[Paper Review] Detecting Correlations with Little Memory and Communication

Yuval Dagan, Ohad Shamir|arXiv (Cornell University)|Mar 4, 2018

Machine Learning and Algorithms17 references3 citations

TL;DR

This paper establishes tight information-theoretic trade-offs between memory/communication complexity and sample complexity for detecting pairwise correlations in high-dimensional data. It proves that detecting correlations with optimal sample complexity requires at least Ω(d²) memory or communication bits, demonstrating that naive streaming or distributed approaches are information-theoretically optimal in this regime.

ABSTRACT

We study the problem of identifying correlations in multivariate data, under information constraints: Either on the amount of memory that can be used by the algorithm, or the amount of communication when the data is distributed across several machines. We prove a tight trade-off between the memory/communication complexity and the sample complexity, implying (for example) that to detect pairwise correlations with optimal sample complexity, the number of required memory/communication bits is at least quadratic in the dimension. Our results substantially improve those of Shamir [2014], which studied a similar question in a much more restricted setting. To the best of our knowledge, these are the first provable sample/memory/communication trade-offs for a practical estimation problem, using standard distributions, and in the natural regime where the memory/communication budget is larger than the size of a single data point. To derive our theorems, we prove a new information-theoretic result, which may be relevant for studying other information-constrained learning problems.

Motivation & Objective

• To understand the fundamental limits of detecting correlations in multivariate data under memory or communication constraints.
• To establish tight trade-offs between memory/communication complexity and sample complexity for a practical statistical estimation problem.
• To analyze the information-theoretic limits of correlation detection in the regime where memory/communication budgets exceed the size of a single data point.
• To develop a novel information-theoretic framework applicable to other learning problems under information constraints.

Proposed method

• Formalizes the problem of detecting pairwise correlations in high-dimensional zero-mean data under memory or communication constraints.
• Uses Hoeffding's inequality and union bounds to establish baseline sample complexity for unconstrained detection.
• Analyzes streaming and distributed algorithms that process data in batches or with limited communication, showing trade-offs between resource usage and sample size.
• Introduces a new information-theoretic result to prove lower bounds on memory and communication complexity.
• Employs concentration inequalities and combinatorial arguments to bound the sum of terms in moment-generating functions, proving that subquadratic memory/communication leads to suboptimal sample complexity.
• Applies the framework to both streaming and distributed settings, showing that the product of memory and total data processed must be Ω(d²/ρ²) for constant success probability.

Experimental results

Research questions

• RQ1 What is the minimum amount of memory required to detect pairwise correlations in high-dimensional data with optimal sample complexity?
• RQ2 How does communication complexity affect the sample complexity in distributed correlation detection?
• RQ3 Can the naive approach of processing one correlation pair at a time be improved upon in terms of the memory-sample complexity trade-off?
• RQ4 What is the fundamental information-theoretic limit on the product of memory and sample size for correlation detection?
• RQ5 How do the results compare to prior work, such as Raz (2016), in terms of model assumptions and regime of applicability?

Key findings

• A tight trade-off is proven between memory/communication complexity and sample complexity: to achieve optimal sample complexity, memory or communication must be Ω(d²).
• The product of memory size and total number of data points processed must be Ω(d²/ρ²) to detect correlations with constant probability, implying that subquadratic memory leads to suboptimal sample complexity.
• The lower bound holds even when the memory/communication budget is larger than the size of a single data point, which is a key distinction from prior work.
• The paper establishes the first provable sample/memory/communication trade-offs for a practical estimation problem using standard distributions.
• The results improve upon Shamir (2014) by considering a broader and more natural regime of memory and communication constraints.
• The analysis shows that the naive approach of processing one pair at a time is information-theoretically optimal, and no algorithm can achieve better trade-offs.