  Backward propagation of errors (backpropagation) is a method to minimize
objective functions (e.g., loss functions) of deep neural networks by
identifying optimal sets of weights and biases. Imposing constraints on weight
precision is often required to alleviate prohibitive workloads on hardware.
Despite the remarkable success of backpropagation, the algorithm itself is not
capable of considering such constraints unless additional algorithms are
applied simultaneously. To address this issue, we propose the constrained
backpropagation (CBP) algorithm based on a pseudo-Lagrange multiplier method to
obtain the optimal set of weights that satisfy a given set of constraints. The
defining characteristic of the proposed CBP algorithm is the utilization of a
Lagrangian function (loss function plus constraint function) as its objective
function. We considered various types of constraints--binary, ternary, one-bit
shift, and two-bit shift weight constraints. As a post-training method, CBP
applied to AlexNet, ResNet-18, ResNet-50, and GoogLeNet on ImageNet, which were
pre-trained using the conventional backpropagation. For all cases, the proposed
algorithm outperforms the state-of-the-art methods on ImageNet, e.g., 66.6%,
74.4%, and 64.0% top-1 accuracy for ResNet-18, ResNet-50, and GoogLeNet with
binary weights, respectively. This highlights CBP as a learning algorithm to
address diverse constraints with the minimal performance loss by employing
appropriate constraint functions.
