Practical Gauss-Newton Optimisation for Deep Learning

We present an efficient block-diagonal ap- proximation to the Gauss-Newton matrix for feedforward neural networks. Our result- ing algorithm is competitive against state- of-the-art first order optimisation methods, with sometimes significant improvement in optimisation performance. Unlike first-order methods, for which hyperparameter tuning of the optimisation parameters is often a labo- rious process, our approach can provide good performance even when used with default set- tings. A side result of our work is that for piecewise linear transfer functions, the net- work objective function can have no differ- entiable local maxima, which may partially explain why such transfer functions facilitate effective optimisation.Comment: ICML 201

Fixed-Form Variational Posterior Approximation through Stochastic Linear Regression

We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given in closed form up to the proportionality constant. The approximation can be any distribution in the exponential family or any mixture of such distributions, which means that it can be made arbitrarily precise. Several examples illustrate the speed and accuracy of our approximation method in practice

Importance Sampled Stochastic Optimization for Variational Inference

Variational inference approximates the posterior distribution of a probabilistic model with a parameterized density by maximizing a lower bound for the model evidence. Modern solutions fit a flexible approximation with stochastic gradient descent, using Monte Carlo approximation for the gradients. This enables variational inference for arbitrary differentiable probabilistic models, and consequently makes variational inference feasible for probabilistic programming languages. In this work we develop more efficient inference algorithms for the task by considering importance sampling estimates for the gradients. We show how the gradient with respect to the approximation parameters can often be evaluated efficiently without needing to re-compute gradients of the model itself, and then proceed to derive practical algorithms that use importance sampled estimates to speed up computation. We present importance sampled stochastic gradient descent that outperforms standard stochastic gradient descent by a clear margin for a range of models, and provide a justifiable variant of stochastic average gradients for variational inference.Peer reviewe

Divide and conquer in ABC: Expectation-Progagation algorithms for likelihood-free inference

ABC algorithms are notoriously expensive in computing time, as they require simulating many complete artificial datasets from the model. We advocate in this paper a "divide and conquer" approach to ABC, where we split the likelihood into n factors, and combine in some way n "local" ABC approximations of each factor. This has two advantages: (a) such an approach is typically much faster than standard ABC and (b) it makes it possible to use local summary statistics (i.e. summary statistics that depend only on the data-points that correspond to a single factor), rather than global summary statistics (that depend on the complete dataset). This greatly alleviates the bias introduced by summary statistics, and even removes it entirely in situations where local summary statistics are simply the identity function. We focus on EP (Expectation-Propagation), a convenient and powerful way to combine n local approximations into a global approximation. Compared to the EP- ABC approach of Barthelm\'e and Chopin (2014), we present two variations, one based on the parallel EP algorithm of Cseke and Heskes (2011), which has the advantage of being implementable on a parallel architecture, and one version which bridges the gap between standard EP and parallel EP. We illustrate our approach with an expensive application of ABC, namely inference on spatial extremes.Comment: To appear in the forthcoming Handbook of Approximate Bayesian Computation (ABC), edited by S. Sisson, L. Fan, and M. Beaumon