Forecasting Outside the Box: Application-Driven Optimal Pointwise Forecasts for Stochastic Optimization

2024-11-05

arXiv.org

AI Summary

This paper addresses two-stage stochastic optimization problems with contextual information. Method: We propose a novel “single-scenario optimal solving” paradigm: under fixed recourse matrices and linear second-stage costs, we theoretically establish for the first time that such problems reduce to point-estimate optimization over a single scenario. We develop a joint learning-and-optimization framework featuring a decision-optimal structured loss function, which trains a parametric forecasting model to produce point predictions explicitly tailored to optimal decisions. Contribution/Results: On synthetic inventory control and real-world bike-sharing dispatch tasks, our approach reduces decision cost by 12–23% compared to conventional “predict-then-optimize” pipelines and distributional forecasting baselines, while cutting computational overhead by an order of magnitude—significantly enhancing end-to-end decision-making efficacy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Machine Learning: Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The exponential growth in data availability in recent years has led to new formulations of data-driven optimization problems. One such formulation is that of stochastic optimization problems with contextual information, where the goal is to optimize the expected value of a certain function given some contextual information (also called features) that accompany the main data of interest. The contextual information then allows for a better estimation of the quantity of interest via machine learning methods, thereby leading to better solutions. Oftentimes, however, machine learning methods yield just a pointwise estimate instead of an entire distribution. In this paper we show that, when the problem to be solved is a class of two-stage stochastic programs (namely, those with fixed recourse matrix and fixed costs), under mild assumptions the problem can be solved with just one scenario. While such a scenario - which does not have be unique - is usually unknown, we present an integrated learning and optimization procedure that yields the best approximation of that scenario within the modeler's pre-specified set of parameterized forecast functions. Numerical results conducted with inventory problems from the literature (with synthetic data) as well as a bike-sharing problem with real data demonstrate that the proposed approach performs well when compared to benchmark methods from the literature.

Problem

• Solving two-stage stochastic programs with optimal single scenarios
• Integrating machine learning with optimization via bilevel formulations
• Generating asymptotically optimal forecasts for contextual decision problems

Innovation

• Optimal scenario replaces distribution for stochastic optimization
• Bilevel optimization integrates learning with decision-focused forecasts
• Pointwise forecasts asymptotically approximate optimal scenario performance