  Sparse deep neural networks have shown their advantages over dense models
with fewer parameters and higher computational efficiency. Here we demonstrate
constraining the synaptic weights on unit Lp-sphere enables the flexibly
control of the sparsity with p and improves the generalization ability of
neural networks. Firstly, to optimize the synaptic weights constrained on unit
Lp-sphere, the parameter optimization algorithm, Lp-spherical gradient descent
(LpSGD) is derived from the augmented Empirical Risk Minimization condition,
which is theoretically proved to be convergent. To understand the mechanism of
how p affects Hoyer's sparsity, the expectation of Hoyer's sparsity under the
hypothesis of gamma distribution is given and the predictions are verified at
various p under different conditions. In addition, the "semi-pruning" and
threshold adaptation are designed for topology evolution to effectively screen
out important connections and lead the neural networks converge from the
initial sparsity to the expected sparsity. Our approach is validated by
experiments on benchmark datasets covering a wide range of domains. And the
theoretical analysis pave the way to future works on training sparse neural
networks with constrained optimization.
