  Pre-conditioning is a well-known concept that can significantly improve the
convergence of optimization algorithms. For noise-free problems, where good
pre-conditioners are not known a priori, iterative linear algebra methods offer
one way to efficiently construct them. For the stochastic optimization problems
that dominate contemporary machine learning, however, this approach is not
readily available. We propose an iterative algorithm inspired by classic
iterative linear solvers that uses a probabilistic model to actively infer a
pre-conditioner in situations where Hessian-projections can only be constructed
with strong Gaussian noise. The algorithm is empirically demonstrated to
efficiently construct effective pre-conditioners for stochastic gradient
descent and its variants. Experiments on problems of comparably low
dimensionality show improved convergence. In very high-dimensional problems,
such as those encountered in deep learning, the pre-conditioner effectively
becomes an automatic learning-rate adaptation scheme, which we also empirically
show to work well.
