L2 Attacks

$L_2$ Attacks are adversarial attacks that constrain the total Euclidean magnitude of the perturbation — $|x - x_{adv}|_2 \leq \epsilon$, allowing larger changes in a few features while keeping the overall perturbation small in the geometric (Euclidean) sense.

Key $L_2$ Attack Methods

• C&W $L_2$: Carlini & Wagner — the strongest $L_2$ attack, using Adam optimization with change-of-variables and margin-based objectives.
• DeepFool: Finds the minimum $L_2$ perturbation to cross the decision boundary — iterative linearization.
• PGD-$L_2$: Projected gradient descent with $L_2$ ball projection.
• DDN: Decoupled direction and norm — separates perturbation direction from magnitude optimization.

Why It Matters

• Natural Metric: $L_2$ distance is the natural geometric distance between images/signals.
• Different From $L_\infty$: $L_2$ robustness does not imply $L_\infty$ robustness (and vice versa).
• Randomized Smoothing: $L_2$ is the natural norm for randomized smoothing certified defenses.

$L_2$ Attacks are the geometric perturbation — finding adversarial examples that are close in Euclidean distance to the original input.