  We develop a line-search second-order algorithmic framework for minimizing
finite sums. We do not make any convexity assumptions, but require the terms of
the sum to be continuously differentiable and have Lipschitz-continuous
gradients. The methods fitting into this framework combine line searches and
suitably decaying step lengths. A key issue is a two-step sampling at each
iteration, which allows us to control the error present in the line-search
procedure. Stationarity of limit points is proved in the almost-sure sense,
while almost-sure convergence of the sequence of approximations to the solution
holds with the additional hypothesis that the functions are strongly convex.
Numerical experiments, including comparisons with state-of-the art stochastic
optimization methods, show the efficiency of our approach.
