Robust low-rank training via approximate orthonormal constraints
Dayana Savostianova Emanuele Zangrando Gianluca Ceruti Francesco Tudisco
With the growth of model and data sizes, a broad effort has been made to
design pruning techniques that reduce the resource demand of deep learning
pipelines, while retaining model performance. In order to reduce both inference
and training costs, a prominent line of work uses low-rank matrix
factorizations to represent the network weights. Although able to retain
accuracy, we observe that low-rank methods tend to compromise model robustness
against adversarial perturbations. By modeling robustness in terms of the
condition number of the neural network, we argue that this loss of robustness
is due to the exploding singular values of the low-rank weight matrices. Thus,
we introduce a robust low-rank training algorithm that maintains the network's
weights on the low-rank matrix manifold while simultaneously enforcing
approximate orthonormal constraints. The resulting model reduces both training
and inference costs while ensuring well-conditioning and thus better
adversarial robustness, without compromising model accuracy. This is shown by
extensive numerical evidence and by our main approximation theorem that shows
the computed robust low-rank network well-approximates the ideal full model,
provided a highly performing low-rank sub-network exists.