Mixture Models, Robustness, and Sum of Squares Proofs (1711.07454v1)

Abstract: We use the Sum of Squares method to develop new efficient algorithms for learning well-separated mixtures of Gaussians and robust mean estimation, both in high dimensions, that substantially improve upon the statistical guarantees achieved by previous efficient algorithms. Firstly, we study mixtures of $k$ distributions in $d$ dimensions, where the means of every pair of distributions are separated by at least $k^{\varepsilon}$. In the special case of spherical Gaussian mixtures, we give a $(dk)^{O(1/\varepsilon^2)}$-time algorithm that learns the means assuming separation at least $k^{\varepsilon}$, for any $\varepsilon > 0$. This is the first algorithm to improve on greedy ("single-linkage") and spectral clustering, breaking a long-standing barrier for efficient algorithms at separation $k^{1/4}$. We also study robust estimation. When an unknown $(1-\varepsilon)$-fraction of $X_1,\ldots,X_n$ are chosen from a sub-Gaussian distribution with mean $\mu$ but the remaining points are chosen adversarially, we give an algorithm recovering $\mu$ to error $\varepsilon^{1-1/t}$ in time $d^{O(t^2)}$, so long as sub-Gaussian-ness up to $O(t)$ moments can be certified by a Sum of Squares proof. This is the first polynomial-time algorithm with guarantees approaching the information-theoretic limit for non-Gaussian distributions. Previous algorithms could not achieve error better than $\varepsilon^{1/2}$. Both of these results are based on a unified technique. Inspired by recent algorithms of Diakonikolas et al. in robust statistics, we devise an SDP based on the Sum of Squares method for the following setting: given $X_1,\ldots,X_n \in \mathbb{R}^d$ for large $d$ and $n = poly(d)$ with the promise that a subset of $X_1,\ldots,X_n$ were sampled from a probability distribution with bounded moments, recover some information about that distribution.

• The paper presents novel algorithms that break the k^(1/4) separation barrier in Gaussian mixtures using the Sum-of-Squares method.
• The paper introduces an efficient robust mean estimation approach that achieves near info-theoretic error rates in adversarial settings.
• The paper unifies theoretical advancements with practical implementations in high-dimensional learning, enhancing both robustness and efficiency.

Overview of Mixture Models, Robustness, and Sum of Squares Proofs

This paper presents the development of new algorithms leveraging the Sum of Squares (SoS) method, targeting high-dimensional learning tasks such as well-separated mixtures of Gaussians and robust mean estimation. The authors Samuel Hopkins and Jerry Li articulate significant improvements in statistical guarantees over prior efficient algorithms and detail the implications for computational learning theory.

Contributions and Key Results

The authors introduce two primary advancements in algorithmic methods for unsupervised learning problems:

1. Learning Mixtures of Separated Gaussians: The paper addresses the challenge of estimating the means of $k$ distributions in $d$ dimensions, specifically in Gaussian mixtures, where pairwise mean separation exceeds a specified threshold. Notably, for spherical Gaussian mixtures, the authors present an algorithm with complexity $(dk)^{O(1/\epsilon^2)}$ to learn means under separation constraints of $k^\epsilon$, surpassing the $k^{1/4}$ separation barrier that has limited prior methods relying on greedy clustering and spectral techniques.
2. Robust Mean Estimation: An algorithm is established for deriving robust mean estimates even when an adversarial fraction of samples does not come from a sub-Gaussian distribution with mean $\mu$. The proposed algorithm achieves error rates approaching the info-theoretic limit for such distributions, characterized by a polynomial runtime $d^{O(t^2)}$, as long as higher moments are ascertainable via a SoS proof.

These advancements are unified by what the authors term novel techniques for understanding and identifying structured subsets within large data sets based on recent approaches in robust statistics and utilizing semidefinite programming (SDP) alongside the SoS method.

Theoretical and Practical Implications

Theoretical Implications: The paper offers a substantive leap in the understanding of structure identification within complex high-dimensional datasets. It confirms that higher moments of distributions can significantly increase the efficacy of polynomial-time algorithms, providing an essential bridge between theoretical statistics and computational efficiency.

Practical Implications: For practitioners in machine learning and data science, particularly those handling high-dimensional data and adversarial environments, these improved algorithms could vastly enhance model robustness and accuracy. The techniques described could lead to better-performing algorithms in fields such as finance, genomics, or image processing, where such data characteristics are prevalent.

Future Directions

The paper lays a foundation for further exploration into the mutual leveraging of higher moments and SoS proofs for diverse statistical settings, including but not limited to non-Gaussian and non-linear distributions. The authors' methodologies could inspire new paradigms for tackling supervised learning tasks and exploring scalability in practical applications across varied domains.

Continued research could focus on refining these techniques to lower computational overhead and enhance applicability, effectively broadening the scope of problems solvable through efficient unsupervised learning algorithms.

Summary

In conclusion, this paper delineates a sophisticated extension of efficient learning algorithms through the innovative use of SoS proofs. It breaks new ground in addressing stubborn statistical separation barriers and optimizing robustness under adversarial conditions. In doing so, it opens up exciting possibilities for both theoretical and real-world advancements in machine learning.