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Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities

Jayanta MandiJames KotarySenne BerdenMaxime MulambaVictor BucareyTias GunsFerdinando Fioretto

article2024JAIR263 citations

Presents a comprehensive taxonomy of decision-focused learning techniques alongside an open-source benchmark evaluating eleven methods across seven optimization problems to guide end-to-end integration of machine learning and constrained optimization under uncertainty.

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Many critical real-world operations—such as supply chain planning, energy scheduling, financial portfolio management, and transportation routing—require solving complex constrained optimization problems where key parameters are uncertain. Traditionally, organizations use a two-stage "predict-then-optimize" approach: a machine learning model first predicts the uncertain parameters (e.g., travel times or future prices) by minimizing standard prediction errors like mean squared error, and an optimization algorithm subsequently prescribes the optimal decision. However, because standard prediction metrics treat all forecasting errors equally regardless of how they affect the final operational outcome, small prediction inaccuracies can lead to severely suboptimal and costly business decisions.

The main objective of the article is to provide a comprehensive evaluation and survey of Decision-Focused Learning (DFL), an emerging paradigm that integrates machine learning and constrained optimization into an end-to-end training system. By embedding the optimization problem directly into the learning loop, DFL trains predictive models to minimize downstream operational losses—such as decision regret—rather than intermediate statistical prediction errors.

To assess the state of the art, the article analyzes both gradient-based and gradient-free DFL methodologies across diverse algorithmic designs. The authors establish a systematic classification of gradient-based techniques into four primary categories: analytical differentiation of optimization mappings, analytical smoothing, smoothing via random perturbations, and differentiation using surrogate loss functions. To evaluate practical performance, the study establishes an open benchmark spanning seven distinct optimization tasks—including shortest path routing, portfolio optimization, energy-cost-aware scheduling, knapsack allocation, and diverse bipartite matching—using both synthetic benchmarks and real-world empirical datasets.

The investigation yields several critical findings. First, DFL consistently produces higher-quality decisions and lower regret than traditional two-stage methods whenever machine learning models are misspecified or subject to real-world data noise and parameter correlations. Second, direct differentiation through optimization problems faces a fundamental technical barrier: linear and combinatorial optimization models produce piecewise-constant solution mappings with zero gradients almost everywhere, necessitating specialized smoothing or surrogate approximations. Third, while DFL substantially improves operational performance, solving and differentiating optimization models during every training iteration introduces significant computational overhead. Finally, benchmark evaluations demonstrate that surrogate loss techniques (such as SPO+) and perturbation-based methods (such as DBB and I-MLE) offer flexible, solver-agnostic implementations that perform well across combinatorial problems without requiring specialized solver modifications.

These findings indicate that adopting DFL can significantly improve operational efficiency, minimize financial suboptimality, and mitigate risk in automated decision systems. However, implementing DFL requires navigating clear engineering trade-offs between model complexity, solver differentiation techniques, and training runtimes. Standard two-stage prediction pipelines remain computationally faster and asymptotically effective only when the predictive model perfectly captures the data distribution, a condition rarely met in practice.

For practitioners and technology leaders, the article supports adopting DFL in high-stakes operational settings where decision regret directly drives costs. For complex combinatorial tasks, teams should leverage solver-agnostic surrogate losses or perturbation methods, or utilize solution-caching techniques that solve the underlying optimization problem only on a subset of iterations to reduce computational bottlenecks by up to 95%. When moving toward deployment, organizations must conduct careful hyperparameter tuning and evaluate model robustness against label noise and data poisoning. Further research and validation remain necessary for problems with uncertain constraints that risk generating infeasible decisions, as well as for scaling end-to-end learning to large, non-linear integer optimization problems.

KU LeuvenUniversidad de O'HigginsUniversity of VirginiaVrije Universiteit Brusselconstrained optimizationdecision modelsmachine learningDecision-Focused Learninggradient-based decision-focused learningcontextual stochastic optimization
arXiv: 2307.13565PredOpt/predopt-benchmarks

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Abstract

Decision-focused learning (DFL) is an emerging paradigm that integrates machine learning (ML) and constrained optimization to enhance decision quality by training ML models in an end-to-end system. This approach shows significant potential to revolutionize combinatorial decision-making in real-world applications that operate under uncertainty, where estimating unknown parameters within decision models is a major challenge. This paper presents a comprehensive review of DFL, providing an in-depth analysis of both gradient-based and gradient-free techniques used to combine ML and constrained optimization. It evaluates the strengths and limitations of these techniques and includes an extensive empirical evaluation of eleven methods across seven problems. The survey also offers insights into recent advancements and future research directions in DFL.

