  Training quantised neural networks (QNNs) is a non-differentiable
optimisation problem since weights and features are output by piecewise
constant functions. The standard solution is to apply the straight-through
estimator (STE), using different functions during the inference and gradient
computation steps. Several STE variants have been proposed in the literature
aiming to maximise the task accuracy of the trained network. In this paper, we
analyse STE variants and study their impact on QNN training. We first observe
that most such variants can be modelled as stochastic regularisations of stair
functions; although this intuitive interpretation is not new, our rigorous
discussion generalises to further variants. Then, we analyse QNNs mixing
different regularisations, finding that some suitably synchronised smoothing of
each layer map is required to guarantee pointwise compositional convergence to
the target discontinuous function. Based on these theoretical insights, we
propose additive noise annealing (ANA), a new algorithm to train QNNs
encompassing standard STE and its variants as special cases. When testing ANA
on the CIFAR-10 image classification benchmark, we find that the major impact
on task accuracy is not due to the qualitative shape of the regularisations but
to the proper synchronisation of the different STE variants used in a network,
in accordance with the theoretical results.
