  We present an adaptive regularization algorithm that can be effectively
applied to the optimization problem in deep learning framework. Our
regularization algorithm aims to take into account the fitness of data to the
current state of model in the determination of regularity to achieve better
generalization. The degree of regularization at each element in the target
space of the neural network architecture is determined based on the residual at
each optimization iteration in an adaptive way. Our adaptive regularization
algorithm is designed to apply a diffusion process driven by the heat equation
with spatially varying diffusivity depending on the probability density
function following a certain distribution of residual. Our data-driven
regularity is imposed by adaptively smoothing a simplified objective function
in which the explicit regularization term is omitted in an alternating manner
between the evaluation of residual and the determination of the degree of its
regularity. The effectiveness of our algorithm is empirically demonstrated by
the numerical experiments in the application of image classification problems,
indicating that our algorithm outperforms other commonly used optimization
algorithms in terms of generalization using popular deep learning models and
benchmark datasets.
