Smart Surrogate Losses for Contextual Stochastic Linear Optimization with Robust Constraints
Authors: Hyungki Im
Wyame Benslimane
Paul Grigas
Venue: NeurIPS 2025 poster ()
Keywords: decision-focused learning
contextual stochastic optimization
robust optimization
Primary area: optimization
TL;DR: We study an extension of contextual stochastic linear optimization (CSLO) that involves uncertain constraints and propose a "Smart predict-then-optimize with Robust Constraint (SPO-RC)" loss function.

Abstract: We study an extension of contextual stochastic linear optimization (CSLO) that, in contrast to most of the existing literature, involves inequality constraints that depend on uncertain parameters predicted by a machine learning model. To handle the constraint uncertainty, we use contextual uncertainty sets constructed via methods like conformal prediction. Given a contextual uncertainty set method, we introduce the "Smart Predict-then-Optimize with Robust Constraints" (SPO-RC) loss, a feasibility-sensitive adaptation of the SPO loss that measures decision error of predicted objective parameters. We also introduce a convex surrogate, SPO-RC+, and prove Fisher consistency with SPO-RC. To enhance performance, we train on truncated datasets where true constraint parameters lie within the uncertainty sets, and we correct the induced sample selection bias using importance reweighting techniques. Through experiments on fractional knapsack and alloy production problem instances, we demonstrate that SPO-RC+ effectively handles uncertainty in constraints and that combining truncation with importance reweighting can further improve performance.

