  Cosmic shear is a primary cosmological probe for several present and upcoming
surveys investigating dark matter and dark energy, such as Euclid or WFIRST.
The probe requires an extremely accurate measurement of the shapes of millions
of galaxies based on imaging data. Crucially, the shear measurement must
address and compensate for a range of interwoven nuisance effects related to
the instrument optics and detector, noise, unknown galaxy morphologies, colors,
blending of sources, and selection effects. This paper explores the use of
supervised machine learning (ML) as a tool to solve this inverse problem. We
present a simple architecture that learns to regress shear point estimates and
weights via shallow artificial neural networks. The networks are trained on
simulations of the forward observing process, and take combinations of moments
of the galaxy images as inputs. A challenging peculiarity of this ML
application is the combination of the noisiness of the input features and the
requirements on the accuracy of the inverse regression. To address this issue,
the proposed training algorithm minimizes bias over multiple realizations of
individual source galaxies, reducing the sensitivity to properties of the
overall sample of source galaxies. Importantly, an observational selection
function of these source galaxies can be straightforwardly taken into account
via the weights. We first introduce key aspects of our approach using toy-model
simulations, and then demonstrate its potential on images mimicking Euclid
data. Finally, we analyze images from the GREAT3 challenge, obtaining
competitively low shear biases despite the use of a simple training set. We
conclude that the further development of ML approaches is of high interest to
meet the stringent requirements on the shear measurement in current and future
surveys. A demonstration implementation of our technique is publicly available.
