Tight Risk Bounds for Gradient Descent on Separable Data

We study the generalization properties of unregularized gradient methods applied to separable linear classification---a setting that has received considerable attention since the pioneering work of Soudry et al. (2018).We establish tight upper and lower (population) risk bounds for gradient descent in this setting, for any smooth loss function, expressed in terms of its tail decay rate.Our bounds take the form $\Theta(r_{\ell,T}^2 / \gamma^2 T + r_{\ell,T}^2 / \gamma^2 n)$, where $T$ is the number of gradient steps, $n$ is size of the training set, $\gamma$ is the data margin, and $r_{\ell,T}$ is a complexity term that depends on the tail decay rate of the loss function (and on $T$).Our upper bound greatly improves the existing risk bounds due to Shamir (2021) and Schliserman and Koren (2022), that either applied to specific loss functions or imposed extraneous technical assumptions, and applies to virtually any convex and smooth loss function.Our risk lower bound is the first in this context and establish the tightness of our general upper bound for any given tail decay rate and in all parameter regimes.The proof technique used to show these results is also markedly simpler compared to previous work, and is straightforward to extend to other gradient methods; we illustrate this by providing analogous results for Stochastic Gradient Descent.