Smart Surrogate Losses for Contextual Stochastic Linear Optimization with Robust Constraints
Hyungki Im
Wyame Benslimane
Paul Grigas
Department of Industrial Engineering and Operations Research University of California, Berkeley
Berkeley, CA 94720
fhyungki. im, wyame imann, pgrigas) @berkelyy edu
Abstract
We study an extension of contextual stochastic linear optimization (CSLO) that, contrast to most of the existing literature, involves inequality constraints that depend on uncertain parameters predicted by a machine learning model. To handle the constraint uncertainty, we use contextual uncertainty constructed via methods like conformal prediction. Given a contextual uncertainty set method, we introduce the "Smart Predict-then-Optimii
with Robust Constraints" (SPO-RC) loss, feasibility-sensitty adaptation of the SPO loss that measures decision error of predicted objective parameters. We also introduce convex surrogate, SPO-RC+, and prove Fisher consistency with SPO-RC. To enhance performance, we train on truncated datasets where true constraint parameters lie within the uncertainty sets, and we correct the induced sample selection bias using importance reweighting techniques. Through experiments on fractional knapsack and alloy production problem instances, we demonstrate that SPO-RC+ effectively handles uncertainty constraints and that combining truncation with importance reweighting can further improve performance.
Introduction
Large-scale optimization problems are prevalent across various domains, from supply chain man- agement and financial planning to energy systems and transportation. These problems often involve uncertain parameters, which may appear in both the objective function and the constraints. In practice, these unknown parameters are often influenced by contextual data, which can be leveraged to improve decision-making. While integrating contextual data to estimate uncertain parameters in optimization models has gained significant interest in recent years (see surveys by (aadana al. (2024) and Mandi et al. 2024) and references therein), most of the existing literature focuses on scenarios where these parameters only appear in the objective function. However, in practice, uncertain parameters often arise within the constraints of the optimization problem well (Wang et al.) 2023 Rahimian and Pagnoncelli 2023). Uncertainty constraints introduces additional layers of complexity, as the feasibility solutions with respect to the true parameters is no longer guaranteed.
In this paper, we study an extension of contextual stochastic linear optimization (CSLO) where both the objective function and constraints involve uncertain parameters, each of which can be predicted from contextual feature data. We propose "Smart (SPO-RC) approach, framework that provides reliable method for handling these uncertainties by using robust constraints. Our approach begins by constructing contextual uncertainty sets for the uncertain constraint parameters using existing techniques such conformal prediction (Romano et al.) 2019 Angelopoulos and Bates 202 Fontana et al. (002)). We then formulate contextual robust optimization problem, which provides a safe approximation to corresponding contextual chance
with Robust Constraints"
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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constrained problem and serves as our mechanism for decision-making Within this framework, we introduce new cost metries and their regret forms the SPO-RC and SPO-RC+ losses. These are direct extensions of the SPO and SPO+ losses, designed to handle changing feasibility sets that vary with contextual data. The SPO-RC loss measures the decision error induced by following a robust predict-hen-oppimizz approach, and the SPO-RC+ loss serves as a convex surrogate. Although we construct uncertainty sets for the constraint parameters, the true parameters may sometimes lie outside these sets, which can potentially lead to infeasible solutions. To address this issue, we truncate data points where the uncertainty sets do not cover the true parameters, focusing our model's capacity on regions where feasibility is guaranteed. However, the truncation introduces sample selection bias, which we mitigate using importance reweighting techniques, such as Kernel Mean Matching (KMM). Our contributions can be summarized as follows:
1. We propose the SPO-RC approach. which formulates a contextual robust optimization problem for gpecison-makkngg and enables us learn the objective cost vector in an integrated manner. We propose a convex surrogate called SPO-RC+, and show that Fisher consistency between the proposed SPO-RC and SPO-RC+ losses holds.
2. We introduce learning with truncation to focus on a region where feasibility is guaranteed, and we propose importance reweighting technique to mitigate sample selection bias.
3. We present numerical results on the fractional knapsack problem and alloy production problem. These results demonstrate the relative merits of truncation and importance reweight- ing, the robustness of our approach, and the potential for combining our methodology with enhanced scalability techniques like solution caching (Mulamba et al. 202I).
1.1 Literature review
Contextual stochastic optimization (CSO) involves making decisions under uncertainty using con- textual information to improve outcomes. Sadana et al. (2024) offers comprehensive review of the methodologies developed for CSO, categorizing them into three primary approaches. The first approach, "decision rule optimization". centers on learning mapping from contextual data to de- cisions (Liyanage and Shanthikumar 2005). Research in this area explores a broad spectrum of methods, including linear decision rules (Ban and Rudin 2019). reproducing kernel Hilbert space (RKHS) based decision rules (Bazier-Matte and Delage, 2020; Bertsimas and Koduri, 2022] Notz and Pibernik 2,22), and non-linear decision rules (Huber et al. Oroojlooollooaaaiii
al. 2020
Rychener et al. 2023 Chen et al., 2023 Zhang et al.] 2023). The second approach, "sequential learning and optimization" "SSaaiiin""",,,,, focuses learning the conditional distribution of outcomes based seggiionmmaiing//ppmmiiaae
on contextual data.
methods are built on top of the learning methods,
and include residual-based methods, (Deng and Sen, 2022 Kannan al.. 2021 2025 2024) and importance weighted methods (Bertsimas and Kallus 2020). To address challenges such as the "optimizzer's s curse", robust methodologies have been introduced, including distributionally robust opti- mization (DRO) and regularization technigues (Nguyen et al.] [024]] Bertsimas and Van Parys] [022] Esteban- Pérez and Morales [2022] Lin et al., (022).. Lastly, the "integrated learning and optimization" (ILO) approach, also known as decision-focusee learning, directly trains predictive models with the goal of improving downstream optimization outcomes. This is achieved by minimizing the expected objective value of the solution induced by the prediction models. However, a key challenge is the computational difficulties of training the model by optimizing a loss function that measures decision error. To overcome this challenges, researchers have introduced implicit differentiation methods (Amos and Kolter 2017: Agrawal et al. 2019; Blondel et al.. 2022 McKenzie et al.) 2023 Butler and K Kwon| 2023) and differentiable surrogate loss functions ( (Elmahhtuub et al. 2020 Elmachtoub
and Grigas 2022 Loke et al. 2022; Kallus and Mao 2023 Estes and Richard 2023). Our work
particularly related to the SPO framework (Elmachtoub and Grigas 2022), we extend this
framework to handle problems with uncertain constraints. As pointed out by Elmachtoub and Grigas (2022). the SPO framework can also be viewed as an extension of structured prediction (Taskar et al. 2005 Nowozin et al., 201T), where the SPO+ loss is an extension of the structured hinge loss. Parmentier 2022) also uses structured prediction to learn a predictor that maps hard instances of optimization problems arising in operations research to easier combinatorial instances.
Constructing uncertainty sets conditional on contextual information is important to guarantee the feasibility of the downstream optimization task. Recent work on learning conditional uncertainty sets has explored various approaches, including partitioning methods (Chenreddy et al. 2022), the
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k-nearest ellipsoid method (Ohmori 2021). differentiable optimization Wang et al. 2023 hhereedy and Delage 2024), and conformal prediction (Johnstone and Cox 2021 Patel al., 2024a b). the latter gaining significant interest due to its distribution--rre, madel-agnostic characteristics (Sun at 2023; Stanton et al. 2023). More recently, end-to-end training methodologies that integrate uncertainty estimation with downstream decision-making objectives have been proposed (Yeh et al.) 2024). However, most of these approaches consider uncertainty only in the objective function rather than in the constraints. We introduce a flexible framework that can incorporate multiple methods for constructing uncertainty sets for constraint parameters, with conformal prediction being just one example.
Notation. Given a distribution D, we denote the marginal distribution of x by Dx. Additionally, use the superscript notation 2 to represent i.idd dataset of size sampled from D. We use Im to represent the m-dimensional one-vector.
2 Smart predict-then-optimizze with robust constraints
To motivate methodology, consider the following optimization problem:
Eic T w/x] wERd
min
s.t. P(h(w; a) S 0/x) 2 1 a, w
(1)
where Rd represents the decision variables, E I2 is the uncertain cost parameter, and R is the uncertain constraint parameter, with h IRd x 1m - RR. The set S C Rdd is compact and convex. Contextual data x RRP is used to predict and a, with D denoting the distribution of (x, c, a). The objective is risk neutral in w, while the constraints must be satisfied with probability least a. Due the intractability of the chance-constrained problem, we adopt a common technique in the literature to reformulatc/approximall problem ( as robust optimization problem by constructing an uncertainty set (x)) such that a E l(x) with probability least 1- a. This is not intended as precise guarantee it serves merely as a motivation for the robust formulation that follows. There exist various approaches construct such an uncertainty set 2((-) from data, including methods inspired by conformal prediction. With a suitable (-)), we reformulate problem ① as robust optimization:
min E[c|x] w s.t. h(w;a) S Va E U(x), w E S. E
(2)
Assuming that tractable robust counterpart exists and that feasible solution can be found, the solution to problem (2) is feasible with probability at least - a. The lincarity of expectation simplifies the objective from Elc² w[x] E[c|x] w. Moreover, since the uncertainty set l(()) depends on x, the feasible set of 2 dynamically changes with x. Note that our methodology. and our proposed SPO-RC and SPO-RC+ losses, build directly the contextual robust optimization problem (2), a more tractable alternative to problem (I.
Remark 2.1. For a precise guarantee that the conditional chance constraint ((((w;a) a) S O[x) Z would require that PPa E [(x) | x) 1- a. While achieving this
1 - holds for each stricter conditional coverage may limit the range of techniques available for constructing U()), ), the approach we present is compatible with both marginal and conditional coverage assumptions.
2.1 Cost metrics
We focus on developing an integrated approach to learn the cost vector by estimating the conditional expectation E(c x within a robust predict- predict-then-opee www-ttheenn-neeee pipeline. To achieve this, we introduce a cost metric that measures the quality of solutions derived from our predictions. We consider a hypothesis class F of cost vector prediction models, where each model f R² - Rd maps a context vector x to predicted cost vector è fx). Given è and ú U(x), we define the robust optimization problem ((), as:
P(e,ū):
min ē w s.t. h(w;a) S Va Eú, S. WERd
We denote W We(2244 ú) the optimal solution set of ((),ú), w" w'(e,i ( an arbitrary element of (e,t. We also define S = S(x) as the feasible set for P(e, ú). Given the predictions è,
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and the realized cost vector we introduce the following notation to record the cost of the decision vector (ê,(() with respect to the realized cost vector c:
cost(C, c; := c w² (2,21)
The cost metric above is minimized when the predicted cost vector è exactly matches the true cost vector However, since computing the cost metric can be intractable due to the non-differentiable and discontinuous nature of the oracle w (., U), we introduce the tractable cost metric inspired by the SPO+ loss Elmachtoub and Grigas (2022).
cost (č, 2; == max {(c - 22) w] 28 "www"c (c,(C) WES
For detailed derivation of the cost metric, we refer the reader to Elmachtoub and Grigas (2022). The cost.+ metric is convex with respect to è, making it learnable using first-order methods. Proposition 2.2. Given e. and U, the cost.+ metric satisfies:
1. cost(ê, c;ú})= I cost (c, c;ú) f for all è E R
(1, (c, w² c,(R)) is subgradient 2. cost.+ (ê, c;l{) is convex function of and 2(w" of cost. metric at è.
Given the U()) and dataset D,, Ci, Cj, a.) learn the predictor f by solving:
fe E argmin JEF
Σ cost. (f(x;), c(;((x)))
2.2 Fisher consistency
We show that many desirable properties from the standard SPO and SPO+ frameworks carry over to our setting. Specifically, we show that the Fisher consistency of the cost(ê, c; and cost. metric holds (Elmachtoub and Grigas 2022).
Assuming full knowledge of the data distribution and no restrictions on the hypothesis class, Fisher consistency implies that minimizing the surrogate loss, the cost. metric, also minimizes the true cost metric. This justifies using the cost., metric when it is challenging to directly minimize the cost metric. To formalize this, let Joost and Sooet, be the functions that minimize the expected cost and c metric, respectively:
foust = arg min (x.c) [cost (f(x), cxx())) f
Söoet: arg min (x,,) [cost c(())))
/
Under the assumption of full knowledge of DD and an unrestricted hypothesis class, we aim to show that:
Sicet - J = EE [c|x]
This equality implies that minimizing the cost and cost metric leads to the same optimal predictor, both resulting in [c)x), which the true optimal solution.
Assumption 2.3. To demonstrate the Fisher consistency of the cost and cost. metrics, we assume the following:
1. Almost surely, W* (B(c(xx)), ú) is a singleton for all x E X, where X denotes the domain of the contextual variables.
2. The distribution of c/x is both centrally symmetric about its mean E[c|x] and continuous over the entire space R.d, for all x E X.
3. The interior of the feasible region S(x) is non-empty for all x E X. Theorem 2.4. Under Assumption 2.3 feost and fčoss, satisfy 4 Therefore, the cost4 metric is Fisher consistent with the cost metric.
The proof of Theorem 2.4 (and other proofs) is provided in Appendix
A A key aspect of Theorem 2.4 the assumption that we are working with a well-specified hypothesis class. This ensures that the learned function can capture the true conditional expectation E[c[x] for every x.

