595 research outputs found
Mixtures of Gaussians are Privately Learnable with a Polynomial Number of Samples
We study the problem of estimating mixtures of Gaussians under the constraint
of differential privacy (DP). Our main result is that samples are sufficient to estimate a
mixture of Gaussians up to total variation distance while
satisfying -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)
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
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