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Experimental Design Using Interlacing Polynomials

Lap Chi Lau, Robert Wang, Hong Zhou

TL;DR

This framework recovers the best-known approximation guarantees for the well-studied D/A/E-design problems with simple analysis and provides an optimal approximation guarantee for a generalized ratio objective that generalizes both D-design and A-design.

Abstract

We present a unified deterministic approach for experimental design problems using the method of interlacing polynomials. Our framework recovers the best-known approximation guarantees for the well-studied D/A/E-design problems with simple analysis. Furthermore, we obtain improved non-trivial approximation guarantee for E-design in the challenging small budget regime. Additionally, our approach provides an optimal approximation guarantee for a generalized ratio objective that generalizes both D-design and A-design.

Experimental Design Using Interlacing Polynomials

TL;DR

This framework recovers the best-known approximation guarantees for the well-studied D/A/E-design problems with simple analysis and provides an optimal approximation guarantee for a generalized ratio objective that generalizes both D-design and A-design.

Abstract

We present a unified deterministic approach for experimental design problems using the method of interlacing polynomials. Our framework recovers the best-known approximation guarantees for the well-studied D/A/E-design problems with simple analysis. Furthermore, we obtain improved non-trivial approximation guarantee for E-design in the challenging small budget regime. Additionally, our approach provides an optimal approximation guarantee for a generalized ratio objective that generalizes both D-design and A-design.

Paper Structure

This paper contains 12 sections, 15 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our Results
  4. Technical Overview
  5. Notations
  6. The Interlacing Family
  7. Interlacing Family Based Algorithms for E/D/A-Design
  8. Analysis for E-Design
  9. Analysis for D-Design
  10. Analysis for A-Design
  11. Experimental Design With Generalized Ratio Objective
  12. Omitted Proof

Key Result

Theorem 1.1

For any $k \geq d$, there is a $(1-\sqrt{(d-1)/k})^{-2}$-approximation deterministic rounding algorithm for E-design using the method of interlacing polynomials. The rounding algorithm runs in $O(kmd^{\omega+1})$ time, where $\omega$ is the matrix-multiplication exponent.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: MSS22
  • Theorem 2.2: MSS22
  • Theorem 2.3: MSS22
  • Lemma 2.4: MSS22
  • Lemma 3.1: MSS22
  • Lemma 3.2: MSS22
  • Lemma 3.3: MSS22
  • ...and 16 more