Improved generalization bounds for binary linear classification via isoperimetry

While algorithmic stability is a central tool for understanding generalization of learning algorithms, existing high-probability guarantees typically rely on uniform boundedness or sub-Gaussian/sub-Weibull tail assumptions, which can be overly restrictive for modern settings with heavy-tailed or unbounded losses. We develop a stability-based framework that requires only a finite $L_p$ moment condition. Our first contribution is sharp concentration inequalities for functions of independent random variables under $L_p$ constraints, extending McDiarmid's bounded-differences techniques beyond the classical regime. Leveraging these results, we derive sharp high-probability generalization bounds across a range of learning paradigms, including empirical risk minimization, transductive regression, and meta-learning. These guarantees show that $L_p$ stability suffices for robust generalization even when boundedness fails, substantially weakening the standard assumptions in the stability literature.

The practical success of deep learning has led to the discovery of several surprising phenomena. One of these phenomena, that has spurred intense theoretical research, is ``benign overfitting'': deep neural networks seem to generalize well in the over-parametrized regime even though the networks show a perfect fit to noisy training data. It is now known that benign overfitting also occurs in various classical statistical models. For linear maximum margin classifiers, benign overfitting has been established theoretically in a class of mixture models with very strong assumptions on the covariate distribution. However, even in this simple setting, many questions remain open. For instance, most of the existing literature focuses on the noiseless case where all true class labels are observed without errors, whereas the more interesting noisy case remains poorly understood. We provide a comprehensive study of benign overfitting for linear maximum margin classifiers. We discover a phase transition in test error bounds for the noisy model which was previously unknown and provide some geometric intuition behind it. We further considerably relax the required covariate assumptions in both the noisy and noiseless cases. Our results demonstrate that benign overfitting of maximum margin classifiers holds in a much wider range of scenarios than was previously known and provide new insights into the underlying mechanisms.

Uniform stability is a classical tool for controlling the generalization error of a learning algorithm. Bousquet, Klochkov, and Zhivotovskiy (2020) showed that the problem can be reduced to a moment inequality for a sum of weakly interacting functions of independent random variables. Their bound contains an additional factor $\log n$, and they asked whether this factor can be removed. We answer this upper-bound question affirmatively. More specifically, let $Z=(Z_1,\ldots,Z_n)$ have independent coordinates and let $g_i(Z)$ satisfy $\mathbb E[g_i(Z)\mid Z_{-i}]=0, \ \left| \mathbb E[g_i(Z)\mid Z_i]\right|\le M, \ \text{for every } i = 1, \dots, n,$ where $Z_{-i}$ denotes all coordinates except $Z_i$. Assume additionally that changing any coordinate $Z_j$, $j\neq i$, changes $g_i$ by at most $β$, we prove that, for every $p\ge2$, for every $p\ge2$, $\left\| \sum_{i=1}^n g_i(Z)\right\|_p \le 16pnβ+M\sqrt{2pn}.$ This removes the $\log n$ factor from the previous bound and matches the lower bound of Bousquet, Klochkov, and Zhivotovskiy up to universal constants in the range covered by their construction. Our proof first establishes the required estimate on the Rademacher cube, then transfers it to arbitrary product distributions by a two-copy randomization argument.