Implicit bias of gradient-descent: fast convergence rate

We consider gradient-flow (GF) and gradient-descent (GD) on linear classification problems in possibly infinite-dimensional and non-hilbertian Banach spaces. For exponential-tailed loss functions, including the usual exponential and logistic loss functions, we establish $\mathcal O (\log (n)/ t)$ convergence rate for the bias in case of GF, and $\widetilde{\mathcal O}(\log (n)/\sqrt{t})$ in case of GD. This is a net improvement on best known rates, namely $\mathcal O(\log (n) / \log (t))$. See Ji and Telgarsky (2019), for example. Upto logarithmic factors, our GD rate matches the very recent parallel work from Ji and Telgarsky (2020) which uses an agressive stepsize schedule. Finally, using the aggressive stepsize schedule proposed py Ji and Telgarsky (2020), we are able to obtain a convergence rate of $\mathcal O(\log (n)/t)$ for the bias. Our methods of analysis are quite general and radically different from the usual techniques used in the literature: we use nonlinear error analysis for convex functions, in the spirit of Kurdyka-\L{}ojasiewicz theory. One major advantage of our method is that it allows us to convert any convergence rate for the margin, to a convergence rate on the bias, which is at least as good as the former. We believe our work will provide an alternative approach for analyzing the implicit bias of gradient-flow / gradient-descent in very general settings.