  Learning an algorithm from examples is a fundamental problem that has been
widely studied. Recently it has been addressed using neural networks, in
particular by Neural Turing Machines (NTMs). These are fully differentiable
computers that use backpropagation to learn their own programming. Despite
their appeal NTMs have a weakness that is caused by their sequential nature:
they are not parallel and are are hard to train due to their large depth when
unfolded.
  We present a neural network architecture to address this problem: the Neural
GPU. It is based on a type of convolutional gated recurrent unit and, like the
NTM, is computationally universal. Unlike the NTM, the Neural GPU is highly
parallel which makes it easier to train and efficient to run.
  An essential property of algorithms is their ability to handle inputs of
arbitrary size. We show that the Neural GPU can be trained on short instances
of an algorithmic task and successfully generalize to long instances. We
verified it on a number of tasks including long addition and long
multiplication of numbers represented in binary. We train the Neural GPU on
numbers with upto 20 bits and observe no errors whatsoever while testing it,
even on much longer numbers.
  To achieve these results we introduce a technique for training deep recurrent
networks: parameter sharing relaxation. We also found a small amount of dropout
and gradient noise to have a large positive effect on learning and
generalization.
