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End-to-end Conditional Robust Optimization

Abhilash Chenreddy, Erick Delage

TL;DR

The paper tackles risk-averse decision-making under uncertainty by unifying machine learning predictions with robust optimization in an end-to-end framework (ECRO). It jointly learns contextual ellipsoidal uncertainty sets and the CRO objective, enabling gradient-based optimization via Fenchel duality and implicit differentiation; it also introduces conditional coverage-focused training (CC loss and regression-based covariate modeling) and two integrated algorithms, TbS and DTbS. Empirical results on synthetic data and US stock portfolios show that end-to-end approaches improve CRO performance (CVaR) while achieving favorable conditional coverage compared to traditional Estimate-Then-Optimize methods. The work demonstrates that incorporating downstream task performance and conditional coverage into the learning objective yields more reliable, risk-aware decisions in high-stakes contexts.

Abstract

The field of Contextual Optimization (CO) integrates machine learning and optimization to solve decision making problems under uncertainty. Recently, a risk sensitive variant of CO, known as Conditional Robust Optimization (CRO), combines uncertainty quantification with robust optimization in order to promote safety and reliability in high stake applications. Exploiting modern differentiable optimization methods, we propose a novel end-to-end approach to train a CRO model in a way that accounts for both the empirical risk of the prescribed decisions and the quality of conditional coverage of the contextual uncertainty set that supports them. While guarantees of success for the latter objective are impossible to obtain from the point of view of conformal prediction theory, high quality conditional coverage is achieved empirically by ingeniously employing a logistic regression differentiable layer within the calculation of coverage quality in our training loss. We show that the proposed training algorithms produce decisions that outperform the traditional estimate then optimize approaches.

End-to-end Conditional Robust Optimization

TL;DR

The paper tackles risk-averse decision-making under uncertainty by unifying machine learning predictions with robust optimization in an end-to-end framework (ECRO). It jointly learns contextual ellipsoidal uncertainty sets and the CRO objective, enabling gradient-based optimization via Fenchel duality and implicit differentiation; it also introduces conditional coverage-focused training (CC loss and regression-based covariate modeling) and two integrated algorithms, TbS and DTbS. Empirical results on synthetic data and US stock portfolios show that end-to-end approaches improve CRO performance (CVaR) while achieving favorable conditional coverage compared to traditional Estimate-Then-Optimize methods. The work demonstrates that incorporating downstream task performance and conditional coverage into the learning objective yields more reliable, risk-aware decisions in high-stakes contexts.

Abstract

The field of Contextual Optimization (CO) integrates machine learning and optimization to solve decision making problems under uncertainty. Recently, a risk sensitive variant of CO, known as Conditional Robust Optimization (CRO), combines uncertainty quantification with robust optimization in order to promote safety and reliability in high stake applications. Exploiting modern differentiable optimization methods, we propose a novel end-to-end approach to train a CRO model in a way that accounts for both the empirical risk of the prescribed decisions and the quality of conditional coverage of the contextual uncertainty set that supports them. While guarantees of success for the latter objective are impossible to obtain from the point of view of conformal prediction theory, high quality conditional coverage is achieved empirically by ingeniously employing a logistic regression differentiable layer within the calculation of coverage quality in our training loss. We show that the proposed training algorithms produce decisions that outperform the traditional estimate then optimize approaches.
Paper Structure (23 sections, 1 theorem, 26 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 23 sections, 1 theorem, 26 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Estimate then Robust Optimize
  4. End-to-End Conditional Robust Optimization
  5. The ECRO training problem
  6. Reducing and solving the robust optimization task
  7. Gradient for problem (\ref{['eq:E2Eloss']})
  8. Task-based Set (TbS) Algorithm
  9. End-to-End CRO with Conditional Coverage
  10. The conditional coverage training problem
  11. Regression-based Conditional Coverage Loss
  12. Dual Task based Set (DTbS) algorithm
  13. Experiments
  14. The portfolio optimization application
  15. CRO performance using synthetic data
  16. ...and 8 more sections

Key Result

Lemma 5.2

A contextual uncertainty set ${\mathcal{U}}(\psi)$ satisfies conditional coverage, at confidence $1-\epsilon$, if and only if

Figures (7)

  • Figure 1: Training pipeline for task-based learning
  • Figure 2: Training pipeline for dual task based learning
  • Figure 3: Comparison of uncertainty set ($\alpha$ = 0.9) coverage for different $\psi$ realizations: (a) $[2.5, -0.2]^T$, (b) $[-2.6, 0.5]^T$, (c) $[2.7, 1.9]^T$. The shade indicate the true conditional distribution.
  • Figure 4: Avg. CVaR of returns across 10 portfolio trajectory simulations. Error bars report 95% CI.
  • Figure 5: Average cumulative distribution of conditional coverage frequency when $\psi$ is sampled uniformly from dataset over 10 simulated environments. Shaded region represent 90% CI
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 5.1
  • Lemma 5.2
  • proof
  • Remark 5.3