Certifying Some Distributional Robustness with Principled Adversarial Training (1710.10571v5)

Abstract: Neural networks are vulnerable to adversarial examples and researchers have proposed many heuristic attack and defense mechanisms. We address this problem through the principled lens of distributionally robust optimization, which guarantees performance under adversarial input perturbations. By considering a Lagrangian penalty formulation of perturbing the underlying data distribution in a Wasserstein ball, we provide a training procedure that augments model parameter updates with worst-case perturbations of training data. For smooth losses, our procedure provably achieves moderate levels of robustness with little computational or statistical cost relative to empirical risk minimization. Furthermore, our statistical guarantees allow us to efficiently certify robustness for the population loss. For imperceptible perturbations, our method matches or outperforms heuristic approaches.

• The paper introduces a principled adversarial training method employing distributionally robust optimization within a Wasserstein ball to secure neural networks against adversarial attacks.
• It reformulates the robust optimization problem using Lagrangian duality and implements a scalable stochastic gradient descent algorithm with convergence guarantees.
• Empirical results on MNIST and Stanford Dogs demonstrate improved robustness against adversarial perturbations while maintaining competitive performance on clean data.

Certifying Some Distributional Robustness With Principled Adversarial Training

The paper "Certifying Some Distributional Robustness With Principled Adversarial Training" by Hongseok Namkoong, Riccardo Volpi, and John Duchi addresses the vulnerability of neural networks to adversarial examples by leveraging a principled approach based on distributionally robust optimization (DRO).

Key Contributions

The authors propose an adversarial training procedure that employs the Lagrangian penalty formulation within a Wasserstein ball to enhance the robustness of models under adversarial input perturbations. The authors derive both statistical and computational guarantees for their method, providing rigorous performance bounds even under worst-case adversarial scenarios.

Theoretical Framework

The theoretical foundation of the method lies in DRO, where the focus is on minimizing the worst-case expected loss over a set of distributions close to the empirical distribution of the training data. The robustness is enforced through a Wasserstein ball, which defines the neighborhood of distributions around the empirical distribution. This approach can be formally expressed as: $\minimize_{\theta \in \Theta} \sup_{P \in \mathcal{P}} E_P[L(\theta; Z)],$ where $\mathcal{P}$ is the set of plausible distributions within a Wasserstein distance from the empirical distribution.

Lagrangian Duality

By reformulating the problem using Lagrangian duality, the authors transform the intractable DRO problem into a more tractable optimization problem. Specifically, they show that for any distribution $Q$, we can write: $$\sup_{P : W_c(P, Q) \le \rho} E_P[L(\theta; Z)] = \inf_{\gamma \ge 0} \left\{ \gamma \rho + E_{Q}[\phi_{\gamma}(\theta; Z)] \right\},$$ where $W_c$ denotes the Wasserstein distance, and $\phi_{\gamma}$ is a robust surrogate loss function defined as: $$\phi_{\gamma}(\theta; z_0) = \sup_{z \in \mathcal{Z}} \left\{L(\theta; z) - \gamma c(z, z_0)\right\}.$$

Algorithm and Computational Guarantees

The paper presents an efficient stochastic gradient descent (SGD) algorithm for solving the adversarial training problem. The key steps involve:

1. Augmenting every update step of model parameters with worst-case perturbations.
2. Applying gradient ascent with respect to these perturbations to find an approximate maximizer.
3. Ensuring convergence guarantees for smooth loss functions under proper conditions.

The algorithm demonstrates that moderate levels of robustness can be achieved with negligible additional computational cost relative to standard empirical risk minimization.

Empirical Validation

The authors extensively validate their method on synthetic and real-world datasets. For instance, evaluations on the MNIST and Stanford Dogs datasets show that their method significantly improves robustness against various adversarial attacks while maintaining competitive performance on clean data. The experiments underscore the following:

• The robustness certificates provided by the method reliably bound the worst-case losses.
• Models trained with the proposed method exhibit higher resilience against adversarial perturbations compared to traditional heuristic-based adversarial training methods.
• The method's performance during adversarial attacks scales favorably with increasing perturbation limits.

Implications and Future Directions

The approach offers a robust and theoretically grounded method for defending neural networks against adversarial attacks. The implications are broad, ranging from safer deployment of machine learning models in security-critical systems to improved generalization in the presence of distributional shifts.

Future research directions could explore:

1. Extending the method to a broader set of loss functions and model architectures, particularly those involving non-smooth activations like ReLUs.
2. Enhancing scalability to accommodate even larger models and datasets.
3. Investigating alternative regularization techniques that could further tighten the bounds and improve generalization without compromising the robustness guarantees.

The work lays a solid foundation for distributionally robust machine learning and opens avenues for further exploration in creating resilient and reliable AI systems.