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2.3 SPO-RC and SPO-RC+ lass functions
natural extension of the cost metric into a loss function, similar to the SPO approach, compare the cost incurred by the predicted solution with the optimal cost achievable in hindsight, assuming perfect knowledge of the true parameter a. We formally introduce the loss function called the SPO Robust Constraint (SPO-RC), defined as follows:
costt(,, c; - c w" (c, 8a), if ú,
EsrO- nc (ê, c; a) = (As(C), otherwise,
where d := {ā: : à = a} is a singleton uncertainty set. The function As(C) provides an upper bound on the maximum possible difference in objective value over all solutions in S throughout the compact vector space C. It is defined as:
As(C) = sup CEc WES
max c T w- c c/w)
weS
The term c cTw" (a) represents the pure optimal cost, which is achievable in hindsight with perfect knowledge of Specifically, when a & Ui the predicted solution w" (è, 24) might outperform the optimal solution in hindsight, resulting in a negative loss. To ensure that the loss remains nonnegative, we set it to the upper bound (C(C) in this case. The definition of the SPO-RC loss can be naturally extended to the SPO-RC+ loss function as follows:
ECPO-RC, (é, c; ú, a) cost. (ê, c; - cTw'
Although the SPO-RC + loss function is continuous and convex, it serves only as a valid upper bound for the SPO-RC loss when a E U7 as the SPO-RC loss is defined as (C(C) when a & Ú. Therefore, when using the SPO-RC+ loss for training, we must truncate the data to regions where a E to ensure that the loss function serves
valid upper bound.
Remark 2.5. Theorem 2.4 assumes that the hypothesis class F is well specified, which means E [c|x] E F, which is rarely true in practice. Therefore, it can be advantageous to focus the predictive capacity of the model on regions where the feasibility of the solution guaranteed (i.e., when a E U).
3 SPO-RC+ with data truncation and importance reweighting
In this section, we introduce our main SPO-RC+ training procedure with data truncation to remove infeasible data points and importance reweighting to correct for the bias that this introduces.
3.1 Conformal prediction
We first introduce conformal prediction as technique to construct the uncertainty set úí for the parameter a based on the context x. Conformal prediction provides prediction sets that contain the true outcome with specified probability 1 - independent of the data distribution and model assumptions. Specifically, we use the split conformal prediction approach, which begins by splitting the data into two sets: training set x and calibration set D The training set is used to fit predictive model 9 that estimates from x. We then compute conforrtryy scores the calibration set to measure the discrepancy between predicted and actual values. For example, using the lg norm, the non-conformity score 8; for calibration point (x,, a;) calculated as 8; = |la, g(x:)ll2 These scores are used to determine a threshold Q2-a, the (M=++))1-a) quantile of the set {8t hipes This threshold defines the uncertainty set for new observations:
U(x) (ā | 0(x)l12 < Q'-
The following proposition shows that the uncertainty set obtained via split conformal prediction achieves the desired marginal coverage.
Proposition 3.1. Suppose the calibration dataset D and the new observation (Neett atest) i.i.d. samples from Dx.a and l is constructed using split conformal prediction. Then, we have
P(atest Ll (xtes)) 2
This result relies on the exchangeability of the calibration dataset and the new observation, which holds when they are i.i.d. samples. Importantly, Proposition 3.1 is true for any distribution and any model g, regardless of its form or assumptions.
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3.2 Importance reweighting
In Section 2.3 we introduced the SPO-RC and SPO-RC+ loss functions and suggested learning with truncated samples where a L. However, the truncation causes distribution shift, making challenging to learn the true optimal solution E[[[x]. Therefore, we use importance reweighting to correct the truncated distribution D to the true distribution 'D by reweighting the data points.
Suppose U/ is constructed using the split conformal prediction method. We consider learning from the [x, a, c) x (x a. c) E U(x)), where 1(-) the indicator function. Notice that the minimizers feast and feust. in (3) remain unchanged even when replacing the cost/cost., metrics with the SPO-RC/SPO-RC+ loss functions. Therefore, we choose to use the minimizers derived from the cost metrics, allowing us focus on the marginal distributions Dx,e and Dx.c- The difference between these marginal distributions can be decomposed into two components: (1) the difference (xx) versus IP (x), and (2) the difference in PD(c|x) versus [-(()x). However, under the assumption that с and a are conditionally independent given x, we find that PD [|x] x) and IP (c)x) are the same.
truncated distribution 2), which satisfies p>(x,
Assumption 3.2. We assume that conditioned on x, C and a are independent. That is,
PD(c, (P(c)x)FD(aax)) X..
Lemma 3.3. Under Assumption 3.2. the conditional distributions of c given x in the original distribution and the truncated distribution DD are the same. That is,
Po(c|x) (Pp()))), E
Lemma 3.3 relies on the fact that the truncation leading to D depends only on x and a, and not directly on Consequently, the only relevant aspect is the marginal distribution of x in D and D, reducing the problem to a simplified covariate shift problem.
PD( c) cost (f(xx),;2((())) Pr c)
E (xx,)) [cost (f(x), cx(((x)) = (x.c)wDDD.
(x(()
E (x.e)
cost (f)xx)
((xo) [f(x)cost (f(x), c;((())))
(x)
where B(x) (x(x)
[x(x] is importance weight. This relationship implies that, with proper importance weights, we can still learn the true optimal predictor E[c|x] despite the truncation. However, since the true underlying distributions D and D are unknown, we estimate 3 using an empirical importance weighting method. While we use Kernel Mean Matching (KMM) (Huang et al. (006)) estimate 3, any other importance weight estimating methods such as logistic regression-bant
classifiers
(Bickel et al., 2009) and adversarial learning approaches (Ganin et al.] 2016) can be used. Given
truncated dataset Dx {(x,, 6 and target dataset Dxx
((),,,,) j=p
KMM estimates
the importance weight vector / E R= by solving:
1 3T 2
min 3
K3 - T s.t.
Σ
3, - nhs S nsc, 0 SS
S 3 Vi (1.... n.),
(5)
where K xn,, a kernel matrix with entries K, = A(x, x,), K. E R is vector with entries K; Nhe E x;) and & is universal kernel. The parameters B and control the range of the importance weights and the tolerance match the distributions, respectively.
We present our learning algorithm within the SPO-RC framework in Algorithm We use KMM compute the importance reweighting vector and the estimator f is then obtained by minimizing a weighted empirical cost over the truncated data. To implement our approach, we partition the data into four distinct sets: two sets are allocated for split conformal prediction, while the other two are used for training and KMM. Notably, the dataset reserved for conformal prediction can also serve as the target dataset for KMM, thereby simplifying the data preparation process. Finally, we note that Appendix B includes generalization bounds that extend results of El Balghiti et al. (2023) to account for our context-dependent loss function structure and the fact that we may train on rmportance-reweighheed truncated data.

