# Decision-Focused Learning with Directional Gradients

arXiv: https://arxiv.org/abs/2402.03256 · HTML: https://mlsanity.com/paper/2402.03256/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 20 of 20)

- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [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)
- [A Primer on Zeroth-Order Optimization in Signal Processing and Machine Learning](https://mlsanity.com/paper/2006.06224/) (2020)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Generalization Bounds in the Predict-then-Optimize Framework](https://mlsanity.com/paper/1905.11488/) (2019)
- [Fast Rates for Contextual Linear Optimization](https://mlsanity.com/paper/2011.03030/) (2020)
- [PyEPO: A PyTorch-based End-to-End Predict-then-Optimize Library for Linear and Integer Programming](https://mlsanity.com/paper/2206.14234/) (2022)
- [Disciplined Convex-Concave Programming](https://mlsanity.com/paper/1604.02639/) (2016)
- [Risk Bounds and Calibration for a Smart Predict-then-Optimize Method](https://mlsanity.com/paper/2108.08887/) (2021)
- [Integrated Conditional Estimation-Optimization](https://mlsanity.com/paper/2110.12351/) (2021)
- [Estimate-Then-Optimize versus Integrated-Estimation-Optimization versus Sample Average Approximation: A Stochastic Dominance Perspective](https://mlsanity.com/paper/2304.06833/) (2023)
- [Landscape Surrogate: Learning Decision Losses for Mathematical Optimization Under Partial Information](https://mlsanity.com/paper/2307.08964/) (2023)
- [Decision-Aware Learning for Optimizing Health Supply Chains](https://mlsanity.com/paper/2211.08507/) (2022)
- [Decision-Focused Learning without Differentiable Optimization: Learning Locally Optimized Decision Losses](https://mlsanity.com/paper/2203.16067/) (2022)
- [Debiasing In-Sample Policy Performance for Small-Data, Large-Scale Optimization](https://mlsanity.com/paper/2107.12438/) (2021)
- [Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization](https://mlsanity.com/paper/2106.11180/) (2021)
- [Off-Policy Evaluation with Policy-Dependent Optimization Response](https://mlsanity.com/paper/2202.12958/) (2022)
- [PASTA: Pessimistic Assortment Optimization](https://mlsanity.com/paper/2302.03821/) (2023)

## Built on by (most cited first, 16 of 16)

- [Dissecting the Impact of Model Misspecification in Data-driven Optimization](https://mlsanity.com/paper/2503.00626/) (2025)
- [Online Decision-Focused Learning](https://mlsanity.com/paper/2505.13564/) (2025)
- [Decision-Focused Bias Correction for Fluid Approximation](https://mlsanity.com/paper/2512.15726/) (2025)
- [Decision-Focused Sequential Experimental Design: A Directional Uncertainty-Guided Approach](https://mlsanity.com/paper/2602.05340/) (2026)
- [A Solver-Free Training Method for Predict-then-Optimize](https://mlsanity.com/paper/2606.19587/) (2026)
- [Prediction Loss Guided Decision-Focused Learning](https://mlsanity.com/paper/2509.08359/) (2025)
- [Decision-aware training of spatiotemporal forecasting models to select a top K subset of sites for intervention](https://mlsanity.com/paper/2503.05622/) (2025)
- [Decision-Focused On-Policy Learning for Contextual Linear Optimization with Partial Feedback](https://mlsanity.com/paper/2606.01081/) (2026)
- [Decision-Focused Learning: When and Why Traditional Prediction Models Fail](https://mlsanity.com/paper/2606.21773/) (2026)
- [End-to-End Fairness Optimization with Fair Decision-Focused Learning](https://mlsanity.com/paper/2607.29441/) (2026)
- [Scalable Decision Focused Learning via Online Trainable Surrogates](https://mlsanity.com/paper/2512.03861/) (2025)
- [CloudCons: A Comprehensive End-to-End Benchmark for Cloud Resource Consolidation](https://mlsanity.com/paper/2606.13513/) (2026)
- [Optimizing Regret](https://mlsanity.com/paper/2607.18866/) (2026)
- [When Do Surrogate Updates Improve Decisions? A Local Theory of Trajectory-Wise Transfer](https://mlsanity.com/paper/2608.01130/) (2026)
- [Decision-Driven Regularization: A Blended Model for Learning and Optimization](https://mlsanity.com/paper/2608.15124/) (2026)
- [PolyStepOR: Learning to Decide Without Optimal Decisions](https://mlsanity.com/paper/2609.32465/) (2026)

## Abstract (arXiv)

We propose a novel family of decision-aware surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-optimize framework. The key idea is to connect the expected downstream decision loss with the directional derivative of a particular plug-in objective, and then approximate this derivative using zeroth order gradient techniques. Unlike the original decision loss which is typically piecewise constant and discontinuous, our new PG losses is a Lipschitz continuous, difference of concave functions that can be optimized using off-the-shelf gradient-based methods. Most importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. Hence, optimizing our surrogate loss yields a best-in-class policy asymptotically, even in misspecified settings. This is the first such result in misspecified settings, and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model is misspecified.