbation concentrates on a coordinate attaining the dual-norm maximum; one method matches the approach of Goodfellow et al. under corresponding assumptions.
4 EXPERIMENTAL EVALUATION
Experiments on MNIST and CIFAR-10 evaluate perturbation-generation methods and adversarial training, finding that LWA substantially improves robustness while retaining high accuracy.
- Finding adversarial examples: Adv Alpha is consistently, but slightly, more effective than Adv Loss, while both are significantly more effective than Adv Loss Sign.Validation accuracy decreases as perturbation magnitude increases.
- MNIST: 19.3% accuracy under Adv Alpha shows that dropout improves robustness but remains weak against that adversarial set.The normal method cannot tolerate validation perturbations, while Goodfellow’s method improves robustness substantially.
- MNIST: 84.4% to 86.7% and 83.6% to 86.2% are the reported improvements on adversarial sets generated by Adv Loss and Adv Alpha, respectively.LWA achieves both the best accuracy and the greatest robustness among the compared methods on the 2-hidden-layer MNIST model.
- MNIST: LWA shows great robustness on LeNet, while representation-layer perturbations do not outperform raw-data perturbations.The authors attribute the difficulty of LWA Rep to perturbation explosion in the representation network and leave avoiding it for future work.
- CIFAR-10: CIFAR-10 experiments compare normal training, dropout, Goodfellow’s method, LWA, and representation-layer LWA in a deeper convolutional network.CIFAR-10 is treated as a more difficult task than MNIST, and batch normalization is used to stabilize LWA Rep.
5 CONCLUSION
The paper formalizes adversarial perturbations through a min-max problem, develops a generic algorithm and an efficient adversarial-example method, and reports improved robustness with high normal accuracy.
- The paper investigates adversarial perturbations within a formal min-max problem setting.
- A generic algorithm derived from the min-max formulation replaces previous heuristic algorithms.
- The paper proposes a more efficient method for finding adversarial examples for a given network.
- Experiments suggest that learning with a strong adversary achieves significantly better robustness while maintaining high normal accuracy relative to literature benchmarks.
A ANALYSIS OF THE REGULARIZATION EFFECT OF LEARNING WITH AN ADVERSARY ON LOGISTIC REGRESSION
The analysis characterizes learning with an adversary as data-dependent regularization: it preserves convexity of the learning problem but induces a non-convex regularizer, while bounded and unique logistic-regression solutions follow under a small-margin condition.
- The adversarial objective remains a convex problem.
- The induced regularization Rz(w) is data-dependent and non-convex, unlike common ℓp regularization.
- A linearly separable dataset with margin less than 2c guarantees bounded logistic-regression solutions under adversarial training.
- The proof uses adversarial perturbation to make the data no longer linearly separable, preventing unbounded optimal solutions.
- Positive definiteness of the Hessian and boundedness yield a unique optimal solution.
- Unlike adding regularization, adversarial training guarantees uniqueness only for linearly separable data with small margin, making it a weaker regularization.