Recent work has identified that using a high learning rate or a small batch
size for Stochastic Gradient Descent (SGD) based training of deep neural
networks encourages finding flatter minima of the training loss towards the end
of training. Moreover, measures of the flatness of minima have been shown to
correlate with good generalization performance. Extending this previous work,
we investigate the loss curvature through the Hessian eigenvalue spectrum in
the early phase of training and find an analogous bias: even at the beginning
of training, a high learning rate or small batch size influences SGD to visit
flatter loss regions. In addition, the evolution of the largest eigenvalues
appears to always follow a similar pattern, with a fast increase in the early
phase, and a decrease or stabilization thereafter, where the peak value is
determined by the learning rate and batch size. Finally, we find that by
altering the learning rate just in the direction of the eigenvectors associated
with the largest eigenvalues, SGD can be steered towards regions which are an
order of magnitude sharper but correspond to models with similar
generalization, which suggests the curvature of the endpoint found by SGD is
not predictive of its generalization properties.
Jastrzębski, Stanisław, Kenton, Zachary, Ballas, Nicolas, Fischer, Asja, Bengio, Yoshua, Storkey, Amos
Recent work has identified that using a high learning rate or a small batch size for Stochastic Gradient Descent (SGD) based training of deep neural networks encourages finding flatter minima of the training loss towards the end of training. Moreover, measures of the flatness of minima have been shown to correlate with good generalization performance. Extending this previous work, we investigate the loss curvature through the Hessian eigenvalue spectrum in the early phase of training and find an analogous bias: even at the beginning of training, a high learning rate or small batch size influences SGD to visit flatter loss regions. In addition, the evolution of the largest eigenvalues appears to always follow a similar pattern, with a fast increase in the early phase, and a decrease or stabilization thereafter, where the peak value is determined by the learning rate and batch size. Finally, we find that by altering the learning rate just in the direction of the eigenvectors associated with the largest eigenvalues, SGD can be steered towards regions which are an order of magnitude sharper but correspond to models with similar generalization, which suggests the curvature of the endpoint found by SGD is not predictive of its generalization properties.
https://arxiv.org/abs/1807.05031