  Stochastic gradient-based optimization is crucial to optimize neural
networks. While popular approaches heuristically adapt the step size and
direction by rescaling gradients, a more principled approach to improve
optimizers requires second-order information. Such methods precondition the
gradient using the objective's Hessian. Yet, computing the Hessian is usually
expensive and effectively using second-order information in the stochastic
gradient setting is non-trivial. We propose using Information-Theoretic Trust
Region Optimization (arTuRO) for improved updates with uncertain second-order
information. By modeling the network parameters as a Gaussian distribution and
using a Kullback-Leibler divergence-based trust region, our approach takes
bounded steps accounting for the objective's curvature and uncertainty in the
parameters. Before each update, it solves the trust region problem for an
optimal step size, resulting in a more stable and faster optimization process.
We approximate the diagonal elements of the Hessian from stochastic gradients
using a simple recursive least squares approach, constructing a model of the
expected Hessian over time using only first-order information. We show that
arTuRO combines the fast convergence of adaptive moment-based optimization with
the generalization capabilities of SGD.
