Tight Risk Bounds for Gradient Descent on Separable Data
This provides foundational theoretical guarantees for gradient methods in machine learning, addressing generalization in a widely studied setting.
The paper establishes tight upper and lower population risk bounds for gradient descent on separable linear classification, with bounds expressed as Θ(r_{ℓ,T}^2 / γ^2 T + r_{ℓ,T}^2 / γ^2 n), matching and extending prior results to any smooth loss function.
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 $Θ(r_{\ell,T}^2 / γ^2 T + r_{\ell,T}^2 / γ^2 n)$, where $T$ is the number of gradient steps, $n$ is size of the training set, $γ$ 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 matches the best known upper bounds due to Shamir (2021); Schliserman and Koren (2022), while extending their applicability to virtually any smooth loss function and relaxing technical assumptions they impose. Our risk lower bounds are the first in this context and establish the tightness of our upper bounds 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.