Topics

constrained optimizationdecision modelsmachine learningoptimizationunknown parameters

Keywords

Decision-Focused Learninggradient-based decision-focused learningcontextual stochastic optimizationsurrogate loss functionsstochastic softmax tricksquadratic programming task lossOWA optimizationsurrogate solvers

Table of Contents

  • 1. Introduction
  • 2. Preliminaries
  • 2.1 Problem Setting
  • 2.2 Learning Paradigms
  • 2.3 Challenges to Decision-Focused Learning
  • 2.4 Optimization Problem Forms
  • 2.4.1 Convex Optimization
  • 2.4.2 Linear Programming
  • 2.4.3 Integer Linear Programming
  • 2.4.4 Integer Nonlinear Programming
  • 3. Review of Decision-Focused Learning Methodologies
  • 3.1 Review of Gradient-Based DFL Methodologies
  • 3.1.1 Analytical Differentiation of Optimization Mappings
  • 3.1.2 Analytical Smoothing of Optimization Mappings
  • 3.1.3 Smoothing by Random Perturbations
  • 3.1.4 Differentiation of Surrogate Loss Functions
  • Discussion
  • 3.2 Review of Gradient-Free DFL Methodologies
  • 3.3 Other Aspects of Decision-Focused Learning
  • 3.3.1 Prediction-Focused vs. Decision-Focused Learning
  • 3.3.2 Multi-task Decision-Focused Learning
  • 3.3.3 Predicting Parameters in the Constraints
  • 3.3.4 Model Robustness in Decision-Focused Learning
  • 3.3.5 Stochastic Optimization
  • 3.3.6 Active Learning Algorithm for DFL
  • 4. Applications of Decision-Focused Learning
  • 5. Experimental Evaluation on Benchmark Problemsets
  • 5.1 Problem Descriptions
  • 5.1.1 Shortest Path Problem on a 5 × 5 grid
  • 5.1.2 Portfolio Optimization Problem
  • 5.1.3 Warcraft Shortest Path Problem
  • 5.1.4 Energy-Cost Aware Scheduling Problem
  • 5.1.5 Knapsack Problem
  • 5.1.6 Diverse Bipartite Matching Problem
  • 5.1.7 Subset Selections
  • 5.2 Experimental Results and Analysis
  • 5.2.1 Comparative Evaluations
  • 5.2.2 Comparison on Runtime
  • 5.2.3 Discussion
  • 6. Future Research Directions
  • 7. Conclusion
  • Acknowledgments
  • Appendix A. Results on All Problem Instances
  • Appendix B. Learning Curves
  • Appendix C. Details about Hyperparameter Configuration
  • References

Knowls

  1. Knowl 1 — Decision-focused learning minimizes downstream decision error

    definition

    In a Predict-Then-Optimize problem, features zz predict unknown objective parameters cc through a model mωm_\omega, and an optimization problem over a known feasible set F\mathcal{F} maps the prediction c^=mω(z)\hat{c}=m_\omega(z) to a decision x∗(c^)x^*(\hat{c}). Prediction-focused learning (PFL) trains the predictor against the observed parameters, typically by minimizing mean squared error. Decision-focused learning (DFL) instead trains for the quality of the resulting decision. For a minimization problem with objective f(x,c)f(x,c), the regret on an example with true parameters cc is

    Regret⁡(x∗(c^),c)=f(x∗(c^),c)−f(x∗(c),c).\operatorname{Regret}(x^*(\hat{c}),c)=f(x^*(\hat{c}),c)-f(x^*(c),c).

    Here x∗(c)x^*(c) is the full-information optimum and x∗(c^)x^*(\hat{c}) is the prescriptive decision. A common DFL training objective is empirical mean regret over NN training pairs (zi,ci)(z_i,c_i), with the inner decision obtained by optimizing under each predicted parameter vector. Zero prediction error implies zero regret, but lower prediction error does not generally imply lower regret: parameter errors that do not change the selected decision can be harmless, while small errors that change it can be costly.

