Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples
International Conference on Algorithmic Learning Theory(2023)
摘要
We study the problem of estimating mixtures of Gaussians under the constraint
of differential privacy (DP). Our main result is that
poly(k,d,1/α,1/ε,log(1/δ)) samples are sufficient
to estimate a mixture of k Gaussians in ℝ^d up to total variation
distance α while satisfying (ε, δ)-DP. This is the
first finite sample complexity upper bound for the problem that does not make
any structural assumptions on the GMMs.
To solve the problem, we devise a new framework which may be useful for other
tasks. On a high level, we show that if a class of distributions (such as
Gaussians) is (1) list decodable and (2) admits a "locally small” cover (Bun
et al., 2021) with respect to total variation distance, then the class of its
mixtures is privately learnable. The proof circumvents a known barrier
indicating that, unlike Gaussians, GMMs do not admit a locally small cover
(Aden-Ali et al., 2021b).
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