Section: New Results
Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression
Participant : Francis Bach.
In this work, we consider supervised learning problems such as logistic regression and study the stochastic gradient method with averaging, in the usual stochastic approximation setting where observations are used only once. We show that after iterations, with a constant step-size proportional to where is the number of observations and is the maximum norm of the observations, the convergence rate is always of order , and improves to where is the lowest eigenvalue of the Hessian at the global optimum (when this eigenvalue is greater than ). Since does not need to be known in advance, this shows that averaged stochastic gradient is adaptive to unknown local strong convexity of the objective function. Our proof relies on the generalized self-concordance properties of the logistic loss and thus extends to all generalized linear models with uniformly bounded features.