Beyond Lipschitz: Sharp Generalization and Excess Risk Bounds for Full-Batch GD

We provide sharp path-dependent generalization and excess risk guarantees for the full-batch Gradient Descent (GD) algorithm on smooth losses (possibly non-Lipschitz, possibly nonconvex), under an interpolation regime. At the heart of our analysis is a new generalization error bound for deterministic symmetric algorithms, which implies that average output stability and a bounded expected optimization error at termination lead to generalization. This result shows that small generalization error occurs along the optimization path, and allows us to bypass Lipschitz or sub-Gaussian assumptions on the loss prevalent in previous works. For nonconvex, Polyak-Lojasiewicz (PL), convex and strongly convex losses, we show the explicit dependence of the generalization error in terms of the accumulated path-dependent optimization error, terminal optimization error, number of samples, and number of iterations. For nonconvex smooth losses, we prove that full-batch GD efficiently generalizes close to any stationary point at termination, under the proper choice of a decreasing step size. Further, if the loss is nonconvex but the objective is PL, we derive quadratically vanishing bounds on the generalization error and the corresponding excess risk, for a choice of a large constant step size. For (resp. strongly-) convex smooth losses, we prove that full-batch GD also generalizes for large constant step sizes, and achieves (resp. quadratically) small excess risk while training fast. In all cases, we close the generalization error gap, by showing matching generalization and optimization error rates. Our full-batch GD generalization error and excess risk bounds are strictly tighter than existing bounds for (stochastic) GD, when the loss is smooth (but possibly non-Lipschitz).