Sample Compression for Real-Valued Learners

We give an algorithmically efficient version of the learner-to-compression scheme conversion in Moran and Yehudayoff (2016). In extending this technique to real-valued hypotheses, we also obtain an efficient regression-to-bounded sample compression converter. To our knowledge, this is the first general compressed regression result (regardless of efficiency or boundedness) guaranteeing uniform approximate reconstruction. Along the way, we develop a generic procedure for constructing weak real-valued learners out of abstract regressors; this may be of independent interest. In particular, this result sheds new light on an open question of H. Simon (1997). We show applications to two regression problems: learning Lipschitz and bounded-variation functions.