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Linear shrinkage for optimization in high dimensions

Naqi Huang, Nestor Parolya, Theresia van Essen

TL;DR

A linear shrinkage method that blends random matrix theory and robust optimization principles is developed, aiming to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables.

Abstract

In large-scale, data-driven applications, parameters are often only known approximately due to noise and limited data samples. In this paper, we focus on high-dimensional optimization problems with linear constraints under uncertain conditions. To find high quality solutions for which the violation of the true constraints is limited, we develop a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both obtaining high quality solutions and adhering to the true constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.

Linear shrinkage for optimization in high dimensions

TL;DR

A linear shrinkage method that blends random matrix theory and robust optimization principles is developed, aiming to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables.

Abstract

In large-scale, data-driven applications, parameters are often only known approximately due to noise and limited data samples. In this paper, we focus on high-dimensional optimization problems with linear constraints under uncertain conditions. To find high quality solutions for which the violation of the true constraints is limited, we develop a linear shrinkage method that blends random matrix theory and robust optimization principles. It aims to minimize the Frobenius distance between the estimated and the true parameter matrix, especially when dealing with a large and comparable number of constraints and variables. This data-driven method excels in simulations, showing superior noise resilience and more stable performance in both obtaining high quality solutions and adhering to the true constraints compared to traditional robust optimization. Our findings highlight the effectiveness of our method in improving the robustness and reliability of optimization in high-dimensional, data-driven scenarios.
Paper Structure (22 sections, 9 theorems, 88 equations, 6 figures, 2 tables)

This paper contains 22 sections, 9 theorems, 88 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Model formulation
  4. Interpretation from robust optimization point of view
  5. Bayesian interpretation
  6. Linear shrinkage method
  7. Finite-sample optimal shrinkage procedure
  8. Asymptotic behaviour of alpha and beta
  9. Consistent estimators of alpha1 and beta1
  10. Extension to column-correlated noise
  11. Asymptotic deterministic equivalents of alpha Co1 and beta Co1
  12. Oracle consistent estimators of alpha C2 and beta C2
  13. Choice of target matrix $\mathbf{U}_m$
  14. Simulation
  15. Conclusion
  16. ...and 7 more sections

Key Result

Theorem 4.1

Consider the optimization problem The optimization problem distance optimization problem has a unique optimal solution, given by where $\bar{\mathbf{A}}_m$ is defined in (def of b_A) and $\mathbf{U}_m$ is the $m \times p$ matrix with all entries being equal to one. We scale each trace term in alpha_m* and beta_m* for the sake of the boundednesses of $\frac{ \text{\rm tr}\left(\bar{\mathbf{A}}_m

Figures (6)

  • Figure 1: Performance of nominal method, linear shrinkage method and robust method for $c = 0.5$
  • Figure 2: Performance of nominal method, linear shrinkage method and robust method for $c = 1$
  • Figure 3: Performance of nominal method, linear shrinkage method and robust method for $c = 2$
  • Figure 4: Performance of nominal method, linear shrinkage method and robust method for $p=200$
  • Figure 5: Performance of nominal method, linear shrinkage method and robust method for $p=500$
  • ...and 1 more figures

Theorems & Definitions (24)

  • Theorem 4.1
  • proof
  • Definition 4.1: Asymptotic deterministic equivalent
  • Theorem 4.2
  • proof
  • Remark 1
  • Lemma 4.3
  • proof
  • Remark 2
  • Theorem 5.1
  • ...and 14 more