Risk and parameter convergence of logistic regression

Gradient descent, when applied to the task of logistic regression, outputs iterates which are biased to follow a unique ray defined by the data. The direction of this ray is the maximum margin predictor of a maximal linearly separable subset of the data; the gradient descent iterates converge to this ray in direction at the rate O(lnln t / ln t). The ray does not pass through the origin in general, and its offset is the bounded global optimum of the risk over the remaining data; gradient descent recovers this offset at a rate O((ln t)² / √t).