  With the increasing application of deep learning in mission-critical systems,
there is a growing need to obtain formal guarantees about the behaviors of
neural networks. Indeed, many approaches for verifying neural networks have
been recently proposed, but these generally struggle with limited scalability
or insufficient accuracy. A key component in many state-of-the-art verification
schemes is computing lower and upper bounds on the values that neurons in the
network can obtain for a specific input domain -- and the tighter these bounds,
the more likely the verification is to succeed. Many common algorithms for
computing these bounds are variations of the symbolic-bound propagation method;
and among these, approaches that utilize a process called back-substitution are
particularly successful. In this paper, we present an approach for making
back-substitution produce tighter bounds. To achieve this, we formulate and
then minimize the imprecision errors incurred during back-substitution. Our
technique is general, in the sense that it can be integrated into numerous
existing symbolic-bound propagation techniques, with only minor modifications.
We implement our approach as a proof-of-concept tool, and present favorable
results compared to state-of-the-art verifiers that perform back-substitution.
