We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions.
In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with
(i) known covariance and sub-gaussian tails, and
(ii) unknown bounded covariance.
Given N samples on R^d, an \eps-fraction of which may be arbitrarily corrupted, our algorithms run in time eO(Nd)/poly(\eps) and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times \Omega(Nd^2), for \eps= O(1)
Our algorithms rely on a natural family of SDPs parameterized by our current guess ν for the unknown mean μ. We give a win-win analysis establishing the following: either a near-optimal solution to the primal SDP yields a good candidate for μ — independent of our current guess ν — or a near-optimal solution to the dual SDP yields a new guess ν0
whose distance from μ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear
time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems.
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Faster Algorithms for High-Dimensional Robust Covariance Estimation.
We study the problem of estimating the covariance matrix of a high-dimensional distribution
when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem with near-optimal error guarantees for several natural structured distributions. Our main contribution is to develop faster algorithms for this problem whose running time nearly matches that of computing the empirical covariance. Given N = Ω(d^2/\eps^2) samples from a d-dimensional Gaussian distribution, an \eps-fraction of which may be arbitrarily corrupted, our algorithm runs in time O(d^{3.26}/ poly(\eps)) and approximates the unknown covariance matrix to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees all have runtimes Ω(d^{2ω}) when \eps = Ω(1), where ω is the exponent of matrix multiplication. We also provide evidence that improving the running time of our algorithm may require new algorithmic techniques.
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- PAR ID:
- 10161651
- Date Published:
- Journal Name:
- The 32’nd Annual Conference on Learning Theory (COLT 2019)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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