  2. Knowl 2 — A four-class taxonomy organizes gradient-based DFL

    model/method

    The survey organizes gradient-based DFL methods by how they obtain a training signal through an optimization problem: (1) analytical differentiation of optimization mappings, which differentiates optimality conditions when the solution map is smooth; (2) analytical smoothing of optimization mappings, which regularizes a discontinuous or combinatorial problem and differentiates the resulting smooth problem; (3) smoothing by random perturbations, which estimates smooth behavior by solving perturbed problems; and (4) differentiation of surrogate loss functions, which replaces regret or another task loss with a tractable differentiable surrogate. The first three families construct or approximate gradients through the decision map; surrogate-loss methods instead provide gradients for a decision-quality proxy. The survey also distinguishes gradient-free DFL, which trains models without gradient-based updates. Differentiable optimization layers from the first three families can in principle be placed inside a larger neural network and used with different task losses, whereas many surrogate-loss methods target regret specifically and are suited to settings where optimization is the final pipeline stage.

  3. Knowl 3 — Analytical differentiation uses optimality conditions for smooth solution maps

    model/method

    When a parameterized optimization problem has a differentiable solution map, its derivatives can be computed implicitly from conditions characterizing the optimum. For unconstrained smooth convex minimization, differentiating the first-order optimality condition yields the derivative of the optimizer; for constrained convex quadratic programs, OptNet differentiates the Karush–Kuhn–Tucker (KKT) conditions and solves the resulting linear system. For conic programs, a homogeneous self-dual embedding supports implicit differentiation, and differentiable convex layers can transform a broad class of convex programs into conic form before solving and differentiating them. Solver unrolling is an alternative: execute an iterative solver and use automatic differentiation through its iterations. Implicit differentiation of a solver’s fixed-point condition can avoid storing the full unrolled computation, under the required differentiability and convergence conditions. These methods can differentiate parameters in constraints as well as in objectives, but exact derivatives do not by themselves solve the main combinatorial obstacle: LP solution maps are often piecewise constant, with zero derivatives almost everywhere and undefined derivatives at transitions.

  4. Knowl 4 — Analytical regularization smooths linear and integer optimization mappings

    model/method

    For a linear program with objective c⊤xc^\top x and feasible polytope F\mathcal{F}, the optimizer can jump between vertices as cc changes. Quadratic Programming Task Loss (QPTL) smooths this map during training by solving min⁡x∈Fc⊤x+μ∥x∥22\min_{x\in\mathcal{F}} c^\top x+\mu\|x\|_2^2, where μ>0\mu>0 controls regularization. This is a projection-like, differentiable surrogate; the quadratic term is removed at test time, so deployment uses the original optimization problem. Other analytical smoothers add entropy or binary-entropy terms. IntOpt instead uses a log-barrier form for a standard LP, minimizing c⊤x−μ∑ilog⁡xic^\top x-\mu\sum_i\log x_i subject to the equality constraints, with xi>0x_i>0; early stopping in an interior-point method yields a smooth surrogate solution. For integer linear programs, one strategy drops integrality and smooths the LP relaxation. A more systematic alternative generates an LP with cutting planes before applying smoothing, but cut generation can be costly and must be repeated across training instances.

  5. Knowl 5 — Perturbation methods estimate gradients through black-box optimizers

    model/method

    Perturbation-based methods treat the optimizer as an oracle and estimate useful gradients from solutions to nearby perturbed problems, without differentiating the solver internals. Differentiation of Blackbox combinatorial solvers (DBB) perturbs predicted objective parameters in the direction of the downstream loss gradient and uses the difference between the perturbed and unperturbed optimizer outputs as a surrogate gradient. A related negative-identity approximation treats the solver’s backward map as a negative identity; for scale-invariant LP and ILP objectives, the cost vector can be normalized to reduce unstable learning. Differentiable Perturbed Optimizers (DPO) average solutions obtained after random perturbations, producing a differentiable Monte Carlo approximation whose smoothness is controlled by perturbation temperature; smaller temperatures approach the unperturbed solution but can increase estimator variance. Implicit Maximum Likelihood Estimation (I-MLE) combines perturb-and-MAP sampling with a finite-difference perturbation aligned to the loss gradient; it can use Sum-of-Gamma noise and has an adaptive-step-size variant. These approaches accommodate linear-objective problems with discrete feasible sets and can use different black-box solvers, but their estimates involve a smoothing-versus-variance trade-off.