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Algorithm 1: SPO-RC+ with Data Truncation and Importance Reweighting
Input: U. D1" :=
(2, D == {(xx cj R
Initialize I= -}:
for i to 71 do
a (x{)) then
(i);
Define Source data Di = {xx) ct at li)z
+- Solve (5) with source dataset Dr and target dataset D2";
+ min Σ B(xx)) cost+ SEFiEI
Return f;
Cost Curves and Intersections
SPO-RC+ (Dy)
SPO-RC+ (Dn)
Item
Item
[[c]]]
Optimal Decision Boundary
Optimal Dediion Boundary
s[c/|]]
Learned Decision Boundery
Learned Decipon Soundary
Betr
ste
11.7%
Neemrast:: 9.8%
-1.00 -0.75 -6.50
0.50 0.5
050 1:00
1.00 -6.75
6.0
6.3
0.50
1.50
-3000 00.75 00.50
+0.55
0.50
0.25
0.50
0.75
1.00
(a) Cost Curves of and 02
(b) SPO-RC+ Model Dr
(c) SPO-RC+ Model DiR
Figure I: Visualization of the importance reweighting toy example
Numerical experiments
In this section, we present computational results from synthetic data on multiple applications. Unless otherwise specified, the f2-norm is used as the conformal prediction score function in most experi- ments. Throughout, models are trained on three different datasets: the original dataset Do, truncated dataset Dj where instances with & U(x,) are removed, and an importance importance-reweighted dataset DIR adjust for the truncation bias. For KMM, we set the constant B 1000 and e
-mm11
vm
where m is the size of the truncated dataset. Although we choose a large value for B, in most cases the probability ratio did not exceed 20. A Gaussian kernel k(x,,x,) exp (- |x, x, x1) is used. While KMM introduces some computational overhead, particularly for very large m, the number of samples in experiments is medium to moderately large (7 5000 samples), and the primary computational cost comes from training the model under our proposed loss function, rather than from reweighting. The computation of KMM weights is relatively minor (about 1-4 seconds for m 1000). Accordingly, our discussion on scalability primarily concerns training time with the SPO-RC+ loss.
Given a testing dataset n = {xx, Cj. ay P we assess model performance using the normalized metric:
c,: El 4SPO C, Eialee w² (c; 5a.)
NormSPORCTest(2"))
If the induced solution w* ((,, is infeasible, the numerator value lc. w" (c;, & )/ thereby penalizing infeasibility. Our code is built on the open source software package PyEPO TTng and Khalil 2024).
the loss fsPO-RC, (ć,, (x;,((x;), a,) to
Toy examples.
To illustrate the effectiveness of importance reweighting, we present simple toy example with two items. The value of each item is function of the context as depicted in Figure la Our goal is to predict which of the two items has the higher value using a linear regression model for each item. Suppose we randomly remove 30% of the data points where |æ| < 0.5, simulating
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(PH)
(1Pg
(Py)
10e)
SPO (0v)
10mm
0.30
0.20
0.30
0.24
Playnomad Deggnes
Praynoma Degres
Figure 2: Out-of-sample test set (size 3000) NormSPORCTest values of linear models with MSE and SPO-RC+ loss functions, as well as random forests, on Eg norm fractional knapsack instances.
Table 1: Percentage of infeasible solutions of fractional knapsack instances (&2 norm) for different
methods test set (size 3000), with deg. 4.
Models Infeasibility (%)
PTO
MSE (DIR)
RF (Dir)
SPO-RC+ (DIR)
45.00
0.02
0.02
0.02
truncation scenario. Figure I illustrates the resulting case where the linear models are trained using the truncated dataset DT. Due to truncation, the predicted value for item consistently exceeds the predicted value for item across all x, leading to incorrect selection of item for certain values of x. This truncation effect, however, can be mitigated by training the models on an importance reweighted dataset DIR- As shown in Figure [c] the learned decision boundary (indicated by the vertical red dotted line) closely aligns with the true optimal decision boundary (indicated by the vertical green dotted line), highlighting the effectiveness of the importance reweighting approach. Additional toy examples and detailed explanations can be found Appendix C.I
Fractional knapsack problem using &g conformity score. problem, with robust formulation:
We consider the fractional knapsack
max c
w
s.t. aT Tw < ba (x)), 1 1, E [0.1]
WERd
where (((x) (ā: l á|l2 I Q' 1 is constructed using conformal prediction with & norm, and à 0(x) is the estimated weight vector from the predictor g. This problem can be reformulated as second-order cone program (SOCP). We generate synthetic data with polynomial kernel models to detailed explanation of the data generation and model
describe the dependencies of and deseription can be found in Appendix C.2
Figure 2 compares the out-of-sample NormSPORCTest performance of three different models, including linear models with either MSE or SPO-RC+ loss functions and random forest models, trained on three datasets Do. Dr, and DIR, across varying levels of the polynomial degree dictating the complexity of the generated cost and constraint vectors. We observe that linear models trained with the SPO-RC+ loss have comparable performance to the non- non-parametri random forest model, and both outperform the linear regression model (MSE). (In this experiment only, we include random forests to illustrate the use of both parametric and nonparametric methods in robust constraints setting. In later experiments, we focus on linear models only to better isolate and evaluate our core contributions.) In terms of dataset variations, models trained on DIR yield better results in some cases, especially when the training size is small (n = 1000). Specifically, we observe that SPO-RC+ on DIR outperforms SPO-RC+ on Dj in nearly every scenario. Table Tableeppresee presents the percentage of infeasible solutions for each model. The PTO model refers to the predict-then-optimzz method, which uses the predicted å directly without uncertainty set. Table[]]shww that all methods that solve the robust constraint problem yield almost no infeasible solutions, while almost half of the solutions from the PTO approach are infeasible.
Fractional knapsack problem using é conformity score. We consider the fractional knapsack problem with (x) constructed using conformal prediction with E norom. The resulting robust
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counterpart, constructed using the estimated à - g(x) and (z) { (āāaa fā-ll1 S Q 3 can be reformulated as a linear program (LP). Figure 3a compares the out-of-sample NormSPORCTest performance of different linear models using either MSE or SPO-RC+ loss functions and across the datasets Do. Dr, and DIR. We observe that for small values of the polynomial degree, the parameter dictating the complexity of the generated cost and constraint vectors, the models trained with MSE slightly outperform those trained with SPO-RC+. However, the performance differences remain marginal in this regime. As expected by observations in the literature (Elmachtoub and Grigas 2022). as the complexity the relationship between features and cost vectors increases, the SPO-RC+ models consistently outperform the MSE models.
We also use this example to demonstrate how our methodology can be effectively combined with techniques to enhance scalability. The main scalability challenge of our method arises primarily during model training, particularly when using the SPO-RC+ loss function. As shown in Table 2 which reports the training times of our method with both MSE and SPO-RC+ losses on small and large datasets, the scalability of our approach is closely tied to the choice of the surrogate training loss employed. Consequently, our method directly benefits from any scalability improvements developed for pocused learning (DFL) methods. To demonstrate this, we adopt a randomized solution caching strategy inspired by Mulamba et al. (202I), who proposed caching solutions to reduce the number of solver calls during training. Specifically, in our case, each iteration of training requires evaluating the +PO-RC+ loss (at least once) and calculating a corresponding gradient by solving robust optimization problem. For each data point, which has its own uncertainty set and corresponding robust problem, maintain a cache of past solutions and, every iteration, whenever needed, either return the best solution from the cache or re-solve the problem. As the fraction of time that perform a fresh re-solve gets smaller, Figures 33 and [3d] demonstraee that the quality of models produced by the SPO-RC+ loss, across all three datasets, only slightly degrades, while the overall training time dramatically improves.
Table 2: Training time comparison (in seconds) for models using MSE and SPO-RC+ on the knapsack and alloy production problem. Results reported for two instance sizes (n and 10), each trained on datasets of 1000 and 5000 training samples.
Method
MSE (DIR)
MSE (DIr)
SPO-RC+ (DRr)
SPO-RC+ (DIR)
Training data
1000
5000
1000
5000
Knapsack (n=5)
6.86
32.90
30.76
157.96
Knapsack (n=10)
8.67
35.65
44.36
214.60
Alloy production (n=5)
5.73
24.01
54.81
280.58
Alloy production (n=10)
6.98
33.24
95.04
427.95
WOND 110-00
Teaiin Sibe 1O0 Tan 2013
Twring Sax 150) Tee 2000
Sox 100 Tex 380
Traniig Soe 10K. Six 200
2.55
014
6.3
30
0.7
622
24
03
60
(a) Knapsack (l1)
(b) Alloy production
(c) Knapsack (21)
(d) Knapsack (e1)
Figure 3: Performance comparison across different tasks and evaluation criteria: (a)-(b) report out-of-sample test set (size 3000) NormSPORCTest performance under varying cost complexities for the fractional knapsack and alloy production problems, respectively. (c)-(d) evaluate the impact of the caching strategy in the knapsack problem with & f--ormm costs, using out-of-sample NormSPORCTest and training time as metrics.
Alloy production problem (Hu et al. (203)). In alloy production process, a plant must produce a target alloy that requires specified minimum quantity of different metals. Suppliers provide ores that contains certain amount of the metals required for production. Since the true chemical

