Decision-Driven Regularization: A Blended Model for Learning and Optimization
Authors: Gar Goei Loke, Qinshen Tang, Yangge Xiao, Xun Zhang
Organizations: Department of Management and Marketing, Durham University Business School, Waterside, Durham DH1 1SL, United Kingdom · Nanyang Business School, Nanyang Technological University, Singapore · Faculty of Business and Economics, The University of Melbourne · International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230036, China
In contextual optimization, the decision-maker seeks optimal decisions to minimize a cost function, that varies based on observed features. This context is common in many business applications ranging from on-demand delivery and retail operations to portfolio optimization and inventory management. In this paper, we study the learning and optimization approach, which first learns how outcomes result from the features, and then selects optimal decisions based on these outcomes. We focus on the integrated learning and optimization literature, and identify that a lack of control for prediction accuracy can lead to overfitting and a loss of decision effectiveness against simple separate learning and optimization models. Instead, we propose a bi-objective formulation that balances prediction accuracy and cost minimization, termed decision-driven regularization. It also addresses ambiguity in the definition of the cost function via a surrogate that depends on a new hyperparameter. We additionally show that alternative perspectives for formulating the problem, namely robust optimization and regret minimization, lead to models that are closely related to our proposed model. As a consequence, our framework generalizes models such as SPO+. Our model is shown to be numerically superior to other benchmarks, such as OLS, Random Forest, XGBoost, SPO+, Perturbation Gradient, and Learning and Rank, in our synthetic studies.
Figures & tables
Figure 1: Visualization of decision for OLS (left) and ILO (right). The green line represents the Ground Truth ( x^1=1.5x^2 ).
Grid Size
(μ,λ)
(0.5,0.25)
(0.5,0.5)
(0.5,0.75)
(0.25,0.5)
(0.75,0.5)
(2,2)
0.02
0.02
0.02
0.02
0.02
(3,3)
0.11
0.11
0.12
0.12
0.11
(4,4)
0.18
0.17
0.18
0.18
0.17
(5,5)
0.31
0.30
0.30
0.31
0.32
Table 1: Average solution time (in seconds) required to compute DDR model.
Figure 2: Performance of DDR against OLS for different parameters of μ and λ
Figure 3: Performance of DDR against SLO models: OLS (left), RF (middle), and XGBoost (right)
Models
In-sample RMSE
Out-of-sample RMSE
5th% -tile
Mean
95th% -tile
5th% -tile
Mean
95th% -tile
OLS
3.8393
3.9416
4.0682
4.1279
4.1846
4.2395
RF
1.5872
1.6531
1.7039
4.3479
4.4255
4.5155
XGBoost
2.9835
3.1140
3.2322
4.4822
4.5833
4.6913
DDR
3.8418
3.9440
4.0705
4.1286
4.1830
4.2352
Table 2: Prediction accuracy of SLO models in terms of root mean square error (RMSE) across multiple datasets
Figure 4: Performance of DDR against ILO models: SPO+ (left), PG (centre) and LTR (right)
Figure 5: Regret reduction of the DDR against OLS under different network structures (left panel) and comparison of DDR against SPO+ for a 5×5 grid network (right panel)
Figure 6: Performance of DDR against ILO models: SPO+ (left), PG (centre) and LTR (right) in the misspecified setting
Figure 7: Performance of DDR against SPO+ in the misspecified setting β=8.0 and N=500 without (left) and with (right) quadratic terms
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Figure B.1: One-dimensional illustrative of centred (dark blue shaded area) and non-centred (broken orange line) confidence intervals
Angles to DDR
Lasso
Ridge
Mean
82°
72°
10th Percentile
80°
72°
90th Percentile
85°
73°
Appendix
Table B.1: Angles of local directions of Lasso and Ridge to DDR in 100 simulations
Models
In-sample RMSE
Out-of-sample RMSE
5th% -tile
Mean
95th% -tile
5th% -tile
Mean
95th% -tile
OLS
3.8393
3.9416
4.0682
4.1279
4.1846
4.2395
DDR
3.8418
3.9440
4.0705
4.1286
4.1830
4.2352
RF(Maximum depth = 2)
4.0068
4.1499
4.2974
4.3693
4.4779
4.5875
RF(Maximum depth = 4)
3.36
3.4682
3.5785
4.3047
4.3904
4.4816
RF(Maximum depth = 6)
2.4572
2.6123
2.782
4.3191
4.3939
4.4766
Appendix
Table D.2: Prediction accuracy of SLO models in terms of root mean square error (RMSE) across multiple datasets
Figure D.2: Performance of DDR against OLS under 3×3 grid and μ=0.75 and λ=0.8 under various settings
Figure D.3: Performance of DDR against SPO+ under 3×3 grid and μ=0.75 and λ=0.8 under various settings
