Decision-Focused Machine Learning

• Decision-focused machine learning is a paradigm that couples predictive modeling with combinatorial optimization to directly minimize decision regret.
• It employs end-to-end differentiable techniques, such as continuous relaxation and KKT-based implicit differentiation, to overcome non-differentiability in optimization solvers.
• The approach has proven effective across domains like resource allocation, energy management, and supply chain optimization, often outperforming traditional two-stage methods.

Decision-focused machine learning is a paradigm in which predictive models are trained end-to-end to directly optimize the quality of downstream decisions produced by combinatorial or constrained optimization solvers. Unlike standard approaches that separate learning (often using metrics such as mean squared error or cross-entropy) from decision-making, decision-focused learning (DFL) tightly couples the inferential and optimization stages, using a loss aligned with the true operational objective rather than prediction accuracy. This structural alignment enables the model to learn to prioritize information most consequential for decision quality, often resulting in improved real-world performance in domains ranging from resource allocation and energy management to supply chain and portfolio optimization.

1. Problem Formulation and Motivation

DFL formalizes the learning problem as an end-to-end pipeline: the predictive model maps contextual features to parameters for an optimization problem, and the optimization module returns a decision. The core objective is to minimize the discrepancy between the quality of the decision made using the predicted parameters and the one that would have been made using the true parameters. This is generally formalized as “regret,” given by: $$\text{Regret}(x^*(\hat{θ}), θ) = f(x^*(\hat{θ}), θ) - f(x^*(θ), θ)$$ where $m(y)$ predicts parameters $\hat{θ}$ from features $y$, and $x^*(\cdot)$ denotes the optimizer over the feasible set.

Traditional two-stage predict-then-optimize approaches decouple learning and optimization, training $m(y)$ for statistical accuracy. However, prediction accuracy does not necessarily translate to good decisions if errors in irrelevant directions outweigh small but critical ones. DFL closes the loop, allowing the model to focus on those predictive features that affect the ultimate decision quality (Wilder et al., 2018, Mandi et al., 2023).

2. End-to-End Differentiability: Relaxation and Backpropagation

A central obstacle in DFL is that combinatorial optimization is generally non-differentiable with respect to its parameters. To enable gradient-based learning, DFL frameworks typically employ one or more of the following techniques:

• Continuous Relaxation: Discrete optimization domains (e.g., subsets in $\{0,1\}^n$) are relaxed to their convex hulls, e.g., $conv(\mathcal{X})$. For submodular maximization, the multilinear extension is used to turn set functions into continuous objectives over $[0,1]^n$ (Wilder et al., 2018).
• Quadratic or Entropic Smoothing: For linear and integer programs (LPs or ILPs), quadratic penalties or entropy regularization terms are added. An LP

$\max_{x \in \mathcal{X}} θ^\top x$

becomes

$$\max_{x} θ^\top x - γ‖x‖^2\;\;\;\text{s.t.}\;Ax=b,\; Gx\leq h$$

yielding a unique solution with nonzero Hessian, so the KKT conditions are differentiable with respect to $\hat{θ}$.

• Gradient Propagation: Using the chain rule, the key is to compute $\partial x^*(\hat{θ}) / \partial \hat{θ}$. For smooth and convex objectives, implicit differentiation through the KKT system enables the backpropagation of decision quality loss to the predictive model parameters: $$\frac{d f(x^*(\hat{θ}), θ)}{d ω} = \frac{∂ f(x, θ)}{∂ x} \frac{d x^*(\hat{θ})}{d \hat{θ}} \frac{d \hat{θ}}{d ω}$$ where $\omega$ are the predictive model parameters. For submodular maximization, projected stochastic gradient ascent is used, allowing each step to be differentiable (Wilder et al., 2018).

Table 1 summarizes relaxation strategies for two problem types:

Problem Type	Relaxation Method	Differentiation Tool
Linear programming	Quadratic regularization ($-\gamma‖x‖^2$)	KKT-based implicit diff.
Submodular maximization	Multilinear extension, constraint relaxation, SGA	Analytical dual variables

3. Surrogate and Proxy Losses

Direct minimization of regret can be challenging due to flat regions (zero gradients) in the loss landscape, even after smoothing. Surrogate loss functions are widely adopted:

• SPO+ Loss: For combinatorial optimization, the SPO+ surrogate is a convex upper bound on regret, producing nonzero gradients even in plateaus (Mandi et al., 2023, Mandi et al., 15 Aug 2025).
• Contrastive and Ranking Losses: DFL can be recast as a learning-to-rank problem, with pointwise, pairwise, or listwise ranking losses constructed over subsets of the feasible solution space. These losses encourage the model to induce the same optimality ordering as in the true parameter setting, typically leading to low decision regret even when statistical prediction error is not minimized (Mandi et al., 2021).
• Adjacent Vertex Alignment: In linear optimization, optimality can be characterized in terms of adjacent vertices on the feasible polytope. By penalizing predicted costs that result in the optimal vertex not being ranked ahead of its neighbors, a solver-free surrogate loss achieves both computational efficiency and decision consistency (Berden et al., 28 May 2025).

