Robustness Within a Fairness Budget: Parity Frontiers and Equal-Opportunity Reversal

Adversarial training protects predictive performance under attack but does not determine how decisions are distributed across protected groups. We formulate binary classification as projected gradient descent (PGD) adversarial-risk minimization under a squared soft demographic-parity (DP) budget, optimized with projected multiplier and one-sided penalty updates. On Adult Income, using the binary \texttt{sex} field supplied by the benchmark for group auditing but excluding it from prediction, we test logistic regression and a one-hidden-layer multilayer perceptron (MLP) across three paired seeds, four PGD strengths, and five budgets. At $\varepsilon=0.3$, PGD training improves balanced accuracy over empirical-risk minimization (ERM) by $7.77$ and $11.45$ percentage points and removes $85.7\%$ and $89.5\%$ of attack-induced growth in hard demographic-parity difference (DPD). At the representative interior value $\tau=0.015$, constrained training cuts DPD relative to PGD training by $52.2\%$ and $49.2\%$, while remaining $3.18$ and $8.34$ balanced-accuracy points above ERM. Across all eight seed-averaged model--attack paths, the sampled budget path strictly orders DPD and accuracy and reverses mean signed equal-opportunity difference (EOpp). A rate theorem gives the exact true-positive-rate shift required for reversal, and the measured group rates satisfy it. Within this controlled feature-space study, robustness and parity admit useful joint operating points, but one fairness metric cannot replace a full group-rate audit.