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 ℓ2 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 O(1/t) rates for both ℓ2 margin maximization and implicit bias, whereas prior work (including all first-order methods for the general hard-margin linear SVM problem) proved widetildeO(1/√t) margin rates, or O(1/t) margin rates to a suboptimal margin, with an implied (slower) bias rate. 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.