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Differentiable Distributionally Robust Optimization Layers

Xutao Ma, Chao Ning, Wenli Du

TL;DR

A differentiable energy-based surrogate is constructed to implement the dual-view methodology and it is proved that such a surrogate enjoys the asymptotic convergency under regularization and develops a novel decision-focused learning pipeline for contextual distributionally robust decision-making tasks.

Abstract

In recent years, there has been a growing research interest in decision-focused learning, which embeds optimization problems as a layer in learning pipelines and demonstrates a superior performance than the prediction-focused approach. However, for distributionally robust optimization (DRO), a popular paradigm for decision-making under uncertainty, it is still unknown how to embed it as a layer, i.e., how to differentiate decisions with respect to an ambiguity set. In this paper, we develop such differentiable DRO layers for generic mixed-integer DRO problems with parameterized second-order conic ambiguity sets and discuss its extension to Wasserstein ambiguity sets. To differentiate the mixed-integer decisions, we propose a novel dual-view methodology by handling continuous and discrete parts of decisions via different principles. Specifically, we construct a differentiable energy-based surrogate to implement the dual-view methodology and use importance sampling to estimate its gradient. We further prove that such a surrogate enjoys the asymptotic convergency under regularization. As an application of the proposed differentiable DRO layers, we develop a novel decision-focused learning pipeline for contextual distributionally robust decision-making tasks and compare it with the prediction-focused approach in experiments.

Differentiable Distributionally Robust Optimization Layers

TL;DR

A differentiable energy-based surrogate is constructed to implement the dual-view methodology and it is proved that such a surrogate enjoys the asymptotic convergency under regularization and develops a novel decision-focused learning pipeline for contextual distributionally robust decision-making tasks.

Abstract

In recent years, there has been a growing research interest in decision-focused learning, which embeds optimization problems as a layer in learning pipelines and demonstrates a superior performance than the prediction-focused approach. However, for distributionally robust optimization (DRO), a popular paradigm for decision-making under uncertainty, it is still unknown how to embed it as a layer, i.e., how to differentiate decisions with respect to an ambiguity set. In this paper, we develop such differentiable DRO layers for generic mixed-integer DRO problems with parameterized second-order conic ambiguity sets and discuss its extension to Wasserstein ambiguity sets. To differentiate the mixed-integer decisions, we propose a novel dual-view methodology by handling continuous and discrete parts of decisions via different principles. Specifically, we construct a differentiable energy-based surrogate to implement the dual-view methodology and use importance sampling to estimate its gradient. We further prove that such a surrogate enjoys the asymptotic convergency under regularization. As an application of the proposed differentiable DRO layers, we develop a novel decision-focused learning pipeline for contextual distributionally robust decision-making tasks and compare it with the prediction-focused approach in experiments.

Paper Structure

This paper contains 40 sections, 6 theorems, 88 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Background
  4. Decision-Making under Uncertainty
  5. Decision-Focused Learning
  6. Distributionally Robust Optimization
  7. Differentiable Distributionally Robust Optimization Layers
  8. Dual-View of Mixed-Integer Decisions
  9. Energy-Based Surrogate Value Function
  10. Gradient Estimation
  11. Application: Contextual Distributionally Robust Decision-Making
  12. Decision-Focused Learning Pipeline
  13. Prediction-Focused Pre-training
  14. Experiments
  15. Toy Example: Multi-item Newsvendor Problem
  16. ...and 25 more sections

Key Result

Theorem 4.5

Suppose $\mathscr{U}(\boldsymbol{\theta})$ is a parameterized SOC ambiguity set and ass_cost_func and ass_sla hold, then the worst-case expectation $f(\boldsymbol{x},\mathscr{U}(\boldsymbol{\theta}))=\max_{\mathbb{P}\in \mathscr{U}(\boldsymbol{\theta})} \mathbb{E}_{\boldsymbol{y}\sim \mathbb{P}}[c(\

Figures (4)

  • Figure 1: Sequential learning and decision-making pipeline.
  • Figure 2: Decision-focused learning pipeline for contextual distributionally robust decision-making.
  • Figure 3: Experimental results on the multi-item newsvendor problem.
  • Figure 4: Wealth evolution on a 40-dimensional continuous portfolio management problem using decision-focused learning, prediction-focused learning, and method proposed in costa2023distributionally. (In the legend, 'DFL' stands for decision-focused learning, and 'PFL' stands for prediction-focused learning.)

Theorems & Definitions (14)

  • Definition 4.1
  • Definition 4.2
  • Theorem 4.5
  • proof
  • Theorem 4.6
  • proof
  • Corollary 4.9
  • Definition 4.12: bonnans2013perturbation, p.41
  • Theorem 4.13
  • proof
  • ...and 4 more