Limitations of the Empirical Fisher Approximation for Natural Gradient Descent

Natural gradient descent, which preconditions a gradient descent update with the Fisher information matrix of the underlying statistical model, is a way to capture partial second-order information. Several highly visible works have advocated an approximation known as the empirical Fisher, drawing connections between approximate second-order methods and heuristics like Adam. We dispute this argument by showing that the empirical Fisher---unlike the Fisher---does not generally capture second-order information. We further argue that the conditions under which the empirical Fisher approaches the Fisher (and the Hessian) are unlikely to be met in practice, and that, even on simple optimization problems, the pathologies of the empirical Fisher can have undesirable effects.Comment: V3: Minor corrections (typographic errors

A Unified Framework for Compositional Fitting of Active Appearance Models

Active Appearance Models (AAMs) are one of the most popular and well-established techniques for modeling deformable objects in computer vision. In this paper, we study the problem of fitting AAMs using Compositional Gradient Descent (CGD) algorithms. We present a unified and complete view of these algorithms and classify them with respect to three main characteristics: i) cost function; ii) type of composition; and iii) optimization method. Furthermore, we extend the previous view by: a) proposing a novel Bayesian cost function that can be interpreted as a general probabilistic formulation of the well-known project-out loss; b) introducing two new types of composition, asymmetric and bidirectional, that combine the gradients of both image and appearance model to derive better conver- gent and more robust CGD algorithms; and c) providing new valuable insights into existent CGD algorithms by reinterpreting them as direct applications of the Schur complement and the Wiberg method. Finally, in order to encourage open research and facilitate future comparisons with our work, we make the implementa- tion of the algorithms studied in this paper publicly available as part of the Menpo Project.Comment: 39 page

Optimization Methods for Inverse Problems

Optimization plays an important role in solving many inverse problems. Indeed, the task of inversion often either involves or is fully cast as a solution of an optimization problem. In this light, the mere non-linear, non-convex, and large-scale nature of many of these inversions gives rise to some very challenging optimization problems. The inverse problem community has long been developing various techniques for solving such optimization tasks. However, other, seemingly disjoint communities, such as that of machine learning, have developed, almost in parallel, interesting alternative methods which might have stayed under the radar of the inverse problem community. In this survey, we aim to change that. In doing so, we first discuss current state-of-the-art optimization methods widely used in inverse problems. We then survey recent related advances in addressing similar challenges in problems faced by the machine learning community, and discuss their potential advantages for solving inverse problems. By highlighting the similarities among the optimization challenges faced by the inverse problem and the machine learning communities, we hope that this survey can serve as a bridge in bringing together these two communities and encourage cross fertilization of ideas.Comment: 13 page

Online Structured Laplace Approximations For Overcoming Catastrophic Forgetting

We introduce the Kronecker factored online Laplace approximation for overcoming catastrophic forgetting in neural networks. The method is grounded in a Bayesian online learning framework, where we recursively approximate the posterior after every task with a Gaussian, leading to a quadratic penalty on changes to the weights. The Laplace approximation requires calculating the Hessian around a mode, which is typically intractable for modern architectures. In order to make our method scalable, we leverage recent block-diagonal Kronecker factored approximations to the curvature. Our algorithm achieves over 90% test accuracy across a sequence of 50 instantiations of the permuted MNIST dataset, substantially outperforming related methods for overcoming catastrophic forgetting.Comment: 13 pages, 6 figure