Adversarially Robust Generalization Requires More Data

Abstract: Machine learning models are often susceptible to adversarial perturbations of their inputs. Even small perturbations can cause state-of-the-art classifiers with high "standard" accuracy to produce an incorrect prediction with high confidence. To better understand this phenomenon, we study adversarially robust learning from the viewpoint of generalization. We show that already in a simple natural data model, the sample complexity of robust learning can be significantly larger than that of "standard" learning. This gap is information theoretic and holds irrespective of the training algorithm or the model family. We complement our theoretical results with experiments on popular image classification datasets and show that a similar gap exists here as well. We postulate that the difficulty of training robust classifiers stems, at least partially, from this inherently larger sample complexity.

• The paper demonstrates that adversarial robustness requires larger sample complexity, with Gaussian models needing up to a √d increase in data.
• It employs Gaussian and Bernoulli models to reveal that robust classification markedly outstrips standard generalization in data requirements.
• The findings underscore the need for innovative algorithms and methods like thresholding to efficiently secure robust accuracy in scarce data settings.

Adversarially Robust Generalization: Insights and Implications

The paper "Adversarially Robust Generalization Requires More Data" by Ludwig Schmidt, Shibani Santurkar, Dimitris Tsipras, Kunal Talwar, and Aleksander Mądry explores the intricacies of adversarial robustness in machine learning models. The key question addressed is whether the sample complexity for adversarially robust generalization differs from that required for standard generalization.

Summary and Key Findings

Modern machine learning models, despite their high accuracy on standard datasets, are notably vulnerable to adversarial perturbations—small, often imperceptible changes in the input data that can lead to incorrect classifications. This vulnerability poses a significant challenge, especially as these models are increasingly deployed in critical environments. The paper investigates if robust generalization inherently demands more data compared to standard generalization.

The authors study adversarially robust learning through the lens of sample complexity. They introduce two theoretical models for their analysis: a mixture of two class-conditional Gaussians and a Bernoulli distribution. Their findings indicate that robust generalization indeed necessitates a larger sample complexity, and this requirement is fundamental, persisting regardless of the learning algorithm or model utilized.

The Gaussian Model

For the Gaussian model:

• Standard Generalization: A single sample suffices to achieve high classification accuracy. This result stems from classical Gaussian concentration inequalities.
• Robust Generalization: Achieving adversarial robustness requires significantly more samples. The authors demonstrate that for perturbations bounded by a constant $\epsilon$, the sample complexity increases by a factor of $\sqrt{d}$, where $d$ is the dimension of the input space.

The theoretical results are bolstered by empirical evidence across widely-used image classification datasets, where robust classifiers consistently demand a larger dataset to bridge the gap between standard and robust accuracy.

Practical Implications

The increase in sample complexity for robust generalization has profound implications:

1. Data Requirement: Adversarial training methods must access substantially more data to achieve robustness. This is critical for areas with data scarcity.
2. Model Evaluation: When evaluating models, it’s imperative to consider not just standard accuracy but robust accuracy, thereby emphasizing the need for larger training sets.
3. Algorithm Development: Novel algorithms that can leverage limited data efficiently for robust learning are necessary. This might involve incorporating domain-specific prior knowledge into the models.

The Bernoulli Model

For a dataset resembling the MNIST dataset, often modeled using a nearly binary Bernoulli distribution, the authors show:

• Standard Generalization: Similar to the Gaussian model, a single sample can yield good classification accuracy.
• Robust Generalization: Linear classifiers struggle with achieving robust accuracy without increasing sample size; however, introducing simple non-linear transformations, such as thresholding, can notably reduce the sample complexity.

Experimental Validation

Experiments on MNIST, CIFAR-10, and SVHN datasets underscore the theoretical findings. Robust classifiers trained on subsets of these datasets reveal a clear trend: achieving robust accuracy similar to standard accuracy requires a significantly larger dataset. Intriguingly, thresholding mechanisms dramatically improve robustness for nearly binary datasets like MNIST, affirming the theoretical predictions.

Discussion and Future Directions

This work establishes that adversarial robustness is not merely a byproduct of model architecture but rather an innate requirement driven by the data distribution and sample complexity. Theoretical insights and experimental results from this paper suggest several future avenues:

• Exploration of Different Perturbation Norms: While the current study focuses on $\ell_\infty$-perturbations, extensions to $\ell_2$ and other norm-based perturbations could offer a more comprehensive understanding of robustness.
• Algorithmic Innovations: Developing algorithms specifically designed to handle the increased sample complexity for robust learning remains an open challenge.
• Broader Distributional Analysis: Further work could explore more complex and realistic data distributions to fully map out the landscape of adversarially robust generalization.
• Transfer Learning: Investigating the potential of transfer learning techniques in improving adversarial robustness without a proportionate increase in training data requirements.

In conclusion, this paper makes a substantial contribution to understanding why achieving adversarial robustness is inherently more challenging in terms of sample complexity, underlining the need for larger datasets and potentially more sophisticated algorithms and network architectures tailored to robust learning.