  In recent studies, several asymptotic upper bounds on generalization errors
on deep neural networks (DNNs) are theoretically derived. These bounds are
functions of several norms of weights of the DNNs, such as the Frobenius and
spectral norms, and they are computed for weights grouped according to either
input and output channels of the DNNs. In this work, we conjecture that if we
can impose multiple constraints on weights of DNNs to upper bound the norms of
the weights, and train the DNNs with these weights, then we can attain
empirical generalization errors closer to the derived theoretical bounds, and
improve accuracy of the DNNs.
  To this end, we pose two problems. First, we aim to obtain weights whose
different norms are all upper bounded by a constant number, e.g. 1.0. To
achieve these bounds, we propose a two-stage renormalization procedure; (i)
normalization of weights according to different norms used in the bounds, and
(ii) reparameterization of the normalized weights to set a constant and finite
upper bound of their norms. In the second problem, we consider training DNNs
with these renormalized weights. To this end, we first propose a strategy to
construct joint spaces (manifolds) of weights according to different
constraints in DNNs. Next, we propose a fine-grained SGD algorithm (FG-SGD) for
optimization on the weight manifolds to train DNNs with assurance of
convergence to minima. Experimental results show that image classification
accuracy of baseline DNNs can be boosted using FG-SGD on collections of
manifolds identified by multiple constraints.
