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Assessing solution quality in risk-averse stochastic programs

E. Ruben van Beesten, Nick W. Koning, David P. Morton

TL;DR

This paper addresses the challenge of reliably estimating the optimality gap in risk-averse stochastic programs, where standard risk-neutral estimators can yield invalid probabilistic guarantees due to bias. It introduces a two-sample estimator G_{n,m} that uses an independent fresh sample to estimate the inner minimizer and demonstrates that this estimator is upward biased, enabling valid upper-bound constructions when combined with risk-neutral replication methods. A general recipe is provided to obtain approximate probabilistic upper bounds by reformulating the risk-averse problem as a risk-neutral problem in a higher-dimensional space and applying standard replication-based techniques to the reformulated problem. The authors establish tractability and convergence results for common risk measures (CVaR, entropic, and spectral risks), showing that the bias of the estimator vanishes as the fresh sample size grows and outlining practical extensions to law-invariant coherent risk measures.

Abstract

In optimization problems, the quality of a candidate solution can be characterized by the optimality gap. For most stochastic optimization problems, this gap must be statistically estimated. We show that for risk-averse problems, standard estimators are optimistically biased, which compromises the statistical guarantee on the optimality gap. We introduce estimators for risk-averse problems that do not suffer from this bias. Our method relies on using two independent samples, each estimating a different component of the optimality gap. Our approach extends a broad class of optimality gap estimation methods from the risk-neutral case to the risk-averse case, such as the multiple replications procedure and its one- and two-sample variants. We show that our approach is tractable and leads to high-quality optimality gap estimates for spectral and quadrangle risk measures. Our approach can further make use of existing bias and variance reduction techniques.

Assessing solution quality in risk-averse stochastic programs

TL;DR

This paper addresses the challenge of reliably estimating the optimality gap in risk-averse stochastic programs, where standard risk-neutral estimators can yield invalid probabilistic guarantees due to bias. It introduces a two-sample estimator G_{n,m} that uses an independent fresh sample to estimate the inner minimizer and demonstrates that this estimator is upward biased, enabling valid upper-bound constructions when combined with risk-neutral replication methods. A general recipe is provided to obtain approximate probabilistic upper bounds by reformulating the risk-averse problem as a risk-neutral problem in a higher-dimensional space and applying standard replication-based techniques to the reformulated problem. The authors establish tractability and convergence results for common risk measures (CVaR, entropic, and spectral risks), showing that the bias of the estimator vanishes as the fresh sample size grows and outlining practical extensions to law-invariant coherent risk measures.

Abstract

In optimization problems, the quality of a candidate solution can be characterized by the optimality gap. For most stochastic optimization problems, this gap must be statistically estimated. We show that for risk-averse problems, standard estimators are optimistically biased, which compromises the statistical guarantee on the optimality gap. We introduce estimators for risk-averse problems that do not suffer from this bias. Our method relies on using two independent samples, each estimating a different component of the optimality gap. Our approach extends a broad class of optimality gap estimation methods from the risk-neutral case to the risk-averse case, such as the multiple replications procedure and its one- and two-sample variants. We show that our approach is tractable and leads to high-quality optimality gap estimates for spectral and quadrangle risk measures. Our approach can further make use of existing bias and variance reduction techniques.
Paper Structure (10 sections, 3 theorems, 15 equations, 1 figure)

This paper contains 10 sections, 3 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Traditional optimality gap estimator
  3. Existing literature: risk-neutral case
  4. Risk-averse case: invalid upper bound due to a downward bias
  5. Conservative optimality gap estimator
  6. A valid probabilistic upper bound on the optimality gap
  7. Recipe for deriving an (approximate) probabilistic upper bound on $G$:
  8. Tractability and quality of $\hat{z}_{n,m}$
  9. Expectation quadrangle risk measures
  10. Tractability

Key Result

lemma thmcounterlemma

Let $\rho$ be a risk measure of the form eq:rho_min, let $\rho^n$ be its empirical analog eqn:empirical_risk_Y with respect to $\mathcal{S}_n$, and assume $\mathcal{S}_n$ satisfies $\mathbb{E}^{\mathcal{S}_n}[\mathbb{E}^n [r(Y,u)]] = \mathbb{E}[r(Y,u)], \ \forall u \in \mathcal{U}$. Then, for all $Y

Figures (1)

  • Figure 1: Illustration of the optimality gap and its estimators. Dashed arrows indicate the expectation of the indicated random object.

Theorems & Definitions (9)

  • lemma thmcounterlemma: Bias of $\rho^n$
  • proof
  • proposition thmcounterproposition: Bias of $z^*$ and $\hat{z}_n$
  • proof
  • theorem 1: Upward bias of $G_{n,m}$
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark