  We consider the question of estimating multi-dimensional Gaussian mixtures
(GM) with compactly supported or subgaussian mixing distributions. Minimax
estimation rate for this class (under Hellinger, TV and KL divergences) is a
long-standing open question, even in one dimension. In this paper we
characterize this rate (for all constant dimensions) in terms of the metric
entropy of the class. Such characterizations originate from seminal works of Le
Cam (1973); Birge (1983); Haussler and Opper (1997); Yang and Barron (1999).
However, for GMs a key ingredient missing from earlier work (and widely
sought-after) is a comparison result showing that the KL and the squared
Hellinger distance are within a constant multiple of each other uniformly over
the class. Our main technical contribution is in showing this fact, from which
we derive entropy characterization for estimation rate under Hellinger and KL.
Interestingly, the sequential (online learning) estimation rate is
characterized by the global entropy, while the single-step (batch) rate
corresponds to local entropy, paralleling a similar result for the Gaussian
sequence model recently discovered by Neykov (2022) and Mourtada (2023).
Additionally, since Hellinger is a proper metric, our comparison shows that GMs
under KL satisfy the triangle inequality within multiplicative constants,
implying that proper and improper estimation rates coincide.
