Learning Solutions of Stochastic Optimization Problems with Bayesian Neural Networks

TL;DR

This paper introduces a Bayesian Neural Network-based framework for predicting uncertain parameters in stochastic optimization problems, improving decision quality by modeling and propagating uncertainty through differentiable end-to-end learning.

Contribution

It proposes a novel approach combining Bayesian Neural Networks with stochastic programming to incorporate prediction uncertainty into optimization decision-making.

Findings

Decoupled learning improves prediction quality of OP parameters.

Combined learning directly minimizes expected OP cost.

Both methods reduce decision regret in synthetic and real datasets.

Abstract

Mathematical solvers use parametrized Optimization Problems (OPs) as inputs to yield optimal decisions. In many real-world settings, some of these parameters are unknown or uncertain. Recent research focuses on predicting the value of these unknown parameters using available contextual features, aiming to decrease decision regret by adopting end-to-end learning approaches. However, these approaches disregard prediction uncertainty and therefore make the mathematical solver susceptible to provide erroneous decisions in case of low-confidence predictions. We propose a novel framework that models prediction uncertainty with Bayesian Neural Networks (BNNs) and propagates this uncertainty into the mathematical solver with a Stochastic Programming technique. The differentiable nature of BNNs and differentiable mathematical solvers allow for two different learning approaches: In the Decoupled learning approach, we update the BNN weights to increase the quality of the predictions' distribution of the OP parameters, while in the Combined learning approach, we update the weights aiming to directly minimize the expected OP's cost function in a stochastic end-to-end fashion. We do an extensive evaluation using synthetic data with various noise properties and a real dataset, showing that decisions regret are generally lower (better) with both proposed methods.