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compositions of the ores are uncertain, we formulate the production problem as robust covering LP:
min C w s.t. a/w 2 h2 Va, E Uxxx), = =1,-. m, w 2 0, WERd
where 6; is the cost per unit of ore from each supplier i, hj is the required quantity of metal j and aj IR is the concentration vector of metal j. where each entry represents the concentration of metal j in the ore across different suppliers. These concentrations may be uncertain but could be predicted based on certain chemical characteristics. We build independent prediction model for each constraint, each using conformal prediction with é2 score, leading to an SOCP reformulation of the robust problem above. detailed explanation of the data generation process and model description can be found in Appendix C.3
We run our experiments required. Training times are reported in Table The results in Figure(3b] show similar pattern as Figure faloor for the knapsack problem, where the MSE and SPO-RC+ models perform comparably for simpler cost functions, while SPO-RC+ consistently outperforms MSE as the complexity of the cost vector increases.
brass production scenario where two metals (Zine and Copper)
Conclusion and future directions
We propose the SPO-RC framework to learn the cost predictor in an integrated way while ensuring solution feasibility. The SPO-RC loss plays central role by penalizing uncertainty sets V that are either overly conservative or fail to capture the true parameters. While effective, the current approach has two key limitations: it applies only to problems with linear objective, and it relies two-step procedure that learns the uncertainty set independently of the cost coefficients. Developing related method address nonlinear objectives and/or fully joint learning procedure are promising directions for future work. Another potential direction for future work involves utilizing contextual uncertainty sets and robust constraints within other structured prediction applications, such the work of Parmentier (2022) on learning simpler approximations of difficult variants of well-known optimization problems.
Acknowledgments
This research was supported by NSF Al Institute for Advances in Optimization Award 2112533.
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A Missing proofs
A.1 Proof of Theorem 2.4
Our proof follows similar steps to those in Elmachtoub and Grigas 2022). Since the hypothesis class F is unrestricted, optimize the function values f(x) individually for each E X. Therefore, solving the problems
Foos E arg min (x, [cost (f(x),
foos E arg min E (x.e)_Dx.e [cost+ (f(x),
is equivalent optimizing each [(x separately. Consequently, for the remainder of the proof, we fix Xp. and also Ug (xx), and consider only the conditional distribution of We define the risks associated with the cost and cost metrics as:
Reoat (c) == Ee cost (c. co:c
(6)
Reost, (2) = Ee cost+ (c. (c,c:
where the Ee denotes the expectation propositions needed to complete the proof of Theorem 2.4.
c. Let define Eee [c/xo]. We first list the
Proposition A.1 (Proposition of Elmachtoub and Grigas (022 If cost vector e a minimizer the other hand, if
of (eoa()), then W"(c", Uu) S WF,Ce.o))
cost vector such that
(" Uo) is singleton and ("(c", (i)) S (2,) then e* is a minimizer of Reou(-).
Proposition A.2 (Proposition 6 of Elmachtoub and Grigas 2022
Under Assumption 2.3, the
unique minimizer of Reost (-)-
Since have fixed to Xo. the uncertainty set Uo is also fixed. Therefore, Propositions A. and A.2 reduce to those presented in Elmachtoub and Grigas (2022), and their proofs follow accordingly. Importantl, these propositions hold true when the constructed uncertainty set satisfies Assumption 2.3, regardless of whether the true parameter a lies within Li, not. This means do not need to be concerned about the quality of to to guarantee consistency when learning with the cost metric or SPO-RC+ loss function. (Indeed, recall the MSE loss will also guarantce consistency.) However, since we do know the true distribution and certain assumptions such having well-defined hypothesis class F often do not hold, this motivates us to focus on the region where feasibility is guaranteed and subsequently cost and SPO-RC+ yield valid upper bounds, as discussed in our main paper. We complete the proof of Theorem 2.4 using the above propositions.
A.2 the expected cost vector E[c[x,] the unique minimizer of Ress, (-). That is, ferst (Xg) is unique and fr (xo)
Proof. Let Xy E X' be given and let úñ ((xx). By Proposition
E[c|x,]. Under Assumption 2.3, the optimal solution set W (Elc Xy)] (b) is singleton. Applying Proposition A. we conclude that E[c|xØ] is a minimizer of Rena( )) Since this holds for every E X, have that almost surely Soeet is unique, fcost [[|[x], and fčost; also minimizes (xxe)) [cost (f(x), cx)({))]. This shows the Fisher consistency between the cost and cost.,
metrics. Moreover, because the füust and füust, remain the same when using SPO-RC and SPO-RC+ loss functions, respectively, this implies Fisher consistency between the SPO-RC and SPO-RC+ loss functions well.
A.2 Proof of Lemma 3.3
Proof. The truncated distribution D satisfies (x a, a) x Pp(x, a c) e U(x)), where I(-) is the indicator function. Therefore, we have
P P a c) (a(
Pplccxx, a) ((x)))()))))))))))))))
=Po(clx, a)),( = =Po(c|x)P, a) (by Assumption 3.2).
21