Contextual decision making uses observed covariates to guide decisions under uncertainty. Two natural approaches are predict-then-optimize (PtO), which learns the conditional expected objective, and generate-then-optimize (GtO), which learns conditional uncertainty and optimizes using generated scenarios. We study when each approach is preferable by examining how the choice of learning target affects statistical accuracy and optimization tractability. Although PtO learns a narrower target, our regret bounds show how differences in smoothness can offset GtO's additional dimensional burden. Exploiting smoothness also raises an optimization tradeoff: within the class of nonnegative-weight PtO methods studied, a lower bound limits the gains from additional smoothness, while more flexible methods may introduce signed weights and lose the guarantee of preserving convexity. GtO preserves scenario-wise convexity, but requires learning a richer object and generating scenarios for optimization. Synthetic experiments and a real case study illustrate how these tradeoffs vary with data availability and problem structure. Together with the scope for reusing a model across tasks, our findings provide practical guidance for choosing between the approaches without identifying a universal winner.
L. Jeff Hong, Weihuan Huang, Xinyao Li
Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN 55455, U.S. · School of Management & Engineering, Nanjing University, Nanjing 210093, China
Many real-world combinatorial problems involve uncertain parameters, which can be predicted given contextual features and historical data. These predict-then-optimize' or contextual optimization' problems have gained significant attention: end-to-end training methods can now minimize the downstream task cost rather than the predictive error. However, despite their effectiveness, these decision-focused learning (DFL) approaches often rely on repeated solving of the underlying combinatorial optimization problem during training, making them computationally expensive and difficult to scale. We reframe the learning problem as a cost-sensitive multi-output regression problem: multi-output due to the combinatorial problem having multiple uncertain parameters, and cost-sensitive due to the downstream task cost being the real target. Our technical contribution is the formalization of multiple loss function components that follow from this reframing: cost-insensitive normalization, decision-aware asymmetric penalization of over- and underpredictions, and instance-based costs that mimic the true downstream task-based loss locally. These components require zero or one solve per training data instance, while requiring no further solves during training. Experiments show that the combination of loss components achieves comparable downstream task quality to the state of the art, while being significantly more efficient, enabling scaling to problem sizes that have not been tackled before with DFL.
Noah Schutte, Senne Berden, Tias Guns +2
Delft University of Technology · KU Leuven · Independent Researcher
Decision-focused learning (DFL) trains predictive models by optimizing downstream decision quality rather than standalone prediction accuracy. For contextual linear optimization, most existing DFL methods assume offline data and full observations of the objective cost vector. We develop an on-policy learning method for sequential contextual linear optimization under partial feedback, generalizing the standard bandit feedback setting. Our method learns a stochastic predict-then-optimize policy that samples a cost-vector prediction from a conditional distribution and solves the resulting downstream linear optimization problem. To update this distributional model, we introduce a two-component hybrid gradient estimator. The first component is a score function estimator, which provides an unbiased but potentially high-variance policy gradient estimate. The second is a decision-focused plug-in component that uses an auxiliary nuisance estimate of the latent cost vector to exploit the downstream optimization structure, becoming more informative as the estimate improves. We prove an O(T^-1/2) bound on the average squared policy-gradient norm, matching the standard non-convex SGD rate. Experiments on top-k selection, shortest path, combinatorial pricing, and a real-data energy-scheduling benchmark show that, under bandit feedback, the hybrid gradient approach achieves lower mean cumulative regret than the contextual-bandit baselines on all four benchmarks and than linear Thompson sampling on three, and that it also works with richer conditional generative cost models. Code is available at https://github.com/Joeyetinghan/on-policy-bandit-dfl.
Wyame Benslimane, Tinghan Ye, Pascal Van Hentenryck +1
Department of Industrial Engineering and Operations Research University of California, Berkeley · H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology