Improved generalization bounds for binary linear classification via isoperimetry

We examine the concentration of uniform generalization errors around their expectation in binary linear classification problems via an isoperimetric argument. In particular, we establish Poincar'e and log-Sobolev inequalities for the joint distribution of the output labels and the label-weighted input vectors, which we apply to derive concentration bounds. The derived results improve upon existing bounds obtained from general unbounded empirical processes, as well as that tailored specifically to logistic regression. In asymptotic analysis, we also show that almost sure convergence of uniform generalization errors to their expectation occurs in very broad settings, such as proportionally high-dimensional regimes. Using this convergence, we establish uniform laws of large numbers under dimension-free conditions.