[Paper Review] Sharp Statistical Guarantees for Adversarially Robust Gaussian Classification

Dan Chen, Yuting Wei+1 more|arXiv (Cornell University)|Jun 29, 2020

Adversarial Robustness in Machine Learning39 references6 citations

TL;DR

This paper establishes the first minimax lower bounds for adversarially robust Gaussian classification under the adversarial signal-to-noise ratio (AdvSNR), proving an excess risk lower bound of order $\Theta(e^{-(\frac{1}{8}+o(1))r^{2}}\frac{d}{n})$ for AdvSNR $r$, dimension $d$, and sample size $n$. It further proposes a computationally efficient estimator achieving this optimal rate, providing sharp statistical guarantees across $\ell_p$-norm perturbations and minimal assumptions on covariance and class separation.

ABSTRACT

Adversarial robustness has become a fundamental requirement in modern machine learning applications. Yet, there has been surprisingly little statistical understanding so far. In this paper, we provide the first result of the optimal minimax guarantees for the excess risk for adversarially robust classification, under Gaussian mixture model proposed by \cite{schmidt2018adversarially}. The results are stated in terms of the Adversarial Signal-to-Noise Ratio (AdvSNR), which generalizes a similar notion for standard linear classification to the adversarial setting. For the Gaussian mixtures with AdvSNR value of $r$, we establish an excess risk lower bound of order $Θ(e^{-(\frac{1}{8}+o(1)) r^2} \frac{d}{n})$ and design a computationally efficient estimator that achieves this optimal rate. Our results built upon minimal set of assumptions while cover a wide spectrum of adversarial perturbations including $\ell_p$ balls for any $p \ge 1$.

Motivation & Objective

• To establish the first minimax lower bounds for excess risk in adversarially robust Gaussian classification under minimal assumptions.
• To characterize the statistical limits of robust classification in terms of the adversarial signal-to-noise ratio (AdvSNR).
• To design a computationally efficient estimator that achieves the optimal minimax rate of convergence.
• To generalize results beyond $\ell_\infty$-type perturbations to arbitrary $\ell_p$-norms for $p \geq 1$.
• To provide a comprehensive understanding of the trade-off between robustness and statistical efficiency in the conditional Gaussian model.

Proposed method

• Derives minimax lower bounds for excess risk using a transformation that maps standard classification problems to robust ones via adversarial perturbations.
• Introduces the Adversarial Signal-to-Noise Ratio (AdvSNR) as a key parameter to characterize robustness and statistical complexity.
• Constructs a robust classifier by solving a constrained quadratic optimization problem that accounts for adversarial perturbations within $\ell_p$-norm balls.
• Uses a change-of-measure argument and a mapping from standard to robust distributions to relate standard and robust excess risks.
• Employs a constructive proof to show that for any standard distribution with AdvSNR $r$, there exists a corresponding robust distribution with the same signal strength.
• Applies first-order optimality conditions to verify that the solution to the adversarial perturbation problem matches the required robust classifier.

Experimental results

Research questions

• RQ1 What is the fundamental statistical limit of adversarially robust classification under the Gaussian mixture model?
• RQ2 How does the adversarial signal-to-noise ratio (AdvSNR) affect the minimax excess risk in robust classification?
• RQ3 Can a computationally efficient estimator achieve the optimal minimax rate of convergence in the robust Gaussian classification setting?
• RQ4 How do the statistical guarantees for robust classification compare to those in the standard (non-adversarial) setting?
• RQ5 To what extent can the results be generalized to arbitrary $\ell_p$-norm adversarial perturbations?

Key findings

• The minimax lower bound for the excess risk in adversarially robust Gaussian classification is $\Omega_P(\exp(-(\frac{1}{8}+o(1))r^2)\frac{d}{n})$ in terms of AdvSNR $r$, dimension $d$, and sample size $n$.
• A computationally efficient estimator is constructed that achieves an excess risk of order $O_P(\exp(-(\frac{1}{8}+o(1))r^2)\frac{d}{n})$, matching the lower bound up to lower-order terms.
• The results hold under minimal assumptions, including unknown and arbitrary covariance matrices and no requirement for class separation.
• The framework generalizes beyond $\ell_\infty$-type perturbations to all $\ell_p$-norms for $p \geq 1$.
• The robust excess risk is lower bounded by the standard excess risk of a transformed distribution, enabling reduction to the standard classification setting.
• The analysis reveals that adversarial robustness incurs a statistical cost that scales exponentially with the square of the AdvSNR, quantifying the intrinsic trade-off between robustness and accuracy.