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Projected gradient descent

Definition

Projected gradient descent (PGD) optimizes a function subject to a constraint set: take an ordinary gradient step, then project the result back onto the set. The projection maps a point to its nearest feasible point; for an ℓ-infinity ball it is coordinate-wise clipping. PGD is the standard first-order method for large-scale constrained optimization.

Papers that use this concept

• Towards Deep Learning Models Resistant to Adversarial Attacks — uses PGD on the input as its attack and argues it is the strongest first-order adversary.
• What Makes and Breaks Safety Fine-tuning? A Mechanistic Study — its continuous-embedding attack optimizes appended embeddings against the model’s gradients, the same first-order gradient-based attack primitive.

Variants and traps

• In adversarial ML, “PGD” almost always means the iterative ℓ-infinity attack popularized by the paper above: projected gradient ascent on the loss with respect to the input, with sign-normalized steps and random restarts. The optimization primitive and the attack named after it are distinct things.

See also

• Stochastic gradient descent
• Cross-entropy loss
• Fast Gradient Sign Method (FGSM)
• ℓp norms