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Uses of Sub-sample Estimates to Reduce Errors in Stochastic Optimization Models

John R. Birge

TL;DR

This paper addresses the challenge that estimation errors in parameters of large-scale stochastic optimization can mislead solutions. It analyzes the SAA framework and shows that while $\sqrt{\nu}(x^\nu-x^*) \to u^*$ under standard conditions, such asymptotics can be uninformative in high dimensions and with finite samples; it then proposes sub-sample (batch) estimates, forming $\bar{x}^{\nu, K}$ by solving $K$ subproblems with $\nu/K$ samples each and averaging, to reduce variance. Through a mean-variance portfolio example, the work derives unbiased estimators and demonstrates that batching can improve objective values and proximity to the true optimum when constraints are present, though gains may vanish in loose or unconstrained settings. The results provide general convergence insights, a universal tail bound, and practical guidance for using sub-sample estimates and debiasing in high-dimensional stochastic programs, with bootstrap-based methods suggested for constructing confidence intervals and determining batch size $K$.

Abstract

Optimization software enables the solution of problems with millions of variables and associated parameters. These parameters are, however, often uncertain and represented with an analytical description of the parameter's distribution or with some form of sample. With large numbers of such parameters, optimization of the resulting model is often driven by mis-specifications or extreme sample characteristics, resulting in solutions that are far from a true optimum. This paper describes how asymptotic convergence results may not be useful in large-scale problems and how the optimization of problems based on sub-sample estimates may achieve improved results over models using full-sample solution estimates. A motivating example and numerical results from a portfolio optimization problem demonstrate the potential improvement. A theoretical analysis also provides insight into the structure of problems where sub-sample optimization may be most beneficial.

Uses of Sub-sample Estimates to Reduce Errors in Stochastic Optimization Models

TL;DR

This paper addresses the challenge that estimation errors in parameters of large-scale stochastic optimization can mislead solutions. It analyzes the SAA framework and shows that while under standard conditions, such asymptotics can be uninformative in high dimensions and with finite samples; it then proposes sub-sample (batch) estimates, forming by solving subproblems with samples each and averaging, to reduce variance. Through a mean-variance portfolio example, the work derives unbiased estimators and demonstrates that batching can improve objective values and proximity to the true optimum when constraints are present, though gains may vanish in loose or unconstrained settings. The results provide general convergence insights, a universal tail bound, and practical guidance for using sub-sample estimates and debiasing in high-dimensional stochastic programs, with bootstrap-based methods suggested for constructing confidence intervals and determining batch size .

Abstract

Optimization software enables the solution of problems with millions of variables and associated parameters. These parameters are, however, often uncertain and represented with an analytical description of the parameter's distribution or with some form of sample. With large numbers of such parameters, optimization of the resulting model is often driven by mis-specifications or extreme sample characteristics, resulting in solutions that are far from a true optimum. This paper describes how asymptotic convergence results may not be useful in large-scale problems and how the optimization of problems based on sub-sample estimates may achieve improved results over models using full-sample solution estimates. A motivating example and numerical results from a portfolio optimization problem demonstrate the potential improvement. A theoretical analysis also provides insight into the structure of problems where sub-sample optimization may be most beneficial.
Paper Structure (7 sections, 4 theorems, 34 equations, 12 figures, 1 table)

This paper contains 7 sections, 4 theorems, 34 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Asymptotic Convergence and Universal Confidence Region Results
  3. Examples of Convergence Behavior and the Use of Sub-samples
  4. Portfolio Optimization Example
  5. General Convergence
  6. Conclusions
  7. Compliance with Ethical Standards

Key Result

Theorem 1

Suppose that $f(\cdot,\xi)$ is convex and twice continuously differentiable , $X=\{x|Ax\le b\}$ is a convex polyhedron, $\nabla f:\Re^n \times\Xi \mapsto \Re^n$: then, for the solution $x^\nu$ to (montecarlo), $\sqrt{\nu}(x^\nu-x^*)$ converges in distribution to $u^*$: where $u^*$ is the solution to: $(x^{*},\pi^{*})$ solve $\nabla\int_{\Xi} f(x^{*},\xi) P(d\xi)+(\pi^{*})^{T}A=0$, $\pi^{*}\ge

Figures (12)

  • Figure 1: Log of probability of error above one for $n=100$, 1000, and 10,000.
  • Figure 2: Log of probability of $\infty$-norm error greater than one for $n=10$ and $n=100$ using multiple ($K=10$) and single batches.
  • Figure 3: Difference in error probability from the $\nu$-sample distribution and the asymptotic distribution for $n=10$, 20, 50, 100, and 200.
  • Figure 4: Histogram of differences between relative expected objective values for sub-sample approximation minus single-sample approximation for $X=[0,1]^{n}$ and $n=10$.
  • Figure 5: Histogram of differences between relative distance from optimum for sub-sample approximation minus single-sample approximation for $X=[0,1]^{n}$ and $n=10$.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2