  Choosing appropriate step sizes is critical for reducing the computational
cost of training large-scale neural network models. Mini-batch sub-sampling
(MBSS) is often employed for computational tractability. However, MBSS
introduces a sampling error, that can manifest as a bias or variance in a line
search. This is because MBSS can be performed statically, where the mini-batch
is updated only when the search direction changes, or dynamically, where the
mini-batch is updated every-time the function is evaluated. Static MBSS results
in a smooth loss function along a search direction, reflecting low variance but
large bias in the estimated "true" (or full batch) minimum. Conversely, dynamic
MBSS results in a point-wise discontinuous function, with computable gradients
using backpropagation, along a search direction, reflecting high variance but
lower bias in the estimated "true" (or full batch) minimum. In this study,
quadratic line search approximations are considered to study the quality of
function and derivative information to construct approximations for dynamic
MBSS loss functions. An empirical study is conducted where function and
derivative information are enforced in various ways for the quadratic
approximations. The results for various neural network problems show that being
selective on what information is enforced helps to reduce the variance of
predicted step sizes.