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To find the marginal distribution Fx(x, c), we sum over all possible a:
P P c) = = ((x)))))(()) x (
Dividing both sides by (x) gives - P
B Extended theoretical results
B.1
Generalization bound
this section, we present additional theoretical results, specifically generalization bounds for the cost metric. This extends the generalization bounds presented in El Balghiti al. (2023) to accommodate context-dependent feasibility sets. We define the population risk of function / with respect to the cost metric as
RD( == E(x.c)-Dx.c [cost cx
and denote its empirical risk over n samples collected in dataset D as
RB
cost (f(xi),
The multivariate Rademacher complexity P (F) class FF with respect to the cost metric is defined
(Bertsimas and Kallus 2020) of the hypothesis
9 cost (F) Ex E,
sup JEF
cost (f(x:), cx,)((x)) ==1
where a; are i.id Rademacher random variables for = 1, We denote the quantity (Cs(C) as an upper bound on the maximum possible objective value over all solutions in S across the entire space C. Specifically, it is given by:
Os (C) sup CEC
max c w wES
By applying the cost metric to the renowned result from Bartlett and Mendelson (002), obtain the following theorem.
Theorem B.1 (Theorem from (Bartlett and Mendelson, 2002)). ) Let F be hypothesis class and let > O. The following inequality holds with probability at least over an i..dd dataset D for all fe
log(1/5)
RD()) S RB(() 2 (F) + (s(C))
(7)
2n
In particular, when the hypothesis class F consist of linear functions, we can further bound the Rademacher complexity R (F) in terms of the sample size n and other relevant quantities. We define $ == (S(x) bound on their radius (())= = max max ||w]] to characterize the size of these sets.
X') as the collection of all possible feasible sets and introduce the upper
SES WES
Proposition B.2 (Corollary 3 of El Balghiti et al. (023 Fi (x + Bx|B E dddxa is the linear hypothesis class, then we have
Anh (Fim) < (C) 2p log (2np(S)d)
1
Notice that the extension can be easily made by adjusting the definition of p(S), which characterizes the size the feasible sets. By incorporating the Rademacher complexity bound from Proposition
B.2 into 77, obtain a generalization bound for the linear hypothesis class in our framework.
In addition to ensuring consistency when truncating, importance reweighting allows us extend the generalization bounds presented in Theorem B. The following lemma provides bound on the difference between the empirical risks calculated under the true distribution and the importance- reweighted truncated distribution.
22

---

Table 3: Comparison of NormSPORCTest and optimal decision boundary across different datasets
Data
R
Rg
Rc
Opt Boundary
Do
8.4%
12%
11%
T = J22
Dr
12.2%
9.2%
15.8%
Not Intersect
DIR
8.4 %
12.2%
11.2%
(22g)
(22a)
Imppotance Rewelghting
Original
Optimal Decision Boundary
Opeimal Decsion Boundary
Leamee Dre wion Boundary
Learned Decision Boundary Bimm
(requency
Itmm
0.00
1,0
-0.25 5.00 0.25
(a) Histograms
(b) SPO-RC+ Model on Do
(c) SPO-RC+ Model DIR
Figure 4: Visualization of the truncation toy example
Lemma B.3 (Lemma 4 from Huang et al. 2006 Suppose have knowledge of 3(x) E [0, B] and let 8 > O. With probability at least 1 over n i.i.d. samples from the true distribution D, and
their corresponding truncations drawn from the truncated distribution D. we have for all f E F:F:
B +1
(f) R R (f)) (1 +1++) /2log(2/8))
where BD represents the truncated distribution adjusted for the importance weight B. Using Lemma
B.3 we can replace R% (J) in the generalization bound 7 with Rn (f") thus ensuring that our
generalization analysis remains valid when using importance-reweighee truncated data.
Proposition B.4. Suppose we have knowledge of B(x) B). let F be a hypothesis class, and let ò > O. The following inequality holds with probability at least 1 - ó over an i.i.d. dataset D and
its corresponding truncated dataset for all f F:F
(V 1og((10)
B² +
RD(l) < RD (f) + 2 cost (F) + Sss(C)
+ (1 (21og(2/0))
2n
71
C Additional experimental details and results
In this section, we provide additional details the numerical experiments presented in Section 4 and some additional results. The experiments were conducted on MacBook Pro equipped with an Intel chip and 16 GB of RAM.
C.1 Additional details and results toy examples
This section provides a more detailed explanation and additional toy examples supplementing the toy example presented in Section 4 To illustrate the effectiveness of importance reweighting and truncation, we consider a simplified version of the fractional knapsack problem. Specifically, we consider case where goal is to predict which of two items has higher value based uniformly distributed context x E [-1, The true relationships between the item values and C2 and the context & given by:
1
Ci = -r ST, O2 g
1)2 2.75 S2.
23

---

where (1 and Le are i..d. normal random variables with mean 0 and standard deviation 0.1. We
generate 1.000 samples of (z, C1, C2). Figure shows the scatter plot of these samples along with their true conditional expectations Ee(e [ar] and E(c2| .r]. The curves intersect at two points, denoted
Ti and 22, partitioning the interval [-1, 1] into three distinet regions: = [-1, B = [.1, F22.
and C = [.2 1.. Our goal is to use linear regression to predict e₁ and c2 and make a decision based on which predicted value is higher. This is a special case of the fractional knapsack problem where
the uncertain weight constraint is replaced with wi + w2 1 without any uncertainty.
C.I.1 Importance reweighting example
In regions A and C, we observe that c2 < CJ. whereas in region B, C₁ < 02. Since the optimal decision changes across these regions and linear models may not capture the non-linear relationships perfectly, the decision made by the models will be incorrect in at least one of these regions. For instance, Figure shows the learned decision boundary (red vertical line) of a linear model, which incorrectly predicts item 1 in region A, even though the true optimal choice is item 2. Notably, the optimal decision boundary (green vertical line in Figure Ic] makes incorrect decisions only in region A, where the difference hetween the two true curves is minimal. In fact, the area between the curves can be quantified using the NormSPORCTes, metric on samples from each region, denoted as R.. RB, and Rc, corresponding to regions A, B, and C, respectively.
Suppose we randomly remove 30% of the data where |x:| < 0.5, reducing samples in region B. This shifts the smallest region from A to (as shown Table 3 thereby changing the optimal decision boundary. However, by applying importance reweighting, we correct for this and realign the decision boundary with the true distribution, as shown in Table 3 Figure 1b and Ic illustrate the learned SPO-RC+ models Dr and DIR- Notice that the learned decision boundary of SPO-RC+ on DIR. the model with importance reweighting, is much closer to the true one than the model without (SPO-RC+ Dr). Additionally, the NormSPORCTest measured with 500 test samples was lower in SPO-RC+ on DiR, indicating better performance.
C.1.2 Truncation example
In this example, we compare SPO-RC+ on Do with SPO-RC+ DIR, to show the effectiveness of truncation. Building on the setting of the previous toy example, we introduce an additional capacity constraint W2 d2 where a2 equals 100 if < 0.8 and d2 equals O otherwise. This makes the decision (Wi, w2) = (0, 1) infeasible when x >> 0.8. We use a linear regression model to predict 02 and construct the uncertainty set l/ using conformal prediction with a = 0.25. In the region where < 0.8, we set d2 large so that the constructed uncertainty set ú is large enough to include the original feasibility set defined by wi + W22 = 1, for all x E [-1, 1]. Thus, this example equivalent to the previous one, except that the induced optimal solution can now be infeasible when x > 0.8.
Figure Cajure4aasher shows the histograms of each dataset, indicating that all samples region C' (including the possible infeasible range [0.8, 1)) are truncated. This means that for æ in the region C, a2 & U. Figures and |4 show the learned linear models from SPO-RC+ on Do and DIR, respectively. The decision boundary learned from DIR closely matches the true optimal boundary, whereas the model trained on Do produces less accurate solution. Evaluated on a dataset sampled from the cussibllit-guurraneed region (x < 0.5), SPO-RC+ on Do performs poorly, achieving a NoRRSPO of 16.7%. In contrast, SPO-RC+ on DiR performs almost perfectly, with NormSPORCTest value of 0.2%. Furthermore, in the region x > 0.8, the induced solution of SPO-RC+ on Do is (wi, wx²) = (1, 0). which is infeasible. These results highlight the effectiveness of truncation when model complexity is limited and suggest focusing our learning capacity on the feasibility- -urolittgg region.
C.2 Additional details on the fractional knapsack instances
Data generation process: We consider a fractional knapsack problem with dl = 5 items and = 10 features. We adapt a popular data generation process in the literature, starting with [llmachtuu and Grigas (2022). We sample each element of the context vector x E TRP from the uniform distribution
Uniform( -1, 1), and generate the cost vector c. as:
( (Bexi),
deg.
5
Cij 3.5deg
+3
+ 10
+ GGg.
vD
24

