Efficient Algorithms and Lower Bounds for Robust Linear Regression (1806.00040v1)

Abstract: We study the problem of high-dimensional linear regression in a robust model where an $\epsilon$-fraction of the samples can be adversarially corrupted. We focus on the fundamental setting where the covariates of the uncorrupted samples are drawn from a Gaussian distribution $\mathcal{N}(0, \Sigma)$ on $\mathbb{R}^d$. We give nearly tight upper bounds and computational lower bounds for this problem. Specifically, our main contributions are as follows: For the case that the covariance matrix is known to be the identity, we give a sample near-optimal and computationally efficient algorithm that outputs a candidate hypothesis vector $\widehat{\beta}$ which approximates the unknown regression vector $\beta$ within $\ell_2$-norm $O(\epsilon \log(1/\epsilon) \sigma)$, where $\sigma$ is the standard deviation of the random observation noise. An error of $\Omega (\epsilon \sigma)$ is information-theoretically necessary, even with infinite sample size. Prior work gave an algorithm for this problem with sample complexity $\tilde{\Omega}(d^2/\epsilon^2)$ whose error guarantee scales with the $\ell_2$-norm of $\beta$. For the case of unknown covariance, we show that we can efficiently achieve the same error guarantee as in the known covariance case using an additional $\tilde{O}(d^2/\epsilon^2)$ unlabeled examples. On the other hand, an error of $O(\epsilon \sigma)$ can be information-theoretically attained with $O(d/\epsilon^2)$ samples. We prove a Statistical Query (SQ) lower bound providing evidence that this quadratic tradeoff in the sample size is inherent. More specifically, we show that any polynomial time SQ learning algorithm for robust linear regression (in Huber's contamination model) with estimation complexity $O(d^{2-c})$, where $c>0$ is an arbitrarily small constant, must incur an error of $\Omega(\sqrt{\epsilon} \sigma)$.

• The paper introduces efficient algorithms for robust linear regression, achieving near-optimal sample complexity under adversarial noise conditions.
• It designs tailored methods for both known and unknown covariance settings, with the known case nearly matching information-theoretic error bounds.
• The study establishes SQ lower bounds that reveal critical computational trade-offs and guide future research in robust machine learning.

Essay on "Efficient Algorithms and Lower Bounds for Robust Linear Regression"

The paper "Efficient Algorithms and Lower Bounds for Robust Linear Regression" introduces significant advancements in the field of robust statistics, particularly in managing adversarial noise within high-dimensional linear regression problems. The research addresses a fundamental challenge within robust machine learning: estimating models in data environments where a fraction of the inputs can be compromised, either through corruption or due to inherent uncertainty in measurements.

Key Contributions

The primary contributions of the paper are twofold: the development of efficient algorithms that achieve near-optimal sample complexities for robust linear regression, and the establishment of computational lower bounds that underscore inherent trade-offs between sample complexity and computational feasibility in robust estimation.

1. Algorithmic Advances:
   • The authors design robust algorithms for linear regression under the condition that the covariates, belonging to the Gaussian space, can be adversarially corrupted. They provide algorithms tailored for two cases:
     • When the covariance matrix ($\Sigma$) is known to be the identity matrix.
     • When $\Sigma$ is unknown.
   • For known covariance, the proposed algorithm is both sample-optimal and computationally efficient, requiring approximately $\tilde{O}(d/^2)$ samples and achieving an $\ell_2$-norm error of $O(\log(1/) \sigma)$. This aligns closely with the information-theoretic minimum error of $\Omega(\sigma)$ achievable with infinite sample size.
   • In the unknown covariance setting, the algorithm remains efficient but requires an additional $\tilde{O}(d^2/^2)$ unlabeled samples.
2. Lower Bounds through Statistical Queries (SQ):
   • The paper rigorously explores the complexity landscape using the Statistical Query (SQ) framework, demonstrating that polynomial time algorithms for this regression task that scale better than $O(d^2/^2)$ are infeasible without incurring a larger error. This is cemented through lower bound proofs that establish boundaries for polynomial time SQ learning algorithms.

Implications and Future Directions

The implications of these findings are profound, as they address a crucial need in industries and disciplines where data corruption is a possibility, such as finance, cybersecurity, and bioinformatics. The algorithms can potentially redefine approaches in preprocessing and data cleaning by offering guaranteed performance bounds even with corrupted data sets.

From a theoretical perspective, the explicit computational and sample size trade-offs highlight critical reflections on the balance between computation and accuracy. It suggests paths forward in algorithm design that not only consider resource constraints but also embrace robustness against anomalies in data.

Future research might explore extending these methods to nonlinear models or optimizing algorithms for distributed or federated learning environments. Understanding robustness in more structured data paradigms, such as those including categorical variables or networks, could also resonate with the ongoing evolution in data types and applications.

In conclusion, this work pioneers the frontier of robust regression modeling amid adversarial conditions, providing a pragmatic yet theoretically backed framework for effective estimation in compromised datasets. This foundational work paves the way for further advancements in robust statistical learning and practical algorithmic applications, fostering greater reliability in machine learning systems deployed in diverse real-world settings.