  Entropy integrals are widely used as a powerful empirical process tool to
obtain upper bounds for the rates of convergence of global empirical risk
minimizers (ERMs), in standard settings such as density estimation and
regression. The upper bound for the convergence rates thus obtained typically
matches the minimax lower bound when the entropy integral converges, but admits
a strict gap compared to the lower bound when it diverges. Birg\'e and Massart
[BM93] provided a striking example showing that such a gap is real with the
entropy structure alone: for a variant of the natural H\"older class with low
regularity, the global ERM actually converges at the rate predicted by the
entropy integral that substantially deviates from the lower bound. The
counter-example has spawned a long-standing negative position on the use of
global ERMs in the regime where the entropy integral diverges, as they are
heuristically believed to converge at a sub-optimal rate in a variety of
models.
  The present paper demonstrates that this gap can be closed if the models
admit certain degree of `set structures' in addition to the entropy structure.
In other words, the global ERMs in such set structured models will indeed be
rate-optimal, matching the lower bound even when the entropy integral diverges.
The models with set structures we investigate include (i) image and edge
estimation, (ii) binary classification, (iii) multiple isotonic regression,
(iv) $s$-concave density estimation, all in general dimensions when the entropy
integral diverges. Here set structures are interpreted broadly in the sense
that the complexity of the underlying models can be essentially captured by the
size of the empirical process over certain class of measurable sets, for which
matching upper and lower bounds are obtained to facilitate the derivation of
sharp convergence rates for the associated global ERMs.
