Recent Advances in Algorithmic High-Dimensional Robust Statistics (1911.05911v1)

Abstract: Learning in the presence of outliers is a fundamental problem in statistics. Until recently, all known efficient unsupervised learning algorithms were very sensitive to outliers in high dimensions. In particular, even for the task of robust mean estimation under natural distributional assumptions, no efficient algorithm was known. Recent work in theoretical computer science gave the first efficient robust estimators for a number of fundamental statistical tasks, including mean and covariance estimation. Since then, there has been a flurry of research activity on algorithmic high-dimensional robust estimation in a range of settings. In this survey article, we introduce the core ideas and algorithmic techniques in the emerging area of algorithmic high-dimensional robust statistics with a focus on robust mean estimation. We also provide an overview of the approaches that have led to computationally efficient robust estimators for a range of broader statistical tasks and discuss new directions and opportunities for future work.

• The paper surveys efficient robust mean estimators offering dimension-independent error guarantees via convex programming and iterative filtering methods.
• The paper introduces key concepts of stability and good sets to ensure estimator accuracy despite adversarial corruption in high-dimensional data.
• The paper extends these robust estimation techniques to covariance estimation, regression, and density estimation, underscoring their broad applicability.

Overview of Recent Advances in Algorithmic High-Dimensional Robust Statistics

The paper under discussion explores the burgeoning field of algorithmic high-dimensional robust statistics, with a particular focus on robust mean estimation. This research area addresses the challenge of learning in the presence of outliers, which can significantly compromise the performance of classical statistical techniques. The paper highlights the progress made in designing efficient algorithms that robustly estimate fundamental statistical parameters, such as mean and covariance, in high-dimensional settings.

Key Contributions and Techniques

1. Robust Mean Estimation: The authors provide a comprehensive survey of algorithmic advances that have led to efficient robust mean estimators. These estimators offer dimension-independent error guarantees under mild assumptions about the distribution of the uncorrupted data. The focus is primarily on two algorithmic paradigms: convex programming and iterative filtering methods.
2. Stability and Good Sets: The concept of stability is central to the success of the algorithms discussed. For a set of samples to be stable, the sample mean and variance, when restricted to large subsets of the data, must exhibit certain regularity properties. This ensures that the robust estimators remain accurate even when a fraction of the data is adversarially corrupted.
3. Convex Programming Approach: The authors describe how robust mean estimation can be formulated as a convex optimization problem, leveraging the stability notion to guide the design of efficient algorithms. This approach is particularly appealing due to its generality and robustness against a wide range of distributional assumptions.
4. Iterative Filtering Approach: The filtering method iteratively removes (or reweights) samples identified as potential outliers by examining the principal components of the sample covariance. This technique can be faster in practice, as it relies primarily on spectral methods, but requires careful handling to ensure convergence to a robust estimate.
5. Extensions and Applications: Beyond robust mean estimation, the paper discusses how these techniques extend to more general estimation tasks, including robust covariance estimation, robust regression, and robust density estimation. The applicability of these methods extends to various domains, such as biology and machine learning, where data contamination is prevalent.

Implications and Future Directions

The implications of this research are both practical and theoretical. Practically, the development of robust statistical methods enhances the reliability of data analysis in fields prone to data contamination. Theoretically, it challenges the notion that robustness inherently leads to computational intractability. Indeed, for multiple fundamental statistical problems, robust solutions with efficient algorithms have been found, inspiring optimism about handling robustness in diverse applications.

However, the paper also acknowledges existing computational-statistical tradeoffs in robust estimation. For some tasks, achieving optimal error guarantees may require super-polynomial time algorithms under certain computational models, highlighting areas where further research is needed to bridge the gap between computational complexity and statistical efficiency.

Conclusion

The paper "Recent Advances in Algorithmic High-Dimensional Robust Statistics" serves as an important reference point for researchers interested in the intersection of statistics, computer science, and data science. It highlights significant progress in understanding and addressing the computational challenges posed by robust high-dimensional learning and provides a foundation for future exploration in this vibrant research area. As algorithmic and theoretical techniques continue to evolve, they offer promising avenues for developing even more effective robust statistical methodologies.