  This paper presents a new dynamic approach to experiment design in settings
where, due to interference or other concerns, experimental units are coarse.
`Region-split' experiments on online platforms are one example of such a
setting. The cost, or regret, of experimentation is a natural concern here. Our
new design, dubbed Synthetically Controlled Thompson Sampling (SCTS), minimizes
the regret associated with experimentation at no practically meaningful loss to
inferential ability. We provide theoretical guarantees characterizing the
near-optimal regret of our approach, and the error rates achieved by the
corresponding treatment effect estimator. Experiments on synthetic and real
world data highlight the merits of our approach relative to both fixed and
`switchback' designs common to such experimental settings.
