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Strong Formulations and Algorithms for Regularized A-optimal Design

Yongchun Li

TL;DR

The paper tackles Regularized A-optimal Design ($RAOD$), proving its NP-hardness and introducing a stronger convex integer programming formulation whose convex relaxation yields bounded optimality gaps for all $k$. It develops an exact cutting-plane algorithm based on a novel convex envelope (conv$\Gamma$) and a corresponding MILP reformulation, plus scalable forward and backward greedy algorithms with data-independent guarantees for different $k$ regimes. Extensive numerical experiments on synthetic and real data (including a user cold-start recommendation setting) demonstrate that the proposed exact method substantially outperforms existing relaxations and that the greedy strategies provide high-quality, scalable solutions. The work offers practical, non-sequential experimental design tools with strong theoretical guarantees, suitable for high-dimensional, small-$k$ scenarios and broader Bayesian-design-inspired applications.

Abstract

We study the Regularized A-optimal Design (RAOD) problem, which selects a subset of $k$ experiments to minimize the inverse of the Fisher information matrix, regularized with a scaled identity matrix. RAOD has broad applications in Bayesian experimental design, sensor placement, and cold-start recommendation. We prove its NP-hardness via a reduction from the independent set problem. By leveraging convex envelope techniques, we propose a new convex integer programming formulation for RAOD, whose continuous relaxation dominates those of existing formulations. More importantly, we demonstrate that our continuous relaxation achieves bounded optimality gaps for all $k$, whereas previous relaxations may suffer from unbounded gaps. This new formulation enables the development of an exact cutting-plane algorithm with superior efficiency, especially in high-dimensional and small-$k$ scenarios. We also investigate scalable forward and backward greedy algorithms for solving RAOD, each with provable performance guarantees for different $k$ ranges. Finally, our numerical results on synthetic and real data demonstrate the efficacy of the proposed exact and approximation algorithms. We further showcase the practical effectiveness of RAOD by applying it to a real-world user cold-start recommendation problem.

Strong Formulations and Algorithms for Regularized A-optimal Design

TL;DR

The paper tackles Regularized A-optimal Design (), proving its NP-hardness and introducing a stronger convex integer programming formulation whose convex relaxation yields bounded optimality gaps for all . It develops an exact cutting-plane algorithm based on a novel convex envelope (conv) and a corresponding MILP reformulation, plus scalable forward and backward greedy algorithms with data-independent guarantees for different regimes. Extensive numerical experiments on synthetic and real data (including a user cold-start recommendation setting) demonstrate that the proposed exact method substantially outperforms existing relaxations and that the greedy strategies provide high-quality, scalable solutions. The work offers practical, non-sequential experimental design tools with strong theoretical guarantees, suitable for high-dimensional, small- scenarios and broader Bayesian-design-inspired applications.

Abstract

We study the Regularized A-optimal Design (RAOD) problem, which selects a subset of experiments to minimize the inverse of the Fisher information matrix, regularized with a scaled identity matrix. RAOD has broad applications in Bayesian experimental design, sensor placement, and cold-start recommendation. We prove its NP-hardness via a reduction from the independent set problem. By leveraging convex envelope techniques, we propose a new convex integer programming formulation for RAOD, whose continuous relaxation dominates those of existing formulations. More importantly, we demonstrate that our continuous relaxation achieves bounded optimality gaps for all , whereas previous relaxations may suffer from unbounded gaps. This new formulation enables the development of an exact cutting-plane algorithm with superior efficiency, especially in high-dimensional and small- scenarios. We also investigate scalable forward and backward greedy algorithms for solving RAOD, each with provable performance guarantees for different ranges. Finally, our numerical results on synthetic and real data demonstrate the efficacy of the proposed exact and approximation algorithms. We further showcase the practical effectiveness of RAOD by applying it to a real-world user cold-start recommendation problem.

Paper Structure

This paper contains 34 sections, 20 theorems, 51 equations, 6 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Summary of the Organization and Contributions
  4. Complexity Analysis
  5. Optimality Gaps of Convex Relaxations
  6. The Conventional Convex Integer Programming Formulation and its Convex Relaxation
  7. A Convex Relaxation of AMESP
  8. A Mixed-integer Second-order Cone Programming Formulation
  9. A Novel Convex Integer Formulation and its Convex Relaxation
  10. A Novel Convex Integer Programming Formulation
  11. A Novel Convex Relaxation
  12. The Cutting-Plane Algorithm
  13. The Forward and Backward Greedy Algorithms
  14. Numerical Results
  15. Evaluations of Convex Relaxations
  16. ...and 19 more sections

Key Result

Proposition 1

aed can be converted into the following: where $\bm C \in {\mathcal{S}}_+^n := \bm A^{\top}\bm A +\lambda \bm I_n$, and $\bm A\in {\mathbb R}^{d\times n}$ is a matrix whose columns are vectors $\{\bm a_i\}_{i\in [n]}$.

Figures (6)

  • Figure 1: The optimality gaps of \ref{['aed-R2']} in \ref{['them:gap']}
  • Figure 2: The approximation ratios of \ref{['algo:forward', 'algo:backward']} in \ref{['them:approx']} with $n=30$
  • Figure 3: Convex relaxations of \ref{['aed']}: Gaps and computation times on synthetic data
  • Figure 4: Convex relaxations of \ref{['aed']}: Gaps and computation times on real data
  • Figure 5: Approximation algorithms of \ref{['aed']}: Gaps and computation times on synthetic data
  • ...and 1 more figures

Theorems & Definitions (58)

  • Proposition 1
  • proof
  • Definition 1: Independent sets and the independent set decision problem
  • Lemma 1: sedrakyan2018algebraic
  • Theorem 1: NP-hardness
  • proof
  • Conjecture 1: hendrych2023solving
  • Remark 1
  • Lemma 2
  • proof
  • ...and 48 more