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Acknowledgements. The authors thank Sergey Markelov, Arseny Akopyan, Roman Fedorov, Rupert Hölzl and their colleagues at Moscow State University and LIRMM (Montpellier) for interesting discussions, and the referees for their comments and suggestions. The work was supported by ANR-15-CE40-0016-01 RaCAF and RFBR 16-01-00... |
# Source: arxiv 1806.00040 # Title: Efficient Algorithms and Lower Bounds for Robust Linear Regression # Sections: all # Downloaded: 2026-03-03T01:44:45.347906+00:00 |
Efficient Algorithms and Lower Bounds for Robust Linear Regression Abstract We study the prototypical problem of high-dimensional linear regression in a robust model where an [MATH] -fraction of the samples can be adversarially corrupted. We focus on the fundamental setting where the covariates of the uncorrupted sampl... |
For the case that the covariance matrix is known to be the identity, we give a sample near-optimal and computationally efficient algorithm that draws [MATH] |
labeled examples and outputs a candidate hypothesis vector [MATH] that approximates the unknown regression vector [MATH] within [MATH] -norm [MATH] , where [MATH] is the standard deviation of the random observation noise. An error of [MATH] is information-theoretically necessary, even with infinite sample size. Hence, ... |
whose error guarantee scales with the [MATH] -norm of [MATH] For the case of unknown covariance [MATH] we show that we can efficiently achieve the same error guarantee of [MATH] as in the known covariance case, using an additional [MATH] unlabeled examples. On the other hand, an error of [MATH] can be information-theor... |
Introduction 1.1 Background and Problem Definition Linear regression is a prototypical problem in statistics with a range of applications in signal processing (e.g., face recognition, time series analysis) and various other data analysis tasks. The reader is referred to RL87 BJK15 and references therein. In the realiza... |
Estimation in the presence of outliers is an important goal in statistics and has been systematically studied within the robust statistics community since Hub64 . Nevertheless, until recently, all known efficient estimators could only tolerate a negligible fraction of outliers in high-dimensional settings, even for the... |
In the remaining of this section, we describe our formal setup. In the realizable setting, the problem of linear regression is defined as follows: We observe a multiset of labeled samples [MATH] , where [MATH] and [MATH] It is assumed that there exists an unknown distribution [MATH] , where [MATH] is a known family of ... |
such that [EQUATION] where [MATH] is some kind of random observation noise. The goal is to compute a hypothesis vector [MATH] such that [MATH] is small. In this work, we study the fundamental setting that [MATH] is [MATH] where the covariance matrix [MATH] is either a priori known or unknown to the algorithm. For simpl... |
We consider the following model of robust estimation that generalizes other existing models, including Huber’s contamination model: |
Definition 1.1 Given [MATH] and a family of probabilistic models [MATH] the adversary operates as follows: The algorithm specifies some number of samples [MATH] The adversary generates [MATH] samples [MATH] from some (unknown) [MATH] The adversary is allowed to inspect the samples, removes [MATH] of them, and replaces ... |
In summary, the adversary is allowed to inspect the samples before corrupting them, both by adding corrupted points and deleting uncorrupted points. In contrast, in Huber’s model the adversary is oblivious to the samples and is only allowed to add corrupted points. |
In the context of robust linear regression studied in this paper, the adversary can change an arbitrary [MATH] -fraction of the labeled samples [MATH] that satisfy the aforementioned definition of linear regression. The goal is to output a hypothesis vector [MATH] such that |
[MATH] is as small as possible. 1.2 Our Results and Techniques 1.2.1 Robust Learning Algorithms We state our positive results for the case of known and identity covariance matrix, [MATH] Our first positive result is a robust learning algorithm for linear regression that has near-optimal sample complexity, runs in polyn... |
Theorem 1.2 (Basic Algorithm for Robust Linear Regression) Let [MATH] be an [MATH] -corrupted set of labeled samples of size [MATH] There exists an efficient algorithm that on input [MATH] and [MATH] , returns a candidate vector [MATH] |
such that with probability at least [MATH] it holds [MATH] where [MATH] ). Roughly speaking, the algorithm establishing Theorem 1.2 relies on the observation that robust linear regression can be reduced to robust mean estimation. The main drawback of this approach is that the error guarantee depends on [MATH] and in pa... |
when [MATH] goes to [MATH] . To eliminate [MATH] from the RHS while retaining a near-optimal sample complexity, we require a more sophisticated approach. Specifically, our main algorithmic contribution is as follows: |
