Deep Neural Network Initialization with Sparsity Inducing Activations
Ilan Price Nicholas Daultry Ball Samuel C. H. Lam Adam C. Jones Jared Tanner
Inducing and leveraging sparse activations during training and inference is a
promising avenue for improving the computational efficiency of deep networks,
which is increasingly important as network sizes continue to grow and their
application becomes more widespread. Here we use the large width Gaussian
process limit to analyze the behaviour, at random initialization, of nonlinear
activations that induce sparsity in the hidden outputs. A previously unreported
form of training instability is proven for arguably two of the most natural
candidates for hidden layer sparsification; those being a shifted ReLU
($\phi(x)=\max(0, x-\tau)$ for $\tau\ge 0$) and soft thresholding ($\phi(x)=0$
for $|x|\le\tau$ and $x-\text{sign}(x)\tau$ for $|x|>\tau$). We show that this
instability is overcome by clipping the nonlinear activation magnitude, at a
level prescribed by the shape of the associated Gaussian process variance map.
Numerical experiments verify the theory and show that the proposed magnitude
clipped sparsifying activations can be trained with training and test
fractional sparsity as high as 85\% while retaining close to full accuracy.