Ensemble Kalman inversion: a derivative-free technique for machine learning tasks

Abstract

The standard probabilistic perspective on machine learning gives rise to empirical risk-minimization tasks that are frequently solved by stochastic gradient descent (SGD) and variants thereof. We present a formulation of these tasks as classical inverse or filtering problems and, furthermore, we propose an efficient, gradient-free algorithm for finding a solution to these problems using ensemble Kalman inversion (EKI). Applications of our approach include offline and online supervised learning with deep neural networks, as well as graph-based semi-supervised learning. The essence of the EKI procedure is an ensemble based approximate gradient descent in which derivatives are replaced by differences from within the ensemble. We suggest several modifications to the basic method, derived from empirically successful heuristics developed in the context of SGD. Numerical results demonstrate wide applicability and robustness of the proposed algorithm.

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

“…The blue dots correspond to the output of this algorithm at the last iteration. The red dots correspond to the last ensemble of the EKI algorithm as presented in (Kovachki and Stuart, 2018). The orange dots depict the RWMH gold standard described above.…”

“…The true distribution, computed by RWMH, is shown in orange. Note that the algorithm of Kovachki and Stuart (2018) collapses to a point (shown in red), unable to escape overfitting, and relating to a form of consensus formation. In contrast, the algorithm of Herty and Visconti (2018), while avoiding overfitting, overestimates the spread of the ensemble members, relative to the gold standard RWMH; this is exhibited by the blue over-dispersed points.…”

“…In this paper, we study a new noisy version of EKI, the ensemble Kalman sampler (EKS), and related mean-field limits, the aim being the construction of methods which lead to approximate posterior samples, without the use of adjoints, and overcoming the issue that the standard noisy EKI does not reproduce the posterior distribution, as highlighted in (Ernst, Sprungk and Starkloff, 2015). We emphasize that the practical derivative-free algorithm that we propose rests on a particle-based approximation of a specific preconditioned gradient flow, as described in section 4.3 of the paper (Kovachki and Stuart, 2018); we add a judiciously chosen noise to this setting and it is this additional noise which enables approximate posterior sampling. Related approximations are also studied in the paper (Pathiraja and Reich, 2019) in which the effect of both time-discretization and particle approximation are discussed when applied to various deterministic interacting particle systems with gradient structure.…”

“…In what follows we instead directly perturb the particles themselves. The benefits of introducing noise on the particles, rather than the data, was demonstrated in (Kovachki and Stuart, 2018), although in that setting only optimization, and not Bayesian inversion, is considered.…”

“…n ∼ N(0, I), Γ 0 is the prior covariance and ∆t n is an adaptive timestep computed as in (Kovachki and Stuart, 2018).…”