A Survey of Contextual Optimization Methods for Decision Making under Uncertainty

TL;DR

This survey consolidates the rapidly growing field of contextual optimization, where covariates are leveraged to make decisions under uncertainty. It codifies three learning paradigms—decision rule optimization, sequential learning and optimization, and integrated learning and optimization—and systematically reviews linear, kernel-based, non-linear, and distributionally robust policy classes, as well as methods for learning conditional distributions and end-to-end training via SPO and related surrogates. The authors synthesize training approaches (unrolling, implicit differentiation, surrogate losses, and differentiable optimizers) and map them to a wide range of applications, while outlining theoretical guarantees, toolboxes, and benchmarking needs. They also chart active research directions, from uncertainty in constraints and risk aversion to privacy, interpretability, and multi-agent settings, highlighting the practical impact and remaining theoretical gaps in contextual optimization. Overall, the paper provides a unifying framework, a comprehensive taxonomy, and a roadmap for advancing the integration of machine learning with stochastic programming in decision-making under uncertainty.

Abstract

Recently there has been a surge of interest in operations research (OR) and the machine learning (ML) community in combining prediction algorithms and optimization techniques to solve decision-making problems in the face of uncertainty. This gave rise to the field of contextual optimization, under which data-driven procedures are developed to prescribe actions to the decision-maker that make the best use of the most recently updated information. A large variety of models and methods have been presented in both OR and ML literature under a variety of names, including data-driven optimization, prescriptive optimization, predictive stochastic programming, policy optimization, (smart) predict/estimate-then-optimize, decision-focused learning, (task-based) end-to-end learning/forecasting/optimization, etc. Focusing on single and two-stage stochastic programming problems, this review article identifies three main frameworks for learning policies from data and discusses their strengths and limitations. We present the existing models and methods under a uniform notation and terminology and classify them according to the three main frameworks identified. Our objective with this survey is to both strengthen the general understanding of this active field of research and stimulate further theoretical and algorithmic advancements in integrating ML and stochastic programming.

Figures (5)

• Figure 1: Decision and training pipelines based on the decision rule paradigm: (left) the decision pipeline and (right) the training pipeline for a given training example $(\boldsymbol{x}_i, \boldsymbol{y}_i)$.
• Figure 2: Decision pipeline for learning and optimization.
• Figure 3: SLO training pipeline for a given training example.
• Figure 4: Predicting $g_{{\boldsymbol{\theta}}_A}(\boldsymbol{x})$ results in the optimal action $z^*(\boldsymbol{x}, g_{{\boldsymbol{\theta}}_A}) = z^*(\boldsymbol{x})$ whereas a small error resulting from predicting $g_{{\boldsymbol{\theta}}_B}(\boldsymbol{x})$ leads to a suboptimal action $z^*(\boldsymbol{x}, g_{{\boldsymbol{\theta}}_B})$ under $c(\boldsymbol{x},\boldsymbol{y}):= -\boldsymbol{y}^\top \boldsymbol{x}$, i.e., $h(\boldsymbol{z}, {\mathbb P}(\boldsymbol{y} \rvert \boldsymbol{x})) = -{\mathbb E}[\boldsymbol{y}\rvert \boldsymbol{x}]^\top \boldsymbol{z}$ (adapted from elmachtoub_smart_2022).
• Figure 5: ILO training pipeline for a given training example.

Theorems & Definitions (6)

• Definition 1: Expected value-based models
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6