  As a popular channel pruning method for convolutional neural networks (CNNs),
network slimming (NS) has a three-stage process: (1) it trains a CNN with
$\ell_1$ regularization applied to the scaling factors of the batch
normalization layers; (2) it removes channels whose scaling factors are below a
chosen threshold; and (3) it retrains the pruned model to recover the original
accuracy. This time-consuming, three-step process is a result of using
subgradient descent to train CNNs. Because subgradient descent does not exactly
train CNNs towards sparse, accurate structures, the latter two steps are
necessary. Moreover, subgradient descent does not have any convergence
guarantee. Therefore, we develop an alternative algorithm called proximal NS.
Our proposed algorithm trains CNNs towards sparse, accurate structures, so
identifying a scaling factor threshold is unnecessary and fine tuning the
pruned CNNs is optional. Using Kurdyka-{\L}ojasiewicz assumptions, we establish
global convergence of proximal NS. Lastly, we validate the efficacy of the
proposed algorithm on VGGNet, DenseNet and ResNet on CIFAR 10/100. Our
experiments demonstrate that after one round of training, proximal NS yields a
CNN with competitive accuracy and compression.
