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1. 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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2. Private High-Dimensional Hypothesis Testing

   Narayanan, S. ( January 2022 , Conference on Learning Theory (COLT 2022))

   We provide improved differentially private algorithms for identity testing of high-dimensional distributions. Specifically, for d-dimensional Gaussian distributions with known covariance Σ, we can test whether the distribution comes from N(μ∗,Σ) for some fixed μ∗ or from some N(μ,Σ) with total variation distance at least α from N(μ∗,Σ) with (ε,0)-differential privacy, using only O~(d1/2α2+d1/3α4/3⋅ε2/3+1α⋅ε) samples if the algorithm is allowed to be computationally inefficient, and only O~(d1/2α2+d1/4α⋅ε) samples for a computationally efficient algorithm. We also provide a matching lower bound showing that our computationally inefficient algorithm has optimal sample complexity. We also extend our algorithms to various related problems, including mean testing of Gaussians with bounded but unknown covariance, uniformity testing of product distributions over {−1,1}d, and tolerant testing. Our results improve over the previous best work of Canonne et al.~\cite{CanonneKMUZ20} for both computationally efficient and inefficient algorithms, and even our computationally efficient algorithm matches the optimal \emph{non-private} sample complexity of O(d√α2) in many standard parameter settings. In addition, our results show that, surprisingly, private identity testing of d-dimensional Gaussians can be done with fewer samples than private identity testing of discrete distributions over a domain of size d \cite{AcharyaSZ18}, which refutes a conjectured lower bound of~\cite{CanonneKMUZ20}. 

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3. High-Dimensional Robust Mean Estimation in Nearly-Linear Time

   Cheng, Yu ; Diakonikolas, Ilias ; Ge, Rong ( April 2019 , Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   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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4. The Full Landscape of Robust Mean Testing: Sharp Separations between Oblivious and Adaptive Contamination

   Canonne, C. L ; Hopkins, S. B. ; Li, J. ; Liu, A. ; Narayanan, S.: ( November 2023 , Foundations of Computer Science (FOCS))

   We consider the question of Gaussian mean testing, a fundamental task in high-dimensional distribution testing and signal processing, subject to adversarial corruptions of the samples. We focus on the relative power of different adversaries, and show that, in contrast to the common wisdom in robust statistics, there exists a strict separation between adaptive adversaries (strong contamination) and oblivious ones (weak contamination) for this task. Specifically, we resolve both the information-theoretic and computational landscapes for robust mean testing. In the exponential-time setting, we establish the tight sample complexity of testing N(0,I) against N(αv,I), where ∥v∥2=1, with an ε-fraction of adversarial corruptions, to be Θ~(max(d√α2,dε3α4,min(d2/3ε2/3α8/3,dεα2))) while the complexity against adaptive adversaries is Θ~(max(d√α2,dε2α4)) which is strictly worse for a large range of vanishing ε,α. To the best of our knowledge, ours is the first separation in sample complexity between the strong and weak contamination models. In the polynomial-time setting, we close a gap in the literature by providing a polynomial-time algorithm against adaptive adversaries achieving the above sample complexity Θ~(max(d−−√/α2,dε2/α4)), and a low-degree lower bound (which complements an existing reduction from planted clique) suggesting that all efficient algorithms require this many samples, even in the oblivious-adversary setting. 

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5. The Full Landscape of Robust Mean Testing: Sharp Separations between Oblivious and Adaptive Contamination

   Canonne, Clément L. ; Hopkins, Samuel B. ; Li, Jerry ; Liu, Allen ; Narayanan, Shyam ( November 2023 , 64th Annual IEEE Symposium on Foundations of Computer Science (FOCS))

   We consider the question of Gaussian mean testing, a fundamental task in high-dimensional distribution testing and signal processing, subject to adversarial corruptions of the samples. We focus on the relative power of different adversaries, and show that, in contrast to the common wisdom in robust statistics, there exists a strict separation between adaptive adversaries (strong contamination) and oblivious ones (weak contamination) for this task. Specifically, we resolve both the information-theoretic and computational landscapes for robust mean testing. In the exponential-time setting, we establish the tight sample complexity of testing N(0,I) against N(αv,I), where ∥v∥2=1, with an ε-fraction of adversarial corruptions, to be Θ~(max(d−−√α2,dε3α4,min(d2/3ε2/3α8/3,dεα2))), while the complexity against adaptive adversaries is Θ~(max(d−−√α2,dε2α4)), which is strictly worse for a large range of vanishing ε,α. To the best of our knowledge, ours is the first separation in sample complexity between the strong and weak contamination models. In the polynomial-time setting, we close a gap in the literature by providing a polynomial-time algorithm against adaptive adversaries achieving the above sample complexity Θ~(max(d−−√/α2,dε2/α4)), and a low-degree lower bound (which complements an existing reduction from planted clique) suggesting that all efficient algorithms require this many samples, even in the oblivious-adversary setting. 

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