Strengthened Information-theoretic Bounds on the Generalization Error

The following problem is considered: given a joint distribution $P_{XY}$ and an event $E$, bound $P_{XY}(E)$ in terms of $P_X{P}_Y(E)$ (where $P_X{P}_Y$ is the product of the marginals of $P_{XY}$) and a measure of dependence of $X$ and $Y$. Such bounds have direct applications in the analysis of the generalization error of learning algorithms, where $E$ represents a large error event and the measure of dependence controls the degree of overfitting. Herein, bounds are demonstrated using several information-theoretic metrics, in particular: mutual information, lautum information, maximal leakage, and $J_{\mathrm{\infty }}$. The mutual information bound can outperform comparable bounds in the literature by an arbitrarily large factor.