  Parameter-specific adaptive learning rate methods are computationally
efficient ways to reduce the ill-conditioning problems encountered when
training large deep networks. Following recent work that strongly suggests that
most of the critical points encountered when training such networks are saddle
points, we find how considering the presence of negative eigenvalues of the
Hessian could help us design better suited adaptive learning rate schemes. We
show that the popular Jacobi preconditioner has undesirable behavior in the
presence of both positive and negative curvature, and present theoretical and
empirical evidence that the so-called equilibration preconditioner is
comparatively better suited to non-convex problems. We introduce a novel
adaptive learning rate scheme, called ESGD, based on the equilibration
preconditioner. Our experiments show that ESGD performs as well or better than
RMSProp in terms of convergence speed, always clearly improving over plain
stochastic gradient descent.
