[Paper Review] Most Correlated Arms Identification

Che-Yu Liu, Sébastien Bubeck|arXiv (Cornell University)|Apr 23, 2014

Mass Spectrometry Techniques and Applications5 references3 citations

TL;DR

This paper introduces adaptive sampling algorithms, SR-C and SE-C, for identifying the h most mutually correlated arms in a Gaussian random vector with unknown correlation matrix. By leveraging a novel difference-based correlation estimator, the algorithms achieve sample complexity of $\widetilde{\mathcal{O}}(\mathrm{H_C} + K)$, where $\mathrm{H_C}$ depends on suboptimality ratios, significantly improving upon non-adaptive uniform sampling in heterogeneous correlation settings.

ABSTRACT

We study the problem of finding the most mutually correlated arms among many arms. We show that adaptive arms sampling strategies can have significant advantages over the non-adaptive uniform sampling strategy. Our proposed algorithms rely on a novel correlation estimator. The use of this accurate estimator allows us to get improved results for a wide range of problem instances.

Motivation & Objective

• To address the challenge of identifying the h most mutually correlated arms in a high-dimensional Gaussian vector with unknown correlation structure.
• To design adaptive sampling strategies that outperform non-adaptive uniform sampling in terms of sample efficiency.
• To introduce a new difference-based correlation estimator that improves estimation accuracy, especially when correlations are close to 1.
• To characterize the sample complexity of adaptive algorithms in both fixed-budget and fixed-confidence settings.
• To establish theoretical bounds on the number of samples required to reliably identify the optimal subset of arms.

Proposed method

• Introduces a difference-based correlation estimator that enhances accuracy, particularly for high correlations, by exploiting the variance structure of pairwise differences.
• Defines the suboptimality ratio $R_{i,\Sigma}$ as a measure of how suboptimal an arm is relative to the optimal set, based on the ratio of uncorrelated pairwise terms.
• Proposes SR-C and SE-C algorithms for fixed-budget and fixed-confidence settings, respectively, using confidence bounds derived from the new estimator.
• Employs a stopping rule in SE-C based on confidence intervals that shrink with sampling, ensuring high-probability correctness.
• Uses the function $\alpha(\theta) = \frac{1}{2}(\log\theta - 1 + \frac{1}{\theta})$ to quantify the difficulty of distinguishing suboptimal arms, with $\alpha(\theta) \sim (1-\theta)^2$ near $\theta=1$.
• Derives sample complexity bounds via concentration inequalities, controlling the probability of error through a union bound over all arm pairs.

Experimental results

Research questions

• RQ1 Can adaptive sampling strategies significantly reduce the number of samples required to identify the h most correlated arms compared to uniform sampling?
• RQ2 How does the accuracy of correlation estimation affect the sample complexity in correlated arm identification?
• RQ3 What is the optimal structure of the sample complexity in terms of the suboptimality ratios $R_{i,\Sigma}$?
• RQ4 Can the $\widetilde{\mathcal{O}}(h \cdot \alpha(R_{(h+1),\Sigma})^{-1})$ term in the upper bound be reduced for general correlation matrices?
• RQ5 Is there a fundamental lower bound on the number of samples required by any adaptive algorithm for this problem?

Key findings

• The proposed algorithms SR-C and SE-C achieve a sample complexity of $\widetilde{\mathcal{O}}(\mathrm{H_C} + K)$, where $\mathrm{H_C} = \frac{h}{\alpha(R_{(h+1),\Sigma})} + \sum_{i=h+1}^{K} \frac{1}{\alpha(R_{(i),\Sigma})}$, significantly improving over non-adaptive strategies.
• The difference-based correlation estimator enables better estimation accuracy when correlations are close to 1, a property not shared by classical estimators.
• In the non-adaptive setting, $\widetilde{\Theta}\left(\frac{K}{\alpha(R_{(h+1),\Sigma})} + K\right)$ samples are necessary and sufficient for reliable identification.
• The term $\mathrm{H_C}$ can be much smaller than $\frac{K}{\alpha(R_{(h+1),\Sigma})}$ in inhomogeneous correlation matrices, demonstrating the advantage of adaptivity.
• The $\widetilde{\mathcal{O}}(h \cdot \alpha(R_{(h+1),\Sigma})^{-1})$ term in the upper bound is suboptimal in some cases, suggesting room for algorithmic improvement.
• The paper identifies a key challenge: early acceptance of optimal arms does not simplify the remaining identification task due to the non-monotonic nature of correlation structure.