Agnostic Estimation of Mean and Covariance (1604.06968v2)

Abstract: We consider the problem of estimating the mean and covariance of a distribution from iid samples in $\mathbb{R}^n$, in the presence of an $\eta$ fraction of malicious noise; this is in contrast to much recent work where the noise itself is assumed to be from a distribution of known type. The agnostic problem includes many interesting special cases, e.g., learning the parameters of a single Gaussian (or finding the best-fit Gaussian) when $\eta$ fraction of data is adversarially corrupted, agnostically learning a mixture of Gaussians, agnostic ICA, etc. We present polynomial-time algorithms to estimate the mean and covariance with error guarantees in terms of information-theoretic lower bounds. As a corollary, we also obtain an agnostic algorithm for Singular Value Decomposition.

• The paper introduces polynomial-time algorithms that robustly estimate distribution parameters despite an η fraction of adversarial noise.
• It provides rigorous error bounds and information-theoretic limits, demonstrating near-optimal performance in one-dimensional Gaussian settings.
• It extends current robust methods by integrating iterative outlier removal and SVD-based techniques, enhancing practical estimation in high-dimensional data.

Overview of "Agnostic Estimation of Mean and Covariance"

This paper addresses the challenge of estimating the mean and covariance of a distribution from independent and identically distributed (iid) samples in the presence of an $\eta$ fraction of malicious noise. Unlike some recent approaches, this work does not assume the noise follows any predictable distribution type. Instead, it considers agnostic settings where noise can be arbitrarily distributed.

Key Contributions

1. Agnostic Estimation Problem: The authors frame their paper within the problem of estimating distribution parameters such as mean and covariance when a fraction of the data can be adversarially corrupted. This general problem also encapsulates tasks like learning Gaussian mixture models and independent component analysis (ICA) under agnostic conditions.
2. Algorithms: The paper introduces polynomial-time algorithms that provide error guarantees grounded in information-theoretic limits. The proposed methods can estimate the mean and covariance even when adversarial noise is present in the data.
3. Theoretical Results: The authors derive bounds on the error of their estimators, supported by information-theoretic lower bounds. Specifically, in one-dimensional Gaussian settings, they establish that any estimation algorithm can only achieve errors constrained by noise-based limits, suggesting their results are near-optimal.
4. Algorithmic Innovation: The work extends current robust estimation techniques by incorporating iterative outlier removal and SVD-based strategies to tolerate a higher fraction of adversarial noise. Their algorithms handle data that is not exactly from the model, addressing a critical limitation of prior methods.
5. Practical Implications: The authors demonstrate that their algorithm is practical, providing implementation details that confirm its efficiency on standard computing hardware. Importantly, they provide an agnostic version of Singular Value Decomposition (SVD) offering applications in data fitting and PCA.
6. Future Directions: The authors note ongoing efforts to explore further optimizations and applications of their algorithms. They also suggest that extending these ideas could improve robustness in various settings beyond those directly addressed.

Implications and Speculative Discussion

This paper opens up new avenues for practical and theoretical advances in robust statistics and machine learning. By eliminating the reliance on specific noise distributions or exact model fits, their approach enhances the applicability of statistical estimation in real-world scenarios where data can be noisy and partially corrupted.

The implications of their work are particularly significant for tasks involving high-dimensional data, where traditional assumptions (like Gaussian noise) may not hold. The algorithms developed could be integral in more resilient machine learning models, potentially impacting areas such as finance, cybersecurity, and healthcare where data integrity is often compromised.

In conclusion, the paper makes a significant step forward in agnostic learning strategies, offering robust statistical tools that operate effectively even under adversarial conditions. Further exploration could yield even more efficient algorithms, leading to broader acceptance and implementation in various AI fields.