Characterizing the implicit bias via a primal-dual analysis

This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For the exponential loss specifically, with $n$ training examples and $t$ gradient descent steps, our dual analysis further allows us to prove an $O(\ln(n)/\ln(t))$ convergence rate to the $\ell_2$ maximum margin direction, when a constant step size is used. This rate is tight in both $n$ and $t$, which has not been presented by prior work. On the other hand, with a properly chosen but aggressive step size schedule, we prove an $O(1/t)$ convergence rate for $\ell_2$ margin maximization, while prior work has only proved an $\tilde{O}(1/\sqrt{t})$ rate, or an $O(1/t)$ convergence rate to a suboptimal margin. Our key observations include that gradient descent on the primal variable naturally induces a mirror descent update on the dual variable, and that the dual objective in this setting is smooth enough to give a faster rate.