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We study the generalization properties of unregularized gradient methodsapplied to separable linear classification -- a setting that has receivedconsiderable attention since the pioneering work of Soudry et al. (2018). Weestablish tight upper and lower (population) risk bounds for gradient descentin this setting, for any smooth loss function, expressed in terms of its taildecay 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, $n$ issize of the training set, $\gamma$ is the data margin, and $r_{\ell,T}$ is acomplexity term that depends on the (tail decay rate) of the loss function (andon $T$). Our upper bound matches the best known upper bounds due to Shamir(2021); Schliserman and Koren (2022), while extending their applicability tovirtually any smooth loss function and relaxing technical assumptions theyimpose. Our risk lower bounds are the first in this context and establish thetightness of our upper bounds for any given tail decay rate and in allparameter regimes. The proof technique used to show these results is alsomarkedly simpler compared to previous work, and is straightforward to extend toother gradient methods; we illustrate this by providing analogous results forStochastic Gradient Descent.

Tight Risk Bounds for Gradient Descent on Separable Data