Residual-Information Generalization Bounds Under Selective Privacy: Extended Version

Abstract

Information-theoretic generalization bounds usually control dependence on the task sample, whereas selective privacy may constrain only a different, private coordinate. We study i.i.d. pairs $(X,Y)$, where a learner observes $(X^n,Y^n)$, leaks at most $\epsilon_n$ nats about $X^n$, and incurs loss on $Y$. A chain-rule transfer converts this mismatched constraint into the closed-form universal certificate $I(W;Y^n)\leq\min\{nH(Y),nH(Y\mid X)+\epsilon_n\}$. Combining the certificate with mutual-information generalization yields a bound governed by residual task uncertainty and normalized privacy leakage. We derive complementary task-compression and individual-sample refinements, identify a sharp obstruction showing why selective privacy alone need not give a vanishing certificate, and invert the bound to obtain the exact leakage allowance for a prescribed generalization tolerance. We prove exact information-level achievability for deterministic-task and independent-residual source families and show that the resulting $\sqrt{\epsilon_n/n}$ generalization rate is tight up to a universal constant. We also give a fixed-horizon adaptive application in which conditional leakage costs add.