• Quasi-Newton Methods: A New Direction

Philipp Hennig, Martin Kiefel
– Accepted
Abstract: Four decades after their invention, quasi-Newton methods are still state of the art in unconstrained numerical optimization. Although not usually interpreted thus, these are learning algorithms that fit a local quadratic approximation to the objective function. We show that many, including the most popular, quasi-Newton methods can be interpreted as approximations of Bayesian linear regression under varying prior assumptions. This new notion elucidates some shortcomings of classical algorithms, and lights the way to a novel nonparametric quasi-Newton method, which is able to make more efficient use of available information at computational cost similar to its predecessors.

• Estimating the Hessian by Back-propagating Curvature

James Martens, Ilya Sutskever, Kevin Swersky

– Accepted

Abstract: In this work we develop Curvature Propagation (CP), a general technique for efficiently computing unbiased approximations of the Hessian of any function that is computed using a computational graph. At the cost of roughly two gradient evaluations, CP can give a rank-1 approximation of the whole Hessian, and can be repeatedly applied to give increasingly precise unbiased estimates of any or all of the entries of the Hessian. Of particular interest is the diagonal of the Hessian, for which no general approach is known to exist that is both efficient and accurate. We show in experiments that CP turns out to work well in practice, giving very accurate estimates of the Hessian of neural networks, for example, with a relatively small amount of work. We also apply CP to Score Matching, where a diagonal of a Hessian plays an integral role in the Score Matching objective, and where it is usually computed exactly using inefficient algorithms which do not scale to larger and more complex models.