  Motivated by neural network training in low-precision arithmetic
environments, this work studies the convergence of variants of SGD using
adaptive step sizes with computational error. Considering a general stochastic
Lipschitz continuous loss function, an asymptotic convergence result to a
Clarke stationary point is proven as well as the non-asymptotic convergence to
an approximate stationary point. It is assumed that only an approximation of
the loss function's stochastic gradient can be computed in addition to error in
computing the SGD step itself. Different variants of SGD are tested
empirically, where improved test set accuracy is observed compared to SGD for
two image recognition tasks.
