Learning with Variational Objectives on Measures.

The theory of machine learning has predominantly focused on variational objectives expressed on functions. Yet, the advent of big data has exposed limitations in these formulations: functions struggle to capture uncertainty arising from data veracity issues; and the assumption of data independence often conflicts with data variety. In this vision paper, we propose a shift towards measures as the basis for variational objectives, highlighting their advantages to quantify uncertainty, to generalize out of training distribution, and to deal with scarce annotations. We aim to inspire and stimulate further research in adapting conventional machine learning algorithms to optimize these new objectives, without compromising statistical efficiency.