  6. Knowl 6 — Surrogate losses train decisions without differentiating regret directly

    model/method

    The Smart Predict-Then-Optimize (SPO) framework addresses the zero-gradient problem of regret for linear-objective optimization by minimizing the convex SPO+ surrogate. For true cost vector cc, predicted cost vector c^\hat{c}, feasible set F\mathcal{F}, and true optimum x∗(c)x^*(c), the surrogate is

    LSPO+(c^,c)=2c^⊤x∗(c)−c⊤x∗(c)+max⁡x∈F{c⊤x−2c^⊤x}.L_{\mathrm{SPO+}}(\hat{c},c)=2\hat{c}^{\top}x^*(c)-c^{\top}x^*(c)+\max_{x\in\mathcal{F}}\{c^{\top}x-2\hat{c}^{\top}x\}.

    A usable subgradient is x∗(c)−x∗(2c^−c)x^*(c)-x^*(2\hat{c}-c). Under stated distributional assumptions, SPO+ is Fisher consistent for regret; published risk bounds relate low excess SPO+ risk to low excess regret risk. Other surrogate approaches avoid solving the predicted optimization problem to compute the training loss. Noise-contrastive estimation contrasts the true optimal solution with negative feasible solutions; learning-to-rank (LTR) variants use pairwise, pairwise-difference, or listwise losses to align the ranking of candidate solutions under predicted and true costs. These methods can use a solution cache containing observed optima and solutions found during training. The cache can also proxy for the full solver, and solving only a fraction of predicted instances (the survey’s experiments use a 5% solve probability) can reduce solver calls substantially. Learned local convex surrogate losses provide another way to approximate regret without an optimization solver in the training loop.

  7. Knowl 7 — Gradient-free DFL directly optimizes regret for restricted model classes

    model/method

    Gradient-free DFL avoids backpropagating through the optimization map and is therefore compatible with predictive models such as trees or explicitly parameterized linear models. Decision trees and tree ensembles can be learned by recursive partitioning directly against regret, or by formulating the learning problem as a mixed-integer linear program. For linear predictors and binary MILP decision problems, an exact MILP reformulation can partition the predictor’s parameter space at points where the lower-level optimizer changes; the resulting solution is globally optimal for that restricted setting, unlike gradient methods that may find only a local solution. Other methods represent the objective as piecewise linear in predictor parameters and use coordinate descent, locating optimizer transition points with dynamic programming. Divide-and-conquer and recursive branch-and-learn variants extend transition-point approaches to broader classes of problems. These methods trade the flexibility of neural-network gradient training for problem- and model-specific formulations.

  8. Knowl 8 — The benchmark evaluates DFL across seven distinct optimization tasks

    experimental setup

    The survey’s benchmark covers seven Predict-Then-Optimize tasks, all with linear optimization objectives: shortest path on a 5×55\times5 grid; continuous portfolio optimization with a quadratic risk constraint; Warcraft-image shortest path, using a CNN to predict pixel costs; energy-cost-aware scheduling as an ILP; knapsack with predicted item values; diverse bipartite matching with diversity constraints; and top-kk subset selection. The grid shortest-path data use 1,000 training and 10,000 test examples per setting, with a linear predictor and five degrees of model misspecification. Portfolio data are synthetic and vary the nonlinearity degree. Scheduling and knapsack use the Irish SEMO electricity-price data; the scheduling instances contain 10, 15, or 20 tasks, and knapsack capacities are 60, 120, or 180. Matching uses 27 CORA-derived instances with 100 nodes each and varies the diversity requirements. Subset-selection instances have dimensions n=25,50,100n=25,50,100, with k=n/5k=n/5 and 1,000 training samples. The comparison includes the prediction-focused baseline and ten DFL approaches: SPO, DBB, I-MLE, Fenchel–Young, HSD, QPTL, three LTR losses, and MAP contrastive loss. Hyperparameters are selected by validation-set grid search; results use 10 random initialization seeds. Relative regret is the main metric, absolute regret is reported for portfolio cases where relative regret is undefined, and subset selection is evaluated by mismatch rate.

