  Deep learning has achieved tremendous success by training increasingly large
models, which are then compressed for practical deployment. We propose a
drastically different approach to compact and optimal deep learning: We
decouple the Degrees of freedom (DoF) and the actual number of parameters of a
model, optimize a small DoF with predefined random linear constraints for a
large model of arbitrary architecture, in one-stage end-to-end learning.
Specifically, we create a recurrent parameter generator (RPG), which repeatedly
fetches parameters from a ring and unpacks them onto a large model with random
permutation and sign flipping to promote parameter decorrelation. We show that
gradient descent can automatically find the best model under constraints with
faster convergence. Our extensive experimentation reveals a log-linear
relationship between model DoF and accuracy. Our RPG demonstrates remarkable
DoF reduction and can be further pruned and quantized for additional run-time
performance gain. For example, in terms of top-1 accuracy on ImageNet, RPG
achieves $96\%$ of ResNet18's performance with only $18\%$ DoF (the equivalent
of one convolutional layer) and $52\%$ of ResNet34's performance with only
$0.25\%$ DoF! Our work shows a significant potential of constrained neural
optimization in compact and optimal deep learning.
