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Risk-averse Decision Making with Contextual Information: Model, Sample Average Approximation, and Kernelization

Yuan Tao, Erick Delage, Huifu Xu

TL;DR

This work develops a unified theory for risk-averse contextual optimization under two uncertainties: problem data uncertainty (PDU) and contextual uncertainty (CU). It introduces nested ex post risk minimization and ex ante joint-risk mappings, identifies conditions for their equivalence, and shows policy optimality across contexts under several risk measures, notably CVaR, optimized certainty equivalents, and entropic risks. The authors formulate a tractable one-stage ex ante model and establish continuity and contextual consistency results, leveraging the interchangeability principle. They extend the analysis to reproducing kernel Hilbert spaces (RKHS), proving SAA consistency and showing that universal kernels can approximate continuous policies; numerical tests on newsvendor and portfolio problems validate theory and demonstrate RKHS-based methods’ practical performance. Overall, the paper provides rigorous foundations and scalable algorithms for risk-averse contextual optimization with data-driven implementation.

Abstract

We consider risk-averse contextual optimization problems where the decision maker (DM) faces two types of uncertainties: problem data uncertainty (PDU) and contextual uncertainty (CU) associated with PDU, the DM makes an optimal decision by minimizing the risk arising from PDU based on the present observation of CU and then assesses the risk of the optimal policy against the CU. A natural question arises as to whether the nested risk minimization/assessment process is equivalent to joint risk minimization/assessment against CU and PDU simultaneously. First, we demonstrate that the equivalence can be established by appropriate choices of the risk measures and give counter examples where such equivalence may fail. One of the interesting findings is that the optimal policies are independent of the choice of the risk measure against the CU under certain conditions. Second, by using the equivalence, we propose computational method for solving the risk-averse contextual optimization problem by solving a one-stage risk minimization problem. The latter is particularly helpful in data-driven environments. We consider a number of risk measures/metrics to characterize the DM's risk preference for PDU and discuss the computational tractability for the resulting risk-averse contextual optimization problem. Third, when the risk-averse contextual optimization problem is defined in the reproducing kernel Hilbert space, we show consistency of the optimal values obtained from solving sample average approximation problems. Some numerical tests, in newsvendor problem and portfolio selection problem, are performed to validate the theoretical results.

Risk-averse Decision Making with Contextual Information: Model, Sample Average Approximation, and Kernelization

TL;DR

This work develops a unified theory for risk-averse contextual optimization under two uncertainties: problem data uncertainty (PDU) and contextual uncertainty (CU). It introduces nested ex post risk minimization and ex ante joint-risk mappings, identifies conditions for their equivalence, and shows policy optimality across contexts under several risk measures, notably CVaR, optimized certainty equivalents, and entropic risks. The authors formulate a tractable one-stage ex ante model and establish continuity and contextual consistency results, leveraging the interchangeability principle. They extend the analysis to reproducing kernel Hilbert spaces (RKHS), proving SAA consistency and showing that universal kernels can approximate continuous policies; numerical tests on newsvendor and portfolio problems validate theory and demonstrate RKHS-based methods’ practical performance. Overall, the paper provides rigorous foundations and scalable algorithms for risk-averse contextual optimization with data-driven implementation.

Abstract

We consider risk-averse contextual optimization problems where the decision maker (DM) faces two types of uncertainties: problem data uncertainty (PDU) and contextual uncertainty (CU) associated with PDU, the DM makes an optimal decision by minimizing the risk arising from PDU based on the present observation of CU and then assesses the risk of the optimal policy against the CU. A natural question arises as to whether the nested risk minimization/assessment process is equivalent to joint risk minimization/assessment against CU and PDU simultaneously. First, we demonstrate that the equivalence can be established by appropriate choices of the risk measures and give counter examples where such equivalence may fail. One of the interesting findings is that the optimal policies are independent of the choice of the risk measure against the CU under certain conditions. Second, by using the equivalence, we propose computational method for solving the risk-averse contextual optimization problem by solving a one-stage risk minimization problem. The latter is particularly helpful in data-driven environments. We consider a number of risk measures/metrics to characterize the DM's risk preference for PDU and discuss the computational tractability for the resulting risk-averse contextual optimization problem. Third, when the risk-averse contextual optimization problem is defined in the reproducing kernel Hilbert space, we show consistency of the optimal values obtained from solving sample average approximation problems. Some numerical tests, in newsvendor problem and portfolio selection problem, are performed to validate the theoretical results.

Paper Structure

This paper contains 43 sections, 15 theorems, 143 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topology of Weak Convergence and Risk Measures
  4. Contextual Stochastic Optimization
  5. Risk Management in Contextual Optimization
  6. Conditional Risk Minimization and Risk Evaluation
  7. Well-definedness of Risk-averse Contextual Optimization Problem
  8. Contextual Consistency
  9. Risk-averse Contextual Optimization and Examples
  10. Optimality of Risk-averse Contextual Optimization
  11. Examples of Ex Ante Risk Mappings
  12. Extensions to General Risk Measures/metrics
  13. Contextual Risk-Averse Optimization in Reproducing Kernel Hilbert Space
  14. Reproducing Kernel Hilbert Space
  15. Consistency of SAA under RKHS
  16. ...and 28 more sections

Key Result

Proposition 3.1

Consider problem (eqn:ex-post-interchange). Under Assumption assumption:well-defined_rho, $\rho^{(2)}_{P_{Y|X=x}} ( c(g(x),Y)): \mathcal{X}\rightarrow \mathbb{R}$ is continuous in $x$ over $\mathcal{X}$.

Figures (8)

  • Figure 1: The policies for newsvendor problem under different risk measures.
  • Figure 2: Relative distance of conditional expected loss, conditional CVaR, and conditional entropic risk measure as a function of context $x$.
  • Figure 3: Optimal policy and SAA policies (sample size $N=800$) for risk-averse contextual optimization problem with expected CVaR.
  • Figure 4: In risk-averse contextual optimization problem with expected CVaR, relative average distance of policies vs. conditional optimal policy under different sample sizes.
  • Figure 5: The policies under different sample sizes
  • ...and 3 more figures

Theorems & Definitions (32)

  • Example 3.1: Shortfall risk
  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.1: Contextual consistency
  • Theorem 4.1
  • Example 4.1: CVaR
  • Example 4.2: Nested-CVaR
  • Example 4.3: Expected optimized certainty equivalent
  • Example 4.4: Entropic risk measure and expected entropic risk measure
  • Example 4.5: Coherent risk measure
  • ...and 22 more