Characterizing the Sample Complexity of Semi-Supervised Adversarially Robust PAC Learning

Core Concepts

There is a significant benefit in semi-supervised robust learning compared to the supervised setting, with the labeled sample complexity being sharply characterized by a different complexity measure (VCU) rather than the standard VC dimension.

Abstract

The paper studies the problem of learning an adversarially robust predictor in the semi-supervised PAC model. The key findings are:

In the simple case where the support of the marginal distribution is known, the labeled sample complexity is Θ(VCU(H)/ε + log(1/δ)/ε).

In the general semi-supervised setting, the authors present a generic algorithm (GRASS) that can be applied to both realizable and agnostic settings.

For the realizable case:

The labeled sample complexity is Õ(VCU(H)/ε + log(1/δ)/ε), which can be significantly smaller than the Ω(RSU(H)/ε + log(1/δ)/ε) lower bound for supervised robust learning.
The unlabeled sample complexity matches the supervised robust learning bound of Õ(VC(H)VC*/ε + log(1/δ)/ε).
For the agnostic case:

If an error of 3η + ε is allowed (where η is the minimal agnostic error), the labeled sample complexity is Õ(VCU(H)/ε^2 + log(1/δ)/ε^2).
Obtaining an error of η + ε requires Ω(RSU(H)/ε^2 + log(1/δ)/ε^2) labeled examples, matching the supervised lower bound.
The authors also show that for any γ > 0, there exists a hypothesis class where using only O(VCU) labeled examples leads to an error of (3/2 - γ)η + ε.
The results demonstrate that there can be a significant benefit in semi-supervised robust learning compared to the supervised setting, with the labeled sample complexity being controlled by the VCU dimension rather than the RSU dimension.

Stats

None

Quotes

None

Key Insights Distilled From

A Characterization of Semi-Supervised Adversarially-Robust PAC Learnability

by Idan Attias,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2202.05420.pdf

Deeper Inquiries

How can the semi-supervised robust learning framework be extended to other loss functions beyond the 0-1 loss, such as regression tasks

In extending the semi-supervised robust learning framework to other loss functions beyond the 0-1 loss, such as regression tasks, several considerations need to be taken into account. One approach could involve adapting the algorithm to optimize for a different loss function, such as mean squared error for regression tasks. This would require modifying the learning objective and updating the algorithm to minimize the chosen loss function while incorporating both labeled and unlabeled data.
For regression tasks, the robust learning algorithm could be adjusted to handle continuous output values instead of discrete classes. The perturbation function U would need to be redefined to account for the nature of the regression problem, potentially considering different types of perturbations that could affect the continuous predictions. The hypothesis class H would also need to be tailored to accommodate regression models, allowing for the prediction of continuous values.
Furthermore, the evaluation metrics for robustness in regression tasks may differ from classification tasks. Instead of focusing on misclassifications, the evaluation could consider the impact of adversarial perturbations on the accuracy of the regression predictions. This could involve measuring the deviation of the predicted values under perturbations and assessing the robustness of the model based on these deviations.
Overall, extending the semi-supervised robust learning framework to regression tasks would involve adapting the algorithm, perturbation function, hypothesis class, and evaluation metrics to suit the characteristics of regression problems and ensure robustness in the face of adversarial attacks.

What are the implications of the findings in this paper for the design of practical semi-supervised robust learning algorithms

The findings in this paper have significant implications for the design of practical semi-supervised robust learning algorithms. By demonstrating the benefits of leveraging unlabeled data in the learning process, the research highlights the potential for improving the robustness of machine learning models in real-world applications.
One key implication is the potential for reducing the labeled sample complexity required for robust learning by incorporating unlabeled data effectively. This can lead to more efficient and cost-effective model training, especially in scenarios where labeled data is scarce or expensive to obtain. Practical algorithms can be designed to leverage this insight by incorporating semi-supervised learning techniques that take advantage of both labeled and unlabeled data to enhance robustness.
Additionally, the findings suggest that improper learning may be necessary in certain scenarios to achieve robustness in semi-supervised settings. This insight can guide the development of algorithms that are capable of learning robust models even when traditional proper learning rules may not suffice. By exploring the trade-offs between proper and improper learning in the context of adversarial robustness, researchers can design more effective and resilient machine learning systems.
Overall, the findings provide a foundation for the development of practical semi-supervised robust learning algorithms that can enhance the robustness and reliability of machine learning models in adversarial settings.

Are there any connections between the VCU dimension and other complexity measures that could provide further insights into the nature of adversarial robustness

The VCU dimension, introduced in the paper as a complexity measure for robust learning, offers insights into the nature of adversarial robustness and its relationship to other complexity measures. By characterizing the complexity of hypothesis classes in terms of their ability to withstand adversarial perturbations, the VCU dimension provides a unique perspective on the challenges and requirements for robust learning.
One potential connection that could be explored is the relationship between the VCU dimension and the Rademacher complexity or the margin-based complexity measures. Understanding how these different complexity measures interact and influence each other could provide further insights into the generalization and robustness properties of machine learning models.
Moreover, investigating the connections between the VCU dimension and other complexity measures, such as the VC dimension or the shattering dimension, could reveal additional nuances in the trade-offs between model complexity, sample complexity, and robustness. By analyzing these relationships, researchers can gain a deeper understanding of the factors that contribute to the robustness of machine learning models and inform the design of more resilient algorithms.