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# Outlier Robust Mean Estimation with Subgaussian Rates via Stability

NIPS 2020, (2020)

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Abstract

We study the problem of outlier robust high-dimensional mean estimation under a finite covariance assumption, and more broadly under finite low-degree moment assumptions. We consider a standard stability condition from the recent robust statistics literature and prove that, except with exponentially small failure probability, there exis...More

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Introduction

- Consider the following problem: For a given family F of distributions on Rd, estimate the mean of an unknown D ∈ F, given access to i.i.d. samples from D.
- This is the problem of mean estimation and is arguably the most fundamental statistical task.
- The authors relax the “i.i.d. assumption” and aim to obtain estimators that are robust to a constant fraction of adversarial outliers

Highlights

- 1.1 Background and Motivation

Consider the following problem: For a given family F of distributions on Rd, estimate the mean of an unknown D ∈ F, given access to i.i.d. samples from D - In the most basic setting where F is the family of high-dimensional Gaussians, the empirical mean is well-known to be an optimal estimator — in the sense that it achieves the best possible accuracy-confidence tradeoff and is easy to compute
- We study high-dimensional mean estimation in the high confidence regime when the underlying family F is only assumed to satisfy bounded moment conditions
- We showed that a standard stability condition from the recent high-dimensional robust statistics literature suffices to obtain near-subgaussian rates for robust mean estimation in the strong contamination model
- An interesting technical question is whether the extra log d factor in Theorem 1.4 is needed. (Our results imply that it is not needed when ǫ = Ω(1).) If not, this would imply that stability-based algorithms achieve subgaussian rates without the pre-processing

Results

- The authors' first main result establishes the stability of a subset of i.i.d. points drawn from a distribution with bounded covariance.

Theorem 1.4. - The authors' first main result establishes the stability of a subset of i.i.d. points drawn from a distribution with bounded covariance.
- Let S be a multiset of n i.i.d. samples from a distribution on.
- Let ǫ′ = O(log(1/τ )/n + ǫ) ≤ c, for a sufficiently small constant c > 0.
- With S′ is (2ǫ′, δ)-stable probability at least 1− with respect to μ and τ, Σ there exists a subset S′ ⊆ S such that , where δ = O( (r(Σ) log r(Σ))/n+

Conclusion

- The authors showed that a standard stability condition from the recent high-dimensional robust statistics literature suffices to obtain near-subgaussian rates for robust mean estimation in the strong contamination model.
- With a simple pre-processing, this leads to efficient outlier-robust estimators with subgaussian rates under only a bounded covariance assumption.
- (The authors' results imply that it is not needed when ǫ = Ω(1).) If not, this would imply that stability-based algorithms achieve subgaussian rates without the pre-processing

Summary

## Introduction:

Consider the following problem: For a given family F of distributions on Rd, estimate the mean of an unknown D ∈ F, given access to i.i.d. samples from D.- This is the problem of mean estimation and is arguably the most fundamental statistical task.
- The authors relax the “i.i.d. assumption” and aim to obtain estimators that are robust to a constant fraction of adversarial outliers
## Objectives:

The authors aim to achieve the best of both worlds. Recall that the aim is to find a w ∈ ∆n,ǫ that satisfies the conditions: (i) μw − μ ≤ δ, and (ii) Σw − I ≤ δ2/ǫ.## Results:

The authors' first main result establishes the stability of a subset of i.i.d. points drawn from a distribution with bounded covariance.

Theorem 1.4.- The authors' first main result establishes the stability of a subset of i.i.d. points drawn from a distribution with bounded covariance.
- Let S be a multiset of n i.i.d. samples from a distribution on.
- Let ǫ′ = O(log(1/τ )/n + ǫ) ≤ c, for a sufficiently small constant c > 0.
- With S′ is (2ǫ′, δ)-stable probability at least 1− with respect to μ and τ, Σ there exists a subset S′ ⊆ S such that , where δ = O( (r(Σ) log r(Σ))/n+
## Conclusion:

The authors showed that a standard stability condition from the recent high-dimensional robust statistics literature suffices to obtain near-subgaussian rates for robust mean estimation in the strong contamination model.- With a simple pre-processing, this leads to efficient outlier-robust estimators with subgaussian rates under only a bounded covariance assumption.
- (The authors' results imply that it is not needed when ǫ = Ω(1).) If not, this would imply that stability-based algorithms achieve subgaussian rates without the pre-processing

Related work

- Since the initials works [DKK+16, LRV16], there has been an explosion of research activity on algorithmic aspects of outlier-robust high dimensional estimation by several communities. See, e.g., [DK19] for a recent survey on the topic. In the context of outlier-robust mean estimation, a number of works [DKK+17, SCV18, CDG18, DHL19] have obtained efficient algorithms under various assumptions on the distribution of the inliers. Notably, efficient high-dimensional outlierrobust mean estimators have been used as primitives for robustly solving machine learning tasks that can be expressed as stochastic optimization problems [PSBR18, DKK+18]. The above works typically focus on the constant probability error regime and do not establish subgaussian rates for their estimators.

Two recent works [DL19, LLVZ19] studied the problem of outlier-robust mean estimation in the additive contamination model (when the adversary is only allowed to add outliers) and gave computationally efficient algorithms with subgaussian rates. Specifically, [DL19] gave an SDP-based algorithm, which is very similar to the algorithm of [CDG18]. The algorithm of [LLVZ19] is a fairly sophisticated iterative spectral algorithm, building on [CFB19]. In the strong contamination model, non-constructive outlier-robust estimators with subgaussian rates were established very recently. Specifically, [LM19b] gave a an exponential time estimator achieving the optimal rate. Our Proposition 1.6 implies that a very simple and practical algorithm – pre-processing followed by iterative filtering [DKK+17, DK19] – achieves this guarantee.

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