Jonathan Wenger

Marvin Pförtner, Jonathan Wenger, Jon Cockayne et al.

Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. But since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.

Jonathan Wenger, Philipp Hennig

Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of uncertainty in the field. Unifying earlier work we propose a class of probabilistic linear solvers which jointly infer the matrix, its inverse and the solution from matrix-vector product observations. This class emerges from a fundamental set of desiderata which constrains the space of possible algorithms and recovers the method of conjugate gradients under certain conditions. We demonstrate how to incorporate prior spectral information in order to calibrate uncertainty and experimentally showcase the potential of such solvers for machine learning.

Jonathan Wenger, Hedvig Kjellström, Rudolph Triebel

Many applications of classification methods not only require high accuracy but also reliable estimation of predictive uncertainty. However, while many current classification frameworks, in particular deep neural networks, achieve high accuracy, they tend to incorrectly estimate uncertainty. In this paper, we propose a method that adjusts the confidence estimates of a general classifier such that they approach the probability of classifying correctly. In contrast to existing approaches, our calibration method employs a non-parametric representation using a latent Gaussian process, and is specifically designed for multi-class classification. It can be applied to any classifier that outputs confidence estimates and is not limited to neural networks. We also provide a theoretical analysis regarding the over- and underconfidence of a classifier and its relationship to calibration, as well as an empirical outlook for calibrated active learning. In experiments we show the universally strong performance of our method across different classifiers and benchmark data sets, in particular for state-of-the art neural network architectures.