  The standard L-BFGS method relies on gradient approximations that are not
dominated by noise, so that search directions are descent directions, the line
search is reliable, and quasi-Newton updating yields useful quadratic models of
the objective function. All of this appears to call for a full batch approach,
but since small batch sizes give rise to faster algorithms with better
generalization properties, L-BFGS is currently not considered an algorithm of
choice for large-scale machine learning applications. One need not, however,
choose between the two extremes represented by the full batch or highly
stochastic regimes, and may instead follow a progressive batching approach in
which the sample size increases during the course of the optimization. In this
paper, we present a new version of the L-BFGS algorithm that combines three
basic components - progressive batching, a stochastic line search, and stable
quasi-Newton updating - and that performs well on training logistic regression
and deep neural networks. We provide supporting convergence theory for the
method.
