SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications (2410.21194v1)

Abstract: We prove that there is a universal constant $C>0$ so that for every $d \in \mathbb N$, every centered subgaussian distribution $\mathcal D$ on $\mathbb R^d$, and every even $p \in \mathbb N$, the $d$-variate polynomial $(Cp)^{p/2} \cdot |v|{2}^p - \mathbb E{X \sim \mathcal D} \langle v,X\rangle^p$ is a sum of square polynomials. This establishes that every subgaussian distribution is \emph{SoS-certifiably subgaussian} -- a condition that yields efficient learning algorithms for a wide variety of high-dimensional statistical tasks. As a direct corollary, we obtain computationally efficient algorithms with near-optimal guarantees for the following tasks, when given samples from an arbitrary subgaussian distribution: robust mean estimation, list-decodable mean estimation, clustering mean-separated mixture models, robust covariance-aware mean estimation, robust covariance estimation, and robust linear regression. Our proof makes essential use of Talagrand's generic chaining/majorizing measures theorem.

• The paper proves that all subgaussian distributions are certifiably subgaussian within the SoS framework, bridging theoretical bounds and computational efficiency.
• It introduces efficient algorithms with optimal error guarantees for high-dimensional robust estimation tasks such as mean and covariance estimation.
• The work leverages novel reductions and generic chaining techniques to adapt Gaussian process methods for subgaussian settings, enhancing both theory and practical applications.

Overview of "SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications"

The paper under discussion addresses a central question in the field of algorithmic statistics, focusing on the relationship between subgaussian distributions and sum of squares (SoS) proofs. It tackles a longstanding open problem by proving that all subgaussian distributions are certifiably subgaussian within the SoS framework. This means that the moment bounds characteristic of subgaussian distributions can indeed be certified using low-degree SoS proofs, a result that has significant implications for computational learning algorithms in high-dimensional statistical tasks.

Main Contributions

1. Certifiability of Subgaussian Distributions: The paper establishes that there exists a universal constant $C > 0$ such that for any $d$-dimensional subgaussian distribution $D$, the $d$-variate polynomial is a sum of squares. This mathematical proof highlights the possibility of certifying moment bounds of subgaussian distributions using SoS certificates, effectively bridging the gap between theoretical probabilistic properties and computational feasibility.
2. Algorithmic Implications: The researchers leverage this certifiability result to propose efficient algorithms for various high-dimensional estimation problems. These include robust mean estimation, list-decodable mean estimation, clustering of mean-separated mixture models, robust covariance estimation, and robust linear regression. The algorithms offer improved error guarantees, and the results are shown to be theoretical optimal under the constraints of common computational models like SQ and low-degree polynomial tests.
3. Technical Innovations: A key technical achievement is the use of a novel reduction that allows for the application of generic chaining results, traditionally associated with Gaussian processes, to subgaussian settings. This is accomplished through a sophisticated analysis relating nonlinear empirical processes to linear ones, enabling the adaptation of classic results from probability theory to prove the certifiability in the SoS context.

Implications of the Research

The implications of certifying subgaussian distributions are multifaceted:

• Practical Implications: Practically, the result implies that data with subgaussian properties can be efficiently handled using existing SoS-based algorithms, which can be made more precise and reliable. This paves the way for new developments in machine learning applications where data often have subgaussian qualities.
• Theoretical Implications: Theoretically, the findings contribute to a better understanding of the nature of subgaussian distributions within the landscape of robust statistical algorithms. It confirms that subgaussian distributions are as amenable to computationally efficient learning as their Gaussian counterparts under analogous conditions.
• Future Directions: The research highlights several avenues for future exploration, including potential extensions to broader classes of distributions like subexponential ones, and the development of faster and more sample-efficient learning algorithms that could further exploit the certifiable properties identified in this work.

Speculation on AI Developments

As machine learning continues to require robust handling of high-dimensional data, the certification of subgaussian properties allows for more sophisticated and reliable algorithms. Future AI systems could leverage these enhanced statistical tools to model uncertainty and noise in data more effectively, leading to advancements in domains like natural language processing, computer vision, and other fields relying heavily on statistical learning from imperfect data.

In conclusion, this paper significantly advances the understanding of subgaussian distributions within the computational framework of sum of squares, providing theoretical robustness and practical tools essential for high-dimensional statistical learning. This foundational work will likely inform both the development of future statistical algorithms and the broader application of AI technologies where robust data handling and inference are critical.