  In this paper, 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 $N$ iterations, with a constant step-size proportional to
$1/R^2 \sqrt{N}$ where $N$ is the number of observations and $R$ is the maximum
norm of the observations, the convergence rate is always of order
$O(1/\sqrt{N})$, and improves to $O(R^2 / \mu N)$ where $\mu$ is the lowest
eigenvalue of the Hessian at the global optimum (when this eigenvalue is
greater than $R^2/\sqrt{N}$). Since $\mu$ does not need to be known in advance,
this shows that averaged stochastic gradient is adaptive to \emph{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.