Theorem 1.3 (Main Algorithm for Robust Linear Regression) Let [MATH] be an [MATH] -corrupted set of labeled samples of size [MATH] There exists an efficient algorithm that on input [MATH] and [MATH] , returns a candidate vector [MATH] |
such that with probability at least [MATH] it holds [MATH] We note that an error of [MATH] is information-theoretically necessary for this problem, even when the sample size is unbounded (see, e.g., Gao17 ). Hence, our second algorithm achieves the minimax optimal error, up to a logarithmic factor in [MATH] |
We note that the unknown covariance case can be easily reduced to the known covariance case as follows: First, we robustly learn the covariance matrix in the appropriate metric using [MATH] samples. Then, we observe that our above algorithms also work when the covariance matrix [MATH] is only approximately known (see A... |
Intuition behind the Algorithm. We now provide some intuition of our algorithm. We start with the first algorithm which works in the simplest setting where [MATH] [MATH] , and [MATH] has mean [MATH] variance [MATH] and achieves estimation accuracy [MATH] The basic algorithm relies on the fact that [MATH] is an unbiased... |
[MATH] -error. However, in the presence of adversarial corruptions, the question is now how to robustly estimate the mean of the distribution of [MATH] . To do this, we apply the filtering technique of DKK 16 The key observation that enables the filtering framework to apply is that an [MATH] -fraction of corruptions ca... |
With [MATH] samples, the empirical covariance matrix of [MATH] also concentrates around the true covariance matrix. Hence, the above key observation provides an indicator to determine whether the mean may be corrupted. However, even if we manage to detect the abnormality by looking at the empirical covariance, it’s sti... |
This approach gives an [MATH] error guarantee due to the fact that the operator norm of the covariance of [MATH] is [MATH] However, this error bound is far from the information-theoretic optimal error of [MATH] |
when [MATH] is large. A natural idea to circumvent this issue is to boost the accuracy by setting [MATH] using the output of the first iteration and again run the basic algorithm. Indeed, this approach yields an estimate with near-optimal error [MATH] |
by running the algorithm [MATH] times. However, this scheme seems to require [MATH] samples, i.e., depending on [MATH] To remove the [MATH] dependence in the sample complexity, one may consider subtracting [MATH] from [MATH] using the same batch of samples repeatedly. The main problem with doing this naively is that, f... |
samples to achieve concentration. Somewhat surprisingly, the only problem that prevents the empirical covariance from achieving the desired concentration is the samples with large [MATH] , as illustrated in the previous example. The concentration property holds if we temporally ignore samples with large [MATH] , thus w... |
Our final modification of the algorithm focuses on removing the [MATH] dependence in the running time. Instead of starting from [MATH] and running the subroutine algorithm repeatedly, we start from the ordinary least squares estimator. Let [MATH] be the set of samples that has large [MATH] We pre-process by running fil... |
does not change the empirical mean of [MATH] by much. Notice that in the case where [MATH] is the ordinary least square estimator, the empirical mean of [MATH] is [MATH] Hence once the the samples passed the pre-process and the filtering step we described in the last paragraph, the algorithm can terminate and output th... |
1.2.2 Statistical Query Lower Bounds In this section, we describe our Statistical Query (SQ) lower bounds establishing a tradeoff between sample complexity and computation complexity for robust linear regression with unknown (bounded) covariance. |
We start with some basic background. A Statistical Query (SQ) algorithm relies on an oracle that given any bounded function on a single domain element provides an estimate of the expectation of the function on a random sample from the input distribution. This computational model was introduced by Kearns Kea98 in the co... |
A recent line of work FGR 13 FGV15 FPV15 Fel16a developed a framework of SQ algorithms for search problems over distributions, which encompasses the linear regression problem studied here. It turns out that one can prove unconditional lower bounds on the computational complexity of SQ algorithms via the notion of Stati... |
As our main negative result in this paper, we prove a Statistical Query (SQ) lower bound giving evidence that if [MATH] has an unknown (bounded) covariance, it is computationally hard to approximate [MATH] well given significantly fewer than [MATH] samples. The reason that this result is interesting is because this lea... |