---

where i = 1,.... indexes the instances, j .,..., d indexes the items, and Ej ~ (0,0)). The matrix B. 0, 1dxp has elements sampled from binomial distribution with probability 0.5. The parameter deg, is fixed integer that determines the complexity of the cost function. Similarly, the weight vector generated as:
5 3.5der
(Baxi) +3
dog. (p- 1xilll)e
atj
where B, and ij are generated in the manner as B, and e respectively. Notice that the noise ag increases with |x:||1. We fix deg, 4 for the weight function, while varying deg. 2, 4, 6, to increase the complexity of the cost function, with 6 = 20.
When the f--norm score is used for conformal prediction, the resulting robust reformulation takes the form of an SOCP:
Model description for 2--n conformal prediction score:
max C W s.t. Q' w|l2 S a w. 1 w = 1, w E [0, 1j
To predict the weight vector å,
one-layer neural network with ReLU activation functions. We set a = 0.2 to compute the conformal prediction quantile. For the cost vector è, we evaluate three models: Linear Regression (MSE), Random Forest (RF), and SPO-RC+ using linear model. All models are trained using the Adam optimizer. The learning rate set to 1 x 10-3 for the MSE and RF models, and 10-3 for the SPO-RC+ model. Training conducted for 50 epochs with early stopping based on validation loss to prevent overfitting.
Model description for -oorr conformal prediction score: When the l f-norm score is used for conformal prediction, the resulting robust reformulation can be formulated as an LP:
max C W S.t. ER
Q!="|[[[le> S b a aTw, w, 1I w = w E 1j.
To predict the weight vector å, we use one-layer neural network with ReLU activation functions, and use a = 0.2 to compute the conformal prediction quantile. For the cost vector è, we evaluate both Linear Regression (MSE) and SPO-RC+ using linear model. For the SPO-RC+ model, warm start our method with MSE for faster training. All models are trained using the Adam optimizer. The learning rate is set to 10 for the MSE and RF models, and x 10 for the SPO-RC+ model. Training is conducted for 50 epochs with early stopping based on validation loss to prevent overfitting.
C.3 Additional details the alloy production instances
Data generation process: For the brass production scenario, we sample each element the context vector and the cost vectors using the same strategy used for the fractional knapsack problem. For each ore, we generate the concentration according to a Gamma distribution
G.
(Gamma P.+) ck; 0} for each supplier =1 1,...,d
where P Rmmxd is base preference matrix independently sampled using uniform distribution [0.1,1]. Since brass production requires around 30% of Zinc and 70% Copper to produce the alloy, set the requirements to [2.9, 7.1].
Model description: When the & -orm score used for conformal prediction, the resulting robust reformulation can be formulated an SOCP:
min c TTw w s.t. Q'~"|| S a h2 Vj 1.... WERd
0,
To predict weight vector å, we use a one-layer neural network with ReLU activation functions. The conformal prediction quantile is computed using two models: Linear Regression (MSE), and SPO-RC+, again using linear model class and with the MSE solution warm start. All models are trained using the Adam optimizer. The learning
0.2. For the cost vector é, we evaluate
25

---

Table 4: Percentage of infeasible solutions of alloy production instances for different methods test set (size 3000), with deg. = 8.
Models Infeasibility (%)
PTO
MSE (DIR)
SPO-RC+ (DIR)
30.894
0.194
0.188
rate is set to 10-2 for MSE and 4 x 10-2 for SPO-RC+. Training is run for 50 epochs with an adaptive learning rate scheduler and early stopping based on validation loss to prevent overfitting, Table shows that all methods that solve the robustly constrained problem yield almost infeasible solutions, while almost 30% of the solutions from the PTO approach are infeasible.
26


