  Artificial neural networks (ANNs) are typically highly nonlinear systems
which are finely tuned via the optimization of their associated, non-convex
loss functions. In many cases, the gradient of any such loss function has
superlinear growth, making the use of the widely-accepted (stochastic) gradient
descent methods, which are based on Euler numerical schemes, problematic. We
offer a new learning algorithm based on an appropriately constructed variant of
the popular stochastic gradient Langevin dynamics (SGLD), which is called tamed
unadjusted stochastic Langevin algorithm (TUSLA). We also provide a
nonasymptotic analysis of the new algorithm's convergence properties in the
context of non-convex learning problems with the use of ANNs. Thus, we provide
finite-time guarantees for TUSLA to find approximate minimizers of both
empirical and population risks. The roots of the TUSLA algorithm are based on
the taming technology for diffusion processes with superlinear coefficients as
developed in \citet{tamed-euler, SabanisAoAP} and for MCMC algorithms in
\citet{tula}. Numerical experiments are presented which confirm the theoretical
findings and illustrate the need for the use of the new algorithm in comparison
to vanilla SGLD within the framework of ANNs.
