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Sample-Efficient "Clustering and Conquer" Procedures for Parallel Large-Scale Ranking and Selection

Zishi Zhang, Yijie Peng

TL;DR

This work tackles the inefficiency of large-scale ranking and selection (R&S) when many alternatives are evaluated in parallel. It introduces Parallel Correlation Clustering and Conquer (P3C), a framework that clusters correlated alternatives and assigns clusters to processors to improve sample efficiency; it also develops a gradient-based analysis of mean-covariance interactions and a few-shot clustering method AC^+ for scalability. It proves that, under a symmetric benchmark, P3C can achieve an $O(p)$ reduction in total samples for sample-optimal R&S procedures, and provides PCC-based guarantees for clustering. It demonstrates substantial empirical gains in fixed-precision drug discovery and fixed-budget neural architecture search (NAS), with P3C reducing required samples and lowering wall-clock time while incurring modest clustering overhead.

Abstract

This work aims to improve the sample efficiency of parallel large-scale ranking and selection (R&S) problems by leveraging correlation information. We modify the commonly used "divide and conquer" framework in parallel computing by adding a correlation-based clustering step, transforming it into "clustering and conquer". Analytical results under a symmetric benchmark scenario show that this seemingly simple modification yields an $\mathcal{O}(p)$ reduction in sample complexity for a widely used class of sample-optimal R&S procedures. Our approach enjoys two key advantages: 1) it does not require highly accurate correlation estimation or precise clustering, and 2) it allows for seamless integration with various existing R&S procedures, while achieving optimal sample complexity. Theoretically, we develop a novel gradient analysis framework to analyze sample efficiency and guide the design of large-scale R&S procedures. We also introduce a new parallel clustering algorithm tailored for large-scale scenarios. Finally, in large-scale AI applications such as neural architecture search, our methods demonstrate superior performance.

Sample-Efficient "Clustering and Conquer" Procedures for Parallel Large-Scale Ranking and Selection

TL;DR

This work tackles the inefficiency of large-scale ranking and selection (R&S) when many alternatives are evaluated in parallel. It introduces Parallel Correlation Clustering and Conquer (P3C), a framework that clusters correlated alternatives and assigns clusters to processors to improve sample efficiency; it also develops a gradient-based analysis of mean-covariance interactions and a few-shot clustering method AC^+ for scalability. It proves that, under a symmetric benchmark, P3C can achieve an reduction in total samples for sample-optimal R&S procedures, and provides PCC-based guarantees for clustering. It demonstrates substantial empirical gains in fixed-precision drug discovery and fixed-budget neural architecture search (NAS), with P3C reducing required samples and lowering wall-clock time while incurring modest clustering overhead.

Abstract

This work aims to improve the sample efficiency of parallel large-scale ranking and selection (R&S) problems by leveraging correlation information. We modify the commonly used "divide and conquer" framework in parallel computing by adding a correlation-based clustering step, transforming it into "clustering and conquer". Analytical results under a symmetric benchmark scenario show that this seemingly simple modification yields an reduction in sample complexity for a widely used class of sample-optimal R&S procedures. Our approach enjoys two key advantages: 1) it does not require highly accurate correlation estimation or precise clustering, and 2) it allows for seamless integration with various existing R&S procedures, while achieving optimal sample complexity. Theoretically, we develop a novel gradient analysis framework to analyze sample efficiency and guide the design of large-scale R&S procedures. We also introduce a new parallel clustering algorithm tailored for large-scale scenarios. Finally, in large-scale AI applications such as neural architecture search, our methods demonstrate superior performance.
Paper Structure (19 sections, 7 theorems, 31 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 19 sections, 7 theorems, 31 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Framework of P3C: From "Divide and Conquer" to "Clustering and Conquer"
  4. Theoretical Analysis: Understanding Mean-Correlation Interactions
  5. Interaction and Impact of Mean and Correlation on PCS
  6. The derivative of PCS with respect to mean and correlation information.
  7. "Separation" effect
  8. Quantifying the Sample Complexity Reduction
  9. Large-Scale Alternative Clustering
  10. Few-shot Alternative Clustering
  11. Computation of $\mathrm{PCC}$
  12. Numerical Experiments
  13. Illustrative Example
  14. Fixed-precision R&S: Drug Discovery
  15. Fixed-budget R&S: Neural Architecture Search
  16. ...and 4 more sections

Key Result

Proposition 1

Let $N_{\mathrm{P3C-KN}}$ and $N_{\mathrm{P3C-KT}}$ denote the required total sample sizes to achieve $\mathrm{PCS_{trad}}\geq 1-\alpha$ using $\mathrm{P3C-KN}$ and $\mathrm{P3C-KT}$, respectively. If the number of alternatives within each cluster is upper bounded by a constant $p_m<\infty$, then $\

Figures (6)

  • Figure 1: Parallel correlation clustering and conquer.
  • Figure 2: An illustrative example with 7 alternatives, where the top 3 alternatives $[1]$, $[2]$ and $[3]$ exhibit high mean performances, and $[5]$, $[6]$ and $[7]$ exhibit low mean performances.
  • Figure 3: Comparison of computational time and required sample size under different numbers of alternatives $p$. The left panel shows the required total sample size ($\times 10^4$) as a function of $p$, while the right panel presents the corresponding wall clock time (seconds).
  • Figure 4: Comparison of $\mathrm{PCC}_{\mathcal{A}\mathcal{C}^+}$ with different support set sizes $p_s$.
  • Figure 5: Comparison of different R&S procedures for different total sample sizes. The left panel shows the $\text{PCS}_{\text{trad}}$ metric comparison, while the right panel presents the corresponding wall clock time (seconds) for different total sample sizes ($\times 10^7$).
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 1: Sample Optimality of P3C-KN and P3C-KT
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Corollary 1
  • Lemma 1
  • Theorem 3
  • Remark 3
  • Proposition 2