[Paper Review] Efficiently Learning Adversarially Robust Halfspaces with Noise

Omar Montasser, Surbhi Goel+2 more|arXiv (Cornell University)|May 15, 2020

Adversarial Robustness in Machine Learning26 references5 citations

TL;DR

This paper establishes necessary and sufficient conditions for efficiently learning adversarially robust halfspaces under $ε$-random classification noise, showing that a computationally efficient algorithm exists for any $\ell_p$-perturbation. It proves that robust empirical risk minimization (RERM) is efficiently solvable when a separation oracle for the perturbation set $\mathcal{U}$ exists, and that an approximate separation oracle is necessary for computing robust loss, enabling efficient robust PAC learning in the noisy realizable setting.

ABSTRACT

We study the problem of learning adversarially robust halfspaces in the distribution-independent setting. In the realizable setting, we provide necessary and sufficient conditions on the adversarial perturbation sets under which halfspaces are efficiently robustly learnable. In the presence of random label noise, we give a simple computationally efficient algorithm for this problem with respect to any $\ell_p$-perturbation.

Motivation & Objective

• To identify necessary and sufficient conditions on perturbation sets $\mathcal{U}$ for efficient robust PAC learning of halfspaces.
• To analyze the computational complexity of robust empirical risk minimization (RERM) in the realizable and noisy settings.
• To establish that an efficient separation oracle for $\mathcal{U}$ enables efficient RERM, while an approximate separation oracle is necessary for computing robust loss.
• To extend results to the random classification noise model, showing efficient robust learning is possible under mild assumptions.
• To clarify the boundary between tractable and intractable robust learning for halfspaces under different noise and perturbation models.

Proposed method

• Proposes a polynomial-time algorithm for RERM using a separation oracle for the perturbation set $\mathcal{U}$, enabling efficient robust learning in the realizable case.
• Introduces a reduction from robust risk minimization to a convex optimization problem via a surrogate loss function $G_\lambda^\gamma({\bm{w}}, {\bm{x}})$ that approximates the robust 0-1 loss.
• Employs Stochastic Mirror Descent (SMD) with a regularized objective to minimize the robust empirical risk, achieving convergence guarantees under random classification noise.
• Uses a margin-based generalization bound that relates the $\gamma/2$-margin error to the robust risk and the surrogate loss.
• Establishes that an efficient approximate separation oracle for $\mathcal{U}$ is necessary to compute the robust loss of a halfspace efficiently.
• Analyzes the sample complexity of the algorithm, showing $\mathcal{O}(1/\varepsilon^2\gamma^2(1-2\eta)^2)$ samples for $q=1$ and $\mathcal{O}(1/\varepsilon^2\gamma^2(1-2\eta)^2(q-1))$ for $q>1$ under mirror descent with $q$-norm regularization.

Experimental results

Research questions

• RQ1 Under what conditions on the perturbation set $\mathcal{U}$ is robust empirical risk minimization (RERM) efficiently solvable for halfspaces in the realizable setting?
• RQ2 Is an efficient separation oracle for $\mathcal{U}$ both necessary and sufficient for efficient robust learning of halfspaces?
• RQ3 Can robust halfspace learning be efficiently achieved under random classification noise, and what is the sample complexity in this setting?
• RQ4 What is the computational cost of computing the robust loss of a halfspace, and how does it relate to the complexity of $\mathcal{U}$?
• RQ5 How does the robust generalization error relate to the margin-based error and surrogate loss under random noise?

Key findings

• An efficient separation oracle for the perturbation set $\mathcal{U}$ is sufficient for polynomial-time robust empirical risk minimization (RERM) of halfspaces.
• An efficient approximate separation oracle for $\mathcal{U}$ is necessary to compute the robust loss of a halfspace efficiently.
• In the random classification noise model, halfspaces are efficiently robustly PAC learnable with respect to any $\ell_p$-perturbation.
• The proposed algorithm achieves a sample complexity of $\mathcal{O}(1/\varepsilon^2\gamma^2(1-2\eta)^2)$ for $q=1$ and $\mathcal{O}(1/\varepsilon^2\gamma^2(1-2\eta)^2(q-1))$ for $q>1$ under Stochastic Mirror Descent.
• The robust risk is bounded in terms of the $\gamma/2$-margin error and the surrogate loss, with the bound depending on $\varepsilon'$, $\lambda$, and $\eta$, ensuring generalization under noise.
• The analysis shows that $\lambda = \frac{\varepsilon\gamma/2 + \eta}{1 + \varepsilon\gamma}$ and $\varepsilon' = \frac{\varepsilon\gamma(1 - 2\eta)}{2(1 + \varepsilon\gamma})$ yield the desired generalization error $\leq \varepsilon$.