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1. Faster Algorithms for High-Dimensional Robust Covariance Estimation.

   Cheng, Yu ; Diakonikolas, Ilias ; Ge, Rong ; Woodruff, David P. ( June 2019 , The 32’nd Annual Conference on Learning Theory (COLT 2019))

   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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2. Robustness Implies Privacy in Statistical Estimation

   Hopkins, Samuel B. ; Kamath, Gautam ; Majid, Mahbod ; Narayanan, Shyam ( June 2023 , 55th Annual ACM Symposium on Theory of Computing (STOC))

   We study the relationship between adversarial robustness and differential privacy in high-dimensional algorithmic statistics. We give the first black-box reduction from privacy to robustness which can produce private estimators with optimal tradeoffs among sample complexity, accuracy, and privacy for a wide range of fundamental high-dimensional parameter estimation problems, including mean and covariance estimation. We show that this reduction can be implemented in polynomial time in some important special cases. In particular, using nearly-optimal polynomial-time robust estimators for the mean and covariance of high-dimensional Gaussians which are based on the Sum-of-Squares method, we design the first polynomial-time private estimators for these problems with nearly-optimal samples-accuracy-privacy tradeoffs. Our algorithms are also robust to a constant fraction of adversarially-corrupted samples. 

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3. Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering

   Diakonikolas, Ilias ; Kane, Daniel ; Karmalkar, Sushrut ; Price, Eric ; Stewart, Alistair ( December 2019 , Advances in neural information processing systems)

   We study high-dimensional sparse estimation tasks in a robust setting where a constant fraction of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust sparse mean estimation and robust sparse PCA. We give the first practically viable robust estimators for these problems. In more detail, our algorithms are sample and computationally efficient and achieve near-optimal robustness guarantees. In contrast to prior provable algorithms which relied on the ellipsoid method, our algorithms use spectral techniques to iteratively remove outliers from the dataset. Our experimental evaluation on synthetic data shows that our algorithms are scalable and significantly outperform a range of previous approaches, nearly matching the best error rate without corruptions. 

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4. Fast, Sample-Efficient, Affine-Invariant Private Mean and Covariance Estimation for Subgaussian Distributions

   Brown, G. ; Hopkins, S. ; Smith, A. ( July 2023 , The Thirty Sixth Annual Conference on Learning Theory (COLT))

   We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation with nearly optimal sample complexity. Only exponential-time estimators were previously known to achieve this guarantee. Given n samples from a (sub-)Gaussian distribution with unknown mean μ and covariance Σ, our (ϵ,δ)-differentially private estimator produces μ~ such that ∥μ−μ~∥Σ≤α as long as n≳dα2+dlog1/δ√αϵ+dlog1/δϵ. The Mahalanobis error metric ∥μ−μ^∥Σ measures the distance between μ^ and μ relative to Σ; it characterizes the error of the sample mean. Our algorithm runs in time O~(ndω−1+nd/\eps), where ω<2.38 is the matrix multiplication exponent.We adapt an exponential-time approach of Brown, Gaboardi, Smith, Ullman, and Zakynthinou (2021), giving efficient variants of stable mean and covariance estimation subroutines that also improve the sample complexity to the nearly optimal bound above.Our stable covariance estimator can be turned to private covariance estimation for unrestricted subgaussian distributions. With n≳d3/2 samples, our estimate is accurate in spectral norm. This is the first such algorithm using n=o(d2) samples, answering an open question posed by Alabi et al. (2022). With n≳d2 samples, our estimate is accurate in Frobenius norm. This leads to a fast, nearly optimal algorithm for private learning of unrestricted Gaussian distributions in TV distance.Duchi, Haque, and Kuditipudi (2023) obtained similar results independently and concurrently. 

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5. Robustly Learning a Gaussian: Getting Optimal Error, Efficiently

   Diakonikolas, Ilias ; Kamath, Ilias ; Kane, Daniel M. ; Li, Jerry ; Moitra, Ankur ; Stewart, Alistair ( January 2018 , Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise — where an ε-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error O(ε) in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of and the running time is polynomial in d and 1/ε. When both the mean and covariance are unknown, the running time is polynomial in d and quasipolynomial in 1/ε. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings. 

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