for a more detailed formal statement): Theorem 1.4 (SQ lower bound, informal statement) No SQ algorithm for robust linear regression for Gaussian covariates with unknown bounded covariance and random noise with [MATH] |
can output a candidate [MATH] with [MATH] on all instances unless it uses [MATH] statistical queries or each query requires [MATH] samples to be simulated. |
We note that [MATH] samples information-theoretically suffice to achieve error [MATH] (see, e.g., Gao17 ) even in the unknown covariance setting. Moreover, as explained in Appendix A.1 with [MATH] samples we can efficiently achieve error [MATH] for unknown covariance as well. Hence, Theorem 1.4 establishes an inherent ... |
To prove this result, we require a generalization of the technique in DKS17c , which was designed for unsupervised learning problems. That work established SQ lower bounds for unsupervised learning problems using a construction consisting of distributions which are standard Gaussians in all except one direction, by sho... |
As already mentioned, DKS17c considers unsupervised learning problems and the distribution in the construction is a suitable model for [MATH] but not for the joint distribution [MATH] , where the direction of the [MATH] coordinate is very different. We instead try to make [MATH] |
conditioned on [MATH] match moments with a Gaussian. When the momets match for all [MATH] , it is hard to learn the direction [MATH] |
where these conditional distributions are different. We show that when [MATH] , we can match three moments and it is hard to find the direction of [MATH] . Thus, we obtain a lower bound that says that we cannot approximate [MATH] to within [MATH] |
with fewer than exponential in [MATH] statistical queries, unless we use queries of precision greater than we could simulate with [MATH] samples. |
1.3 Comparison to Prior Work Since the initial works DKK 16 LRV16 there have been a considerable number of papers on a wide range of topics related to robust high-dimensional estimation, including: learning graphical models DKS16 understanding computation-robustness tradeoffs DKS17c DKK 18a giving applications to explo... |
In the context of the estimation task studied in this paper, BJK15 BJKK17 have proposed efficient algorithms for “robust” linear regression. However, these works consider a restrictive corruption model that only allows adversarial corruptions to the responses (but not the covariates). |
BDLS17 studies (sparse) linear regression under Huber’s contamination model. The three main differences between the results of BDLS17 and our work are as follows: (1) The error guarantee provided in BDLS17 scales with [MATH] the [MATH] -norm of the regression vector. This multiplicative dependence on [MATH] |
is not information-theoretically necessary (see, e.g., Gao17 ). In contrast, the error guarantee of our algorithm has no dependence on [MATH] , matching the information-theoretic bound |
[MATH] up to a [MATH] factor. (2) The algorithm employed in BDLS17 building on the convex programming method of DKK 16 makes essential use of the ellipsoid method (whose separation oracle is another convex program), hence is not scalable in high dimensions. (3) Moreover, the BDLS17 algorithm assumes a priori knowledge ... |
1.4 Concurrent and Independent Works Three recent works PSBR18 DKK 18b KKM18 provide robust efficient algorithms for linear regression in various settings. These works make weaker distributional assumptions on the uncorrupted data and as a result provide weaker error guarantees, in most cases scaling with [MATH] as opp... |
succeed in the oversampled regime in the sense that their sample complexities are at least quadratic in the dimension. The algorithm in KKM18 relies on the SOS convex programming hierarchy. |
1.5 Structure of this Paper In Section , we describe our robust algorithms and in Section we give our SQ lower bounds. For the clarity of the presentation, most proofs are deferred to the appendix. |
Robust Algorithm for Linear Regression 2.1 Notation Before introducing our algorithm, we define the necessary notations in this subsection. Let [MATH] be a random variable and [MATH] be a distribution. We use [MATH] to denote that [MATH] is drawn from distribution [MATH] For [MATH] being a multiset of examples, we writ... |
Under our corruption model where an [MATH] -fraction of the samples can be arbitrarily corrupted, we will typically use [MATH] to denote the set of samples before being corrupted by the adversary. Given a set of samples, [MATH] , we denote the set [MATH] to be [MATH] (which contains the samples added by the adversary),... |
2.2 Basic Robust Linear Regression Algorithm The algorithm that achieves the performance guarantee stated in Theorem 1.2 is an iterative algorithm that invokes the following algorithm, Algorithm , multiple times as a subroutine. Every time Algorithm gets called, it either returns an estimate of [MATH] or returns a set ... |
1: procedure Filter-LR-Identity covariance 2: input: A multiset [MATH] such that there exists an [MATH] -good [MATH] with [MATH] |
