Worst-Case Control of Fairness for Continuous Sensitive Attributes

Fair representation learning (FRL) seeks a representation that is informative for downstream prediction yet independent of a sensitive attribute. For a continuous sensitive attribute, existing methods measure the discrepancy between conditional and marginal distributions at each sensitive value and *average* it over the distribution of the sensitive attribute. Such an average can be small even when a fairness violation is confined to a low-probability range of sensitive values, and thus provides no uniform guarantee over the sensitive domain. We therefore propose the *supremum integral probability metric* (**supIPM**) to measure the worst-case discrepancy between the conditional and marginal distributions of the representation over sensitive values. supIPM controls the worst-case disparity of predictions, which we call $L_{\infty }$-generalized demographic parity. We then propose **supFR**, an algorithm for learning representations that control worst-case fairness violations in downstream predictions. We establish a uniform convergence rate of the empirical supIPM and show that it yields a bound on downstream disparity for prediction heads of bounded Hölder norm. Experiments on tabular and graph benchmarks show that supFR attains prediction performance comparable to existing methods while substantially reducing the worst-case fairness violation.