  In this paper, we develop convergence analysis of a modified line search
method for objective functions whose value is computed with noise and whose
gradient estimates are inexact and possibly random. The noise is assumed to be
bounded in absolute value without any additional assumptions. We extend the
framework based on stochastic methods from [Cartis and Scheinberg, 2018] which
was developed to provide analysis of a standard line search method with exact
function values and random gradients to the case of noisy functions. We
introduce two alternative conditions on the gradient which when satisfied with
some sufficiently large probability at each iteration, guarantees convergence
properties of the line search method. We derive expected complexity bounds to
reach a near optimal neighborhood for convex, strongly convex and nonconvex
functions. The exact dependence of the convergence neighborhood on the noise is
specified.
