  Training deep neural networks with an $L_0$ regularization is one of the
prominent approaches for network pruning or sparsification. The method prunes
the network during training by encouraging weights to become exactly zero.
However, recent work of Gale et al. reveals that although this method yields
high compression rates on smaller datasets, it performs inconsistently on
large-scale learning tasks, such as ResNet50 on ImageNet. We analyze this
phenomenon through the lens of variational inference and find that it is likely
due to the independent modeling of binary gates, the mean-field approximation,
which is known in Bayesian statistics for its poor performance due to the crude
approximation. To mitigate this deficiency, we propose a dependency modeling of
binary gates, which can be modeled effectively as a multi-layer perceptron
(MLP). We term our algorithm Dep-$L_0$ as it prunes networks via a
dependency-enabled $L_0$ regularization. Extensive experiments on CIFAR10,
CIFAR100 and ImageNet with VGG16, ResNet50, ResNet56 show that our Dep-$L_0$
outperforms the original $L_0$-HC algorithm of Louizos et al. by a significant
margin, especially on ImageNet. Compared with the state-of-the-arts network
sparsification algorithms, our dependency modeling makes the $L_0$-based
sparsification once again very competitive on large-scale learning tasks. Our
source code is available at https://github.com/leo-yangli/dep-l0.