3: output: Multiset [MATH] or mean vector [MATH] satisfying Proposition 2.1 4: Robustly estimate [MATH] , the variance of [MATH] Denote the estimation as [MATH] |
5: Compute the sample mean [MATH] and the sample covariance matrix [MATH] 6: Compute approximations for the largest absolute eigenvalue of |
[MATH] [MATH] and the associated unit eigenvector [MATH] 7: if [MATH] 8: return [MATH] 9: end 10: Let [MATH] Find [MATH] such that |
[EQUATION] 11: return the multiset [MATH] Algorithm 1 Filter algorithm for LR with identity covariance The following proposition formalizes the guarantee of Algorithm |
that it either returns a cleaner dataset or an estimate of [MATH] with [MATH] -error at most [MATH] , where [MATH] Proposition 2.1 |
Let [MATH] and [MATH] Let [MATH] be an [MATH] -good set with respect to [MATH] Let [MATH] be any multiset with [MATH] . The algorithm Filter-LR-Identity covariance |
runs in polynomial time and, given [MATH] and [MATH] returns one of the following: (i) A vector [MATH] such that [MATH] (ii) A multiset [MATH] such that [MATH] |
where [MATH] is the size of the symmetric difference of multisets [MATH] and [MATH] divided by the cardinality of [MATH] The proof of Proposition 2.1 is deferred to Appendix . Assuming Proposition 2.1 holds, we are now ready to show Theorem 1.2 , which is restated below for convenience: |
Theorem 1.2 Let [MATH] be an [MATH] -corrupted set of labeled samples of size [MATH] There exists an efficient algorithm that on input [MATH] and [MATH] , returns a candidate vector [MATH] |
such that with probability at least [MATH] it holds [MATH] (recalling that [MATH] ). Proof of Theorem 1.2 By the definition of [MATH] since [MATH] has been obtained from [MATH] |
by corrupting an [MATH] -fraction of the points in [MATH] we have that [MATH] By Proposition 2.3 (see below), the set [MATH] of uncorrupted samples is [MATH] -good with respect to [MATH] with probability at least [MATH] |
We henceforth condition on this event. We iteratively apply the Filter-LR-Identity covariance procedure of Proposition 2.1 until it terminates returning a vector [MATH] with [MATH] |
We claim that we need at most [MATH] iterations for this to happen, simply because the sequence of iterations results in a sequence of sets [MATH] satisfy [MATH] |
To better illustrate how Algorithm works, we provide a proof sketch of Proposition 2.1 Our algorithm succeeds under a set of deterministic conditions that are satisfied by an uncorrupted set of samples with high probability. |
Definition 2.2 Let [MATH] [MATH] , and [MATH] We say that a multiset [MATH] of elements in [MATH] is [MATH] -good (with respect to [MATH] if the following conditions are satisfied: |
(i) For all [MATH] , we have [MATH] and [MATH] (ii) For every [MATH] with [MATH] we have that [EQUATION] (iii) We have that [MATH] |
(iv) We have that [MATH] Roughly speaking, condition (i) claims that none of the uncorrupted samples is too big in magnitude, condition (ii) establishes the empirical tail bound of the set of samples, condition (iii) and (iv) guarantee that the empirical mean and empirical covariance converge well to the true mean and ... |
Proposition 2.3 Let [MATH] and [MATH] If the multiset [MATH] consists of [MATH] labeled samples [MATH] , where [MATH] and [MATH] , where [MATH] then [MATH] is [MATH] -good with respect to [MATH] with probability at least [MATH] |
We note that the sample size in the above proposition is optimal, up to logarithmic factors, and is crucial in establishing the near-optimal sample complexity of our algorithm. The proof of Proposition 2.3 is deferred to Appendix |
Given that the deterministic conditions hold for the uncorrupted data, our algorithm simply computes the sample mean and covariance (Step 5). Notice that condition (iii) and (iv) also establishes a connection between the sample mean and covariance, in the sense that for uncorrupted data, sample covariance can be predic... |
2.3 Main Algorithm: Removing the Dependence on [MATH] in the Error In this section, we describe an algorithm establishing Theorem 1.3 which we restate below for completeness. |
Theorem 1.3 Let [MATH] be an [MATH] -corrupted set of labeled samples of size [MATH] There exists an efficient algorithm that on input [MATH] and [MATH] , returns a candidate vector [MATH] |
such that with probability at least [MATH] it holds [MATH] Similarly to the basic algorithm of the previous subsection, the algorithm that achieves the performance guarantee of Theorem 1.3 is iterative and invokes Algorithm multiple times as a subroutine. Every time Algorithm gets called, it either returns an estimate ... |
1: procedure Filter-LR-Identity covariance-2 2: input: A multiset [MATH] such that there exists an [MATH] -representative [MATH] with [MATH] |
3: output: Multiset [MATH] or mean vector [MATH] satisfying Proposition 2.4 4: Let [MATH] 5: Robustly estimate the standard deviation of [MATH] as [MATH] using its interquartile range. |