4. Empirical Evaluation and Real-World Applications

Empirical studies validate that DFL achieves superior decision quality compared to prediction-focused or two-stage approaches, sometimes at the expense of worse statistical metrics such as mean squared error. Notable findings include:

• Budget Allocation (Synthetic): Decision-focused models improve objective values by at least 37% over the two-stage approach, even when MSE is worse.
• Bipartite Matching (LP): DFL yields up to 70% more matches than two-stage or standard accuracy-trained models.
• Recommendation with Submodular Maximization: DFL-trained models produce higher value recommendations measured on the true decision metric, even with lower predictive AUC or cross-entropy.

DFL has been successfully deployed in domains such as supply chain inventory routing, strategic storage arbitrage in power systems, marketing budget allocation with uncertainty, and closed-loop HVAC management. Experimental protocols commonly report regret or normalized regret, decision quality, and feasibility rates where relevant (Wilder et al., 2018, Islam et al., 2023).

5. Advances, Robustness, and Generality

Recent works extend DFL to handle broader uncertainty models and operational constraints:

• Robustness: Systematic studies show that models that achieve low regret are, paradoxically, sometimes more vulnerable to adversarial attacks carefully crafted to move them away from the optimum, even if their statistical errors remain low. This indicates a nuanced trade-off between decision quality and robustness (Farhat, 2023).
• Uncertainty and Proxy Generalization: Methods have emerged to integrate epistemic uncertainty via non-probabilistic models (intervals, contamination, p-boxes), and to design decision proxies that are sufficient for optimality, e.g., via scenario-based sample average approximation or surjective quadratic proxies (Shariatmadar et al., 25 Feb 2025, Schutte et al., 6 May 2025). These approaches guarantee that optimal or near-optimal decisions can be recovered in settings where a single deterministic plug-in prediction would fail.
• Online and Dynamic Settings: Algorithms have been proposed for DFL in online environments, regularizing the optimization mapping and addressing the challenge of non-stationary or time-varying cost functions, with accompanying dynamic regret bounds (Capitaine et al., 19 May 2025).

6. Model Architecture and Implementation Considerations

Implementation of DFL in practice involves several system design choices:

• Choice of optimization solver and differentiability: For large-scale LPs and ILPs, regularization, interior point methods, or differentiable solver layers (e.g., Cvxpylayers, DYS-Net) are adopted. In black-box or non-differentiable contexts, finite-difference or surrogate gradients are employed.
• Computational efficiency: Techniques such as precomputing adjacent vertices (Berden et al., 28 May 2025) or using cached sets for ranking losses (Mandi et al., 2021) alleviate the bottleneck of repeatedly solving large optimization problems.
• Feasibility control in constraints: When parameters in constraints must be predicted, feasibility-aware loss functions combine penalizations for infeasibility and for suboptimal exclusion of the optimal solution. A tunable parameter enables explicit trade-off between these objectives (Mandi et al., 6 Oct 2025).
• Regularization and abstraction: Output space folding and regularized losses reduce overfitting, especially in high-dimensional or limited data regimes, by focusing model complexity on decision-relevant information (Poli et al., 2023).

7. Limitations, Open Problems, and Future Directions

While DFL delivers significant improvements in several decision-centric tasks, ongoing challenges and research directions include:

• Scalability: While surrogate losses and solver-free approaches have improved efficiency, scaling to extremely large combinatorial problems remains nontrivial.
• Unobserved parameters: In many real-world applications, true cost vectors or constraint parameters are not directly observed, motivating the design of surrogate losses that are compatible with only revealed outcomes (Mandi et al., 2023).
• Bilevel formulations and broader optimization landscapes: Deeper theoretical connections with bilevel and multi-objective optimization, as well as more expressive uncertainty modeling (conditional generative models and imprecise decision theory), are active areas for extending DFL’s scope (Wang et al., 8 Feb 2025, Shariatmadar et al., 25 Feb 2025).
• Integration into domain-specific pipelines: Ongoing deployments in fields such as marketing, energy, and supply chain point to practical issues—such as handling counterfactuals and operational constraints—that require further system-level advances (Zhou et al., 18 Jul 2024, Mandi et al., 6 Oct 2025).

A summary of key methodological advances and their typical use cases is as follows:

Technique/Advance	Main Use Case	Reference
Quadratic relaxation/KKT	LP/ILP parameter prediction	(Wilder et al., 2018)
Surrogate/ranking losses	Combinatorial, non-diff. objectives	(Mandi et al., 2021)
Solver-free lava loss	Large-scale LP with geometric structure	(Berden et al., 28 May 2025)
Online DFL, dynamic regret	Non-stationary sequential optimization	(Capitaine et al., 19 May 2025)
Feasibility–aware loss	Constraint parameter prediction	(Mandi et al., 6 Oct 2025)

DFL represents a maturation of the machine learning-for-operations paradigm by aligning statistical learning directly with the metrics that matter in optimization-based systems. Ongoing research continues to broaden its methodological base and range of applicability, especially to problems with complex uncertainty, constraint-driven feasibility, and adversarial robustness requirements.