Robustness of classifiers: from adversarial to random noise (1608.08967v1)

Abstract: Several recent works have shown that state-of-the-art classifiers are vulnerable to worst-case (i.e., adversarial) perturbations of the datapoints. On the other hand, it has been empirically observed that these same classifiers are relatively robust to random noise. In this paper, we propose to study a \textit{semi-random} noise regime that generalizes both the random and worst-case noise regimes. We propose the first quantitative analysis of the robustness of nonlinear classifiers in this general noise regime. We establish precise theoretical bounds on the robustness of classifiers in this general regime, which depend on the curvature of the classifier's decision boundary. Our bounds confirm and quantify the empirical observations that classifiers satisfying curvature constraints are robust to random noise. Moreover, we quantify the robustness of classifiers in terms of the subspace dimension in the semi-random noise regime, and show that our bounds remarkably interpolate between the worst-case and random noise regimes. We perform experiments and show that the derived bounds provide very accurate estimates when applied to various state-of-the-art deep neural networks and datasets. This result suggests bounds on the curvature of the classifiers' decision boundaries that we support experimentally, and more generally offers important insights onto the geometry of high dimensional classification problems.

• The paper introduces a semi-random noise regime that combines random noise with adversarial perturbations to better reflect real-world conditions.
• The paper quantifies classifier robustness by deriving bounds based on decision boundary curvature, revealing a square root relationship with data dimensionality.
• The paper empirically validates its theoretical findings using deep neural networks, underscoring practical insights for designing robust classifiers.

Robustness of Classifiers: From Adversarial to Random Noise

The paper "Robustness of classifiers: from adversarial to random noise" investigates the resilience of state-of-the-art classifiers, particularly deep neural networks, against adversarial and random noise. The authors address a gap between these two types of perturbations by exploring a semi-random noise regime, which introduces a middle ground between purely random and worst-case adversarial attacks.

Key Contributions and Findings

1. Semi-Random Noise Regime: The paper introduces a novel semi-random noise regime that blends random noise and adversarial perturbations. This regime involves projecting noise of random direction onto a specified subspace within which adversarial perturbations are maximized. This innovative approach enriches the spectrum of noise models to better reflect real-world scenarios where purely random or fully adversarial noise may not always be present.
2. Quantitative Analysis of Nonlinear Classifiers: By establishing robust theoretical bounds, the authors quantify the resilience of nonlinear classifiers within the semi-random regime. These bounds are derived based on the curvature of a classifier's decision boundary and offer predictive insights into how variations in boundary curvature influence robustness.
3. Dimensionality Insights: A significant finding is that robustness to random noise enhances with data dimensionality. Specifically, robustness is shown to increase proportionately to the square root of the dimensionality of the data, given small curvature decision boundaries. This provides a theoretical underpinning for the empirical observation that classifiers appear more susceptible to adversarial perturbations than to random noise.
4. Interpolation Between Noise Regimes: The proposed bounds effectively interpolate between the extremes of adversarial and random noise, emphasising that even when the semi-random noise is predominantly random, classifiers can still be susceptible to perturbations if they contain a small adversarial component. This insight underscores the necessity for robust classifier designs that consider a spectrum of noise regimes, particularly in high-dimensional scenarios.
5. Empirical Validation: Through rigorous experiments with various state-of-the-art deep neural networks, the derived bounds demonstrate high accuracy in predicting classifier robustness across multiple datasets. This empirical validation not only affirms the theoretical contributions but also enhances understanding of the geometric properties of decision boundaries.

Theoretical and Practical Implications

The primary theoretical implication of this work is the illumination of the relationship between decision boundary curvature and classifier robustness across different noise regimes. This relationship suggests that imposing constraints that tame decision boundary curvature could potentially enhance classifier robustness, especially in high-dimensional settings.

From a practical standpoint, this research informs the design of classifiers capable of withstanding diverse noise. Enhanced robustness to a mix of noise types is crucial, particularly in real-world applications where data may not always conform to worst-case or purely random perturbation models.

Speculations on Future Developments

Future developments could explore efficient algorithms for real-time curvature estimation and adjustment in classifier design. Another promising direction is the exploration of adaptive algorithms that dynamically adjust decision boundary properties based on the detected noise environment, thus optimizing robustness under varying operational conditions.

By shedding light on the nuanced interactions between dimensionality, curvature, and noise types, this paper provides a comprehensive framework for future work aimed at developing next-generation robust classifiers in an ever-evolving landscape of cyber threats and application domains.