# Score Function Gradient Estimation to Widen the Applicability of Decision-Focused Learning

arXiv: https://arxiv.org/abs/2307.05213 · HTML: https://mlsanity.com/paper/2307.05213/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 11 Jul 2023.

## Builds on (most cited first, 19 of 19)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems](https://mlsanity.com/paper/1911.10092/) (2019)
- [Interior Point Solving for LP-based prediction+optimisation](https://mlsanity.com/paper/2010.13943/) (2020)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Differentiating Through a Cone Program](https://mlsanity.com/paper/1904.09043/) (2019)
- [Decision-Focused Learning: Through the Lens of Learning to Rank](https://mlsanity.com/paper/2112.03609/) (2021)
- [Decision Trees for Decision-Making under the Predict-then-Optimize Framework](https://mlsanity.com/paper/2003.00360/) (2020)
- [Contrastive Losses and Solution Caching for Predict-and-Optimize](https://mlsanity.com/paper/2011.05354/) (2020)
- [CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints](https://mlsanity.com/paper/2105.02343/) (2021)
- [Predict+Optimize for Packing and Covering LPs with Unknown Parameters in Constraints](https://mlsanity.com/paper/2209.03668/) (2022)
- [Learning with Differentiable Algorithms](https://mlsanity.com/paper/2209.00616/) (2022)
- [Branch & Learn with Post-hoc Correction for Predict+Optimize with Unknown Parameters in Constraints](https://mlsanity.com/paper/2303.06698/) (2023)

## Built on by (most cited first, 11 of 11)

- [Robust Losses for Decision-Focused Learning](https://mlsanity.com/paper/2310.04328/) (2023)
- [Diffusion-DFL: Decision-focused Diffusion Models for Stochastic Optimization](https://mlsanity.com/paper/2510.11590/) (2025)
- [Sufficient Decision Proxies for Decision-Focused Learning](https://mlsanity.com/paper/2505.03953/) (2025)
- [Decision-Focused On-Policy Learning for Contextual Linear Optimization with Partial Feedback](https://mlsanity.com/paper/2606.01081/) (2026)
- [Learning From Scenarios for Stochastic Repairable Scheduling](https://mlsanity.com/paper/2312.03492/) (2023)
- [Decision-Focused Learning for Complex System Identification: HVAC Management System Application](https://mlsanity.com/paper/2501.14708/) (2025)
- [Scalable Decision Focused Learning via Online Trainable Surrogates](https://mlsanity.com/paper/2512.03861/) (2025)
- [Contextual Preference Distribution Learning](https://mlsanity.com/paper/2603.17139/) (2026)
- [PolyStepOR: Learning to Decide Without Optimal Decisions](https://mlsanity.com/paper/2609.32465/) (2026)
- [RL-PaO: Prediction as Action in Decision Making under Uncertainty](https://mlsanity.com/paper/2609.37065/) (2026)
- [Variance-Optimal Control Variates for Learning with Black-box Feedback](https://mlsanity.com/paper/2610.04766/) (2026)

## Abstract (arXiv)

Many real-world optimization problems contain parameters that are unknown before deployment time, either due to stochasticity or to lack of information (e.g., demand or travel times in delivery problems). A common strategy in such cases is to estimate said parameters via machine learning (ML) models trained to minimize the prediction error, which however is not necessarily aligned with the downstream task-level error. The decision-focused learning (DFL) paradigm overcomes this limitation by training to directly minimize a task loss, e.g. regret. Since the latter has non-informative gradients for combinatorial problems, state-of-the-art DFL methods introduce surrogates and approximations that enable training. But these methods exploit specific assumptions about the problem structures (e.g., convex or linear problems, unknown parameters only in the objective function). We propose an alternative method that makes no such assumptions, it combines stochastic smoothing with score function gradient estimation which works on any task loss. This opens up the use of DFL methods to nonlinear objectives, uncertain parameters in the problem constraints, and even two-stage stochastic optimization. Experiments show that it typically requires more epochs, but that it is on par with specialized methods and performs especially well for the difficult case of problems with uncertainty in the constraints, in terms of solution quality, scalability, or both.