6: if [MATH] 7: Find [MATH] such that [EQUATION] 8: return [MATH] 9: end 10: Let [MATH] be the subset of [MATH] with [MATH] 11: Let [MATH] , where [MATH] |
12: if [MATH] 13: Find [MATH] such that [EQUATION] 14: return [MATH] 15: end 16: Find the top eigenvalue [MATH] , and corresponding eigenvector [MATH] , of [MATH] |
17: if [MATH] 18: [MATH] Find [MATH] such that [EQUATION] 19: return [MATH] 20: end 21: return [MATH] Algorithm 2 Filter algorithm for LR with identity covariance with no [MATH] dependence. |
More specifically, in order to estimate [MATH] within [MATH] -error [MATH] , as stated in Theorem 1.3 , we repeatedly execute Algorithm until it returns an estimate of [MATH] . Our Algorithm , which has no dependency on [MATH] , is a combination of three filter algorithms, which first computes the ordinary least square... |
Proposition 2.4 Let [MATH] and [MATH] Let [MATH] be [MATH] -representative with respect to [MATH] Let [MATH] be any multiset with [MATH] . There exists a polynomial time algorithm Filter-LR-Identity covariance-2 |
that, given [MATH] and [MATH] returns one of the following: (i) A vector [MATH] such that [MATH] (ii) A multiset [MATH] such that [MATH] |
where [MATH] is the size of the symmetric difference of multisets [MATH] and [MATH] divided by the cardinality of [MATH] Like the basic algorithm we discussed in the previous subsection (which has dependency on [MATH] ), the success of our new algorithm relies on the deterministic conditions which hold with high probab... |
in Appendix lists the conditions that need to hold for our algorithm to work, which are similar to Definition 2.2 for the basic algorithm. As usual, each condition consists of four sub-conditions, which guarantee that the set of uncorrupted samples are bounded, satisfy certain tail bounds and have mean and covariance c... |
Statistical Query Lower Bounds In this section, we formally describe our main lower bound result and provide a high-level proof sketch. Consider the joint distribution of [MATH] in a linear regression problem without corruptions when the covariance of [MATH] is unknown. Formally, let [MATH] be the distribution of [MATH... |
well without needing precision stronger than is possible with a strongly sub-quadratic number of samples: Theorem 3.1 No algorithm given statistical query access to [MATH] , defined as above with unknown noise and unknown variances [MATH] and [MATH] gives an output [MATH] with [MATH] on all instances unless it uses mor... |
[EQUATION] oracles for any [MATH] The detailed proof of Theorem 3.1 is given in Appendix Informally speaking, the theorem shows that no Statistical Query algorithm can approximate [MATH] to within [MATH] with fewer than exponential in [MATH] queries, unless using queries of precision greater than we could simulate with... |
In the proof of Theorem 3.1 , we use the construction in Proposition 3.3 of DKS17c which intuitively says that if we have a distribution which is standard Gaussian in all except one direction, then if the low-degree moments match the standard Gaussian, then that direction is hard to find with an SQ algorithm. The idea ... |
Lemma 3.2 Let [MATH] be the joint distribution of [MATH] with [MATH] and [MATH] , where [MATH] is unknown an [MATH] Then [MATH] , where [MATH] |
and [MATH] Proof. Notice that [MATH] is a [MATH] dimensional Gaussian distribution with covariance [EQUATION] By the mean and covariance formula of the conditional distribution of a Gaussian, we have that |
[MATH] Notice that [MATH] is indeed standard Gaussian in all except the [MATH] direction. By adding corruptions, we can make the distribution of [MATH] projected onto [MATH] |
agree with the first three moments of [MATH] and, like the construction of DKS17c still be a standard Gaussian in all the other orthogonal directions. Then we can show that we cannot find the direction of [MATH] with an SQ algorithm. Lemma E.4 establishes the upper bound of the statistical correlation between a pair of... |
The further the mean of [MATH] conditioned on [MATH] is from [MATH] , the more noise needs to be added to match the first three moments. Lemma E.2 , which is the main lemma of the lower bound proof, shows that we can match the first three moments by adding [MATH] fraction of noise when the [MATH] has mean [MATH] in the... |
Acknowledgements. We would like to thank Jason Lee for his contributions to the early stages of this work. I.D. and A.S. thank Daniel Kane for numerous discussions on robust high-dimensional estimation over the last five years. |
Appendix A Proof of Proposition 2.3 : Deterministic Regularity Conditions for Algorithm This section establishes Proposition 2.3 |
Technical Facts. We will require a couple of technical facts. We start with the following basic Gaussian concentration result: Fact A.1 |
Let [MATH] . Then for any unit vector [MATH] we have that [MATH] We will make essential use of the following concentration inequality for quadratic forms: |
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