  9. Knowl 9 — Benchmark performance depends on the optimization task and model fit

    empirical result

    No evaluated method performs best across all seven tasks. SPO is comparatively robust across the benchmark, but its ranking changes by task. On synthetic grid shortest path, the prediction-focused baseline performs best when the linear predictor is correctly specified; as misspecification increases, DFL methods can improve on it, with Fenchel–Young performing best in the most misspecified reported setting. On portfolio optimization, SPO is slightly worse than the prediction-focused baseline when the predictor matches the data generator, while several DFL methods—including SPO and Fenchel–Young—are more competitive under misspecification; DBB, I-MLE, Fenchel–Young, and QPTL perform poorly relative to the baseline in this quadratic-constraint problem. On Warcraft shortest path, all tested DFL methods outperform the prediction-focused model, whose regret worsens as image size grows; HSD and QPTL were not run because of computational burden. On scheduling, SPO, MAP, Fenchel–Young, I-MLE, and pairwise-difference LTR are among the strongest, while QPTL and HSD perform poorly on the LP relaxation and standard pairwise/listwise LTR do not stabilize well. On knapsack, QPTL is strongest at capacity 60 and DBB at capacities 120 and 180. Matching results vary with diversity constraints and generally have high relative regret. In subset selection, QPTL and I-MLE have a marginal advantage, while DBB performs worst. The results support task-specific evaluation rather than assuming one universally superior DFL method.

  10. Knowl 10 — Solver calls determine much of DFL training cost

    empirical result

    In the energy-cost-aware scheduling runtime comparison, prediction-focused training is fastest because it does not solve the optimization problem during training. SPO, DBB, I-MLE, and Fenchel–Young take roughly 100 times as long per epoch as the prediction-focused baseline in the reported experiments. Cached MAP and LTR approaches, run with a 5% probability of solving a predicted instance, are substantially faster than the other DFL approaches. QPTL and HSD are not invariably faster than black-box methods despite using LP relaxations: their primal-dual differentiation involves matrix factorization, whereas black-box methods need only an optimizer solution. For scale, the paper reports that Gurobi solves a knapsack instance in about 0.001 seconds and the hardest scheduling instance in about 0.1 seconds. Absolute training times are affected by system overhead, but the comparison shows a practical trade-off between decision quality and repeated solver cost.

  11. Knowl 11 — Open challenges include robustness, generalization, and broader optimization settings

    limitation

    The survey’s principal scope is Predict-Then-Optimize learning in which uncertainty is in the objective and the feasible constraints are known. Predicting constraint parameters introduces the additional risk that a prescribed decision is infeasible under the true parameters, so the task loss must account for correction or infeasibility as well as suboptimality. The paper identifies several wider research needs: generalizing trained models across related optimization instances; handling discrete problems with nonlinear objectives; developing risk-sensitive or distributionally robust DFL rather than optimizing expected regret alone; reducing variance in zeroth-order gradient estimates; scaling bilevel formulations, solution caches, and surrogate solvers; and establishing stronger regret guarantees, particularly with nonlinear constraints. Further open settings include multistage decision problems, optimization layers between multiple ML stages, and multimodal data that predict several interdependent optimization parameters. These are presented as opportunities and unresolved limitations, not as capabilities established by the benchmark.

Coverage note — The survey’s catalog of application case studies is not itemized separately because it primarily illustrates uses of reviewed methods rather than adding a common technical result; the seven benchmark tasks and the broader open challenges are retained.

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Citation

MLA
Mandi, J., et al. “Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities”. Journal of Artificial Intelligence Research, vol. 80, 2024, pp. 1623–701, https://doi.org/10.1613/JAIR.1.15320.
APA
Mandi, J., Kotary, J., Berden, S., Mulamba, M., Bucarey, V., Guns, T., & Fioretto, F. (2024). Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities. Journal of Artificial Intelligence Research, 80, 1623–1701. https://doi.org/10.1613/JAIR.1.15320
Chicago
Mandi, J., J. Kotary, S. Berden, et al. 2024. “Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities”. Journal of Artificial Intelligence Research 80: 1623–1701. https://doi.org/10.1613/JAIR.1.15320.
Harvard
Mandi, J. et al. (2024) “Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities”, Journal of Artificial Intelligence Research, 80, pp. 1623–1701. Available at: https://doi.org/10.1613/JAIR.1.15320.
Vancouver
1. Mandi J, Kotary J, Berden S, Mulamba M, Bucarey V, Guns T, Fioretto F (2024) Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities. Journal of Artificial Intelligence Research 80:1623–1701

BibTeX

@article{Mandi_2024, title={Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities}, volume={80}, ISSN={1076-9757}, url={http://dx.doi.org/10.1613/JAIR.1.15320}, DOI={10.1613/jair.1.15320}, journal={Journal of Artificial Intelligence Research}, publisher={AI Access Foundation}, author={Mandi, Jayanta and Kotary, James and Berden, Senne and Mulamba, Maxime and Bucarey, Victor and Guns, Tias and Fioretto, Ferdinando}, year={2024}, month=Aug, pages={1623–1701} }
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License: https://creativecommons.org/licenses/by/4.0/