Replicability in High Dimensional Statistics (2406.02628v1)

Abstract: The replicability crisis is a major issue across nearly all areas of empirical science, calling for the formal study of replicability in statistics. Motivated in this context, [Impagliazzo, Lei, Pitassi, and Sorrell STOC 2022] introduced the notion of replicable learning algorithms, and gave basic procedures for $1$-dimensional tasks including statistical queries. In this work, we study the computational and statistical cost of replicability for several fundamental high dimensional statistical tasks, including multi-hypothesis testing and mean estimation. Our main contribution establishes a computational and statistical equivalence between optimal replicable algorithms and high dimensional isoperimetric tilings. As a consequence, we obtain matching sample complexity upper and lower bounds for replicable mean estimation of distributions with bounded covariance, resolving an open problem of [Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sivakumar, and Sorrell, STOC2023] and for the $N$-Coin Problem, resolving a problem of [Karbasi, Velegkas, Yang, and Zhou, NeurIPS2023] up to log factors. While our equivalence is computational, allowing us to shave log factors in sample complexity from the best known efficient algorithms, efficient isoperimetric tilings are not known. To circumvent this, we introduce several relaxed paradigms that do allow for sample and computationally efficient algorithms, including allowing pre-processing, adaptivity, and approximate replicability. In these cases we give efficient algorithms matching or beating the best known sample complexity for mean estimation and the coin problem, including a generic procedure that reduces the standard quadratic overhead of replicability to linear in expectation.

• The paper establishes a novel equivalence between replicability and isoperimetric tilings, providing new insights into sample complexity in high-dimensional settings.
• The research derives matching upper and lower bounds for replicable mean estimation in distributions with bounded covariance, addressing key open problems.
• The work introduces relaxed algorithmic paradigms that enable adaptivity and efficiency, reducing sample complexity for large-scale, high-dimensional analyses.

Replicability in High Dimensional Statistics: An Overview

The paper "Replicability in High Dimensional Statistics" by Max Hopkins et al. addresses the issue of replicability in statistical analyses, which is of prime importance across empirical sciences. The concept of replicability is taken further by embedding it into the field of high-dimensional statistics, specifically tackling problems such as multi-hypothesis testing and high-dimensional mean estimation. This work expands upon the notion of "replicable learning algorithms" first introduced by Impagliazzo et al. and explores its implications and applications in higher dimensions.

Key Contributions and Findings

1. Equivalence of Replicability and Isoperimetric Tilings: The authors establish a novel connection between replicable algorithms and the geometric concept of isoperimetric tilings in high-dimensional spaces. They demonstrate a computational and statistical equivalence between optimal replicable algorithms and high-dimensional isoperimetric tilings, leading to significant insights into sample complexity and algorithm design.
2. Sample Complexity Bounds: By leveraging the above equivalence, the paper derives matching upper and lower bounds on sample complexity for replicable mean estimation in distributions with bounded covariance, such as Gaussian distributions. These results resolve previously open questions about the $N$-Coin problem and multi-hypothesis testing.
3. Relaxed Paradigms for Efficient Algorithms: The authors introduce several relaxed paradigms that allow for efficient algorithms without fully solving the tiling problem. These include allowing pre-processing, adaptivity, and approximate replicability, which considerably reduces the sample complexity in scenarios where perfectly efficient tilings are not available.
4. Algorithm Design: The paper provides explicit constructions for replicable learning algorithms based on these relaxed paradigms, showing improved sample complexities over existing methods. It also outlines strategies for adaptively composing multiple replicable processes to handle large-scale statistical tasks efficiently.

Theoretical and Practical Implications

The work opens new directions in both the theoretical understanding and practical implementation of replicable statistical methods. The equivalence with isoperimetric tilings suggests that replicability isn't just a statistical characteristic but has deep geometric connections. This insight could influence how future algorithms are developed, particularly those requiring rigorous statistical guarantees.

From a practical standpoint, the ability to achieve replicability with reduced sample and computational complexity has immediate applications in fields such as epidemiology and genomics, where multi-dimensional data is ubiquitous. By enabling more efficient and reliable data analyses, this work paves the way for more robust scientific findings.

Future Directions in AI and Beyond

Looking forward, this research could catalyze advancements in artificial intelligence, particularly in areas requiring model reproducibility and robustness, such as clinical trials or climate modeling. Additionally, exploring these geometric underpinnings further may lead to better understanding of the underlying structures governing high-dimensional data analysis. Future work might explore optimizing tile configurations related to specific family distributions or exploring entirely new domains where these principles could apply.

In conclusion, this paper not only provides substantial advancements in understanding replicability through a geometric lens but also equips researchers with practical methodologies to enhance the reliability of high-dimensional statistical analyses.