  In this paper, we examine an important problem of learning neural networks
that certifiably meet certain specifications on input-output behaviors. Our
strategy is to find an inner approximation of the set of admissible policy
parameters, which is convex in a transformed space. To this end, we address the
key technical challenge of convexifying the verification condition for neural
networks, which is derived by abstracting the nonlinear specifications and
activation functions with quadratic constraints. In particular, we propose a
reparametrization scheme of the original neural network based on loop
transformation, which leads to a convex condition that can be enforced during
learning. This theoretical construction is validated in an experiment that
specifies reachable sets for different regions of inputs.