Reviews:
[{"review_id": "rqDWv44MQj", "reviewer": "Reviewer_Gr2F", "summary": "The paper introduces an extension to contextual stochastic linear optimization (CSLO) that incorporates uncertain constraints predicted by machine learning models. The authors propose a \"Smart Predict-then-Optimize with Robust Constraints\" (SPO-RC) approach, which uses contextual uncertainty sets to handle constraint uncertainty. This, in its turn extends SPO for problems with uncertainty in the constraints. This is done through the convex surrogate loss, SPO-RC+, and authors demonstrate its effectiveness through experiments on problems like fractional knapsack and alloy production. The approach includes data truncation and importance reweighting to address sample selection bias, showing improved performance in handling uncertainty.", "questions": "please see \"Strengths And Weaknesses\"", "limitations": "please see \"Strengths And Weaknesses\"", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Overall, I think this is a nice addition to the literature of both domains (SPO and stochastic optimization), that potnetially can have a big impact...\n\nStrengths\n\n- The SPO-RC framework effectively incorporates uncertainty in constraints, which is a significant advancement over traditional methods that primarily focus on uncertainties in the objective function.\n\n- The introduction of the SPO-RC+ loss as a convex surrogate allows for efficient optimization using first-order methods, making the approach computationally feasible for large-scale problems (similar to non-stochastic version of SPO+).\n\n- The approach is validated through experiments on practical problems like fractional knapsack and alloy production, demonstrating its effectiveness in real-world applications. Although having large scale experiments would be nice.\n\nWeaknesses\n\n- The need for data truncation and importance reweighting adds layers of complexity to the implementation, which might be challenging for practitioners without a strong background in these techniques.\n\n- My guess is that the use of importance reweighting to correct for sample selection bias could lead to overfitting, especially if the reweighting is not carefully calibrated.\n\n- While the method is designed to be scalable, the additional computational overhead from constructing uncertainty sets and performing reweighting might limit its applicability to very large datasets or real-time decision-making scenarios. Would be great to validate this experimentally. \n\n- The approach assumes certain properties of the data distribution (e.g., conditional independence of cost and constraint parameters), which may not hold in all practical situations, potentially affecting the robustness of the results.", "was_revised": true, "final_justification": "My concerns were addressed during rebuttal.", "rebuttal": "We first thank you for reading our paper and giving us valuable comments. Let us now answer your questions.\n\n\n1. **The need for data truncation and importance reweighting adds layers of complexity to the implementation, which might be challenging for practitioners without a strong background in these techniques. My guess is that the use of importance reweighting to correct for sample selection bias could lead to overfitting, especially if the reweighting is not carefully calibrated.**\n\n**A:** We agree that data truncation and importance reweighting may seem to add complexity to the implementation. However, in practice, these steps can be implemented with only a few additional lines of code and do not significantly increase the computational burden for most standard learning pipelines.\nWe would also like to clarify that our motivation for introducing truncation and importance reweighting is primarily theoretical: these components verify that our SPO-RC+ training loss is a valid upper bound on the true regret and they allow us to derive meaningful generalization bounds under sample selection bias. That said, as shown in our experimental results, the performance of the truncated and reweighted model (denoted as $\\mathcal{D}\\_{\\text{IR}}$) does not always outperform its non-reweighted ($\\mathcal{D}\\_T$) and non-reweighted/non-truncated counterparts ($\\mathcal{D}\\_o$).\nThis suggests that while the theoretical advantages are valid, practical performance may vary due to factors such as overfitting or inaccurate estimation of the density ratio used for reweighting -- issues that the reviewer rightly pointed out.\nIn practice, one can choose the best among $\\mathcal{D}\\_o$, $\\mathcal{D}\\_T$, and $\\mathcal{D}\\_{\\text{IR}}$ based on validation performance, which provides a pragmatic way to mitigate the potential drawbacks of reweighting or truncation when they are not beneficial.\n\n2. **While the method is designed to be scalable, the additional computational overhead from constructing uncertainty sets and performing reweighting might limit its applicability to very large datasets or real-time decision-making scenarios. It would be great to validate this experimentally**\n\n**A:** Thank you for raising this important point. We agree that the reweighting step may introduce some computational overhead, particularly for very large datasets (e.g., with over 100,000 samples). However, in our experiments involving medium to moderately large datasets, we observed that the primary computational cost comes from training the model using our proposed objective, rather than from the reweighting step itself. The time required to compute the importance weights (generally taking around 1 to 4 seconds for 1000 samples) is relatively minor by comparison.\n\n\n3. **The approach assumes certain properties of the data distribution (e.g., conditional independence of cost and constraint parameters), which may not hold in all practical situations, potentially affecting the robustness of the results.**\n\n**A:** Thank you for pointing out this important consideration. It is true that our method applies importance reweighting under the assumption of conditional independence between the cost and constraint parameters given the contextual variable $x$. However, this assumption is reasonable in settings where $x$ sufficiently captures the information that determines both $c$ and $a$—a common scenario in many practical applications where rich contextual features are available.\nWe would also like to clarify that this assumption is only required for the theoretical justification of the importance reweighting step. It is not necessary for the convergence or validity of the core SPO-RC method itself. Even without importance reweighting, SPO-RC remains applicable."}, {"review_id": "JA7LXR5Kmn", "reviewer": "Reviewer_NQYV", "summary": "Building on the SPO framework, the paper studies the SPO-RC loss and its convex surrogate to address CSLO problems with inequality constraints that depend on uncertain parameters. The paper establishes the Fisher consistency of the estimator and addresses bias caused by data truncation through importance reweighting techniques.", "questions": "Does the estimation error of the density ratio affect the coverage guarantee?\n\nCould existing results on SPO be applied to approximately solve the problem by introducing an additional penalty term in the objective function?", "limitations": "The paper discusses limitations of the linear objective and the two-step procedure. The issue of sample inefficiency could merit further consideration.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper tackles the interesting problem of solving CSLO with inequality constraints that involve uncertain parameters. It proposes valid methodological components, including the use of conformal prediction for constructing uncertainty sets and importance reweighting for bias correction.\n\nHowever, the paper lacks sufficient explanation of the SPO framework for readers unfamiliar with the concept. The approach may suffer from inefficiency, as the amount of data available for model training is reduced due to data truncation in SPO-RC+ and data splitting in conformal prediction.", "was_revised": true, "final_justification": "Most concerns are addressed. One concern remains that the convex surrogate, the data truncation, and the subsequent importance-reweighting can degrade performance in finite samples, motivating the need for a more elegant solution. The manuscript is a valuable contribution to the community. The score is raised to 4.", "rebuttal": "We first thank you for reading our paper and giving us valuable comments. Let us now answer your questions.\n\n\n1. **However, the paper lacks sufficient explanation of the SPO framework for readers unfamiliar with the concept.**\n\n**A:** Thank you for this helpful suggestion. Due to space constraints, we were unable to provide a full explanation of the SPO framework in the current version, and we acknowledge that this may make the paper less self-contained for readers unfamiliar with the concept. As the areas of decision-focused learning and contextual stochastic optimization are maturing, we hope that many readers will be already be familiar with the framework and/or that our provided references will be helpful to gain the background knowledge.\n\n\n2. **The approach may suffer from inefficiency, as the amount of data available for model training is reduced due to data truncation in SPO-RC+ and data splitting in conformal prediction.**\n\n**A:** Thank you for your insightful comment. We acknowledge that using split conformal prediction to construct the uncertainty set can reduce the effective size of the training dataset. However, we present split conformal prediction as one possible approach to build context-dependent uncertainty sets, chosen for its simplicity and well-established theoretical guarantees.\nWe agree that exploring alternatives—such as full conformal prediction or other data-efficient methods—would be a valuable direction for future work to mitigate this issue.\nAdditionally, we would like to clarify that the data truncation step in SPO-RC+ reduces the dataset by only a small constant fraction $\\alpha$ in expectation (e.g., about $\\alpha = 5\\%$), which is relatively minor overall and compared to the reduction from data splitting in conformal prediction.\n\n\n3. **Does the estimation error of the density ratio affect the coverage guarantee?**\n\n**A:** Thank you for the thoughtful question. We interpret the “coverage guarantee” in this context as referring to the convergence guarantee of our method—please feel free to correct us if this interpretation is inaccurate.\nUnder this interpretation, we note that the estimation error of the density ratio does indeed affect the convergence guarantee of the importance-weighted variant of SPO-RC. Specifically, inaccurate estimation of the density ratio can lead to biased learning and the generalization bound guarantee might not hold.\nHowever, this does not impact the general convergence guarantees of the core SPO-RC method itself, which does not rely on importance reweighting.\n\n4. **Could existing results on SPO be applied to approximately solve the problem by introducing an additional penalty term in the objective function?**\n\n**A:** We interpret your suggestion as referring to an exact or inexact penalty method, where the constrained optimization problem is approximately solved by incorporating the constraint into the objective function via a penalty term. This is an interesting suggestion, and something to possibly consider as a baseline, however we did not consider it due to its difficulties with ensuring feasibility when there is uncertainty in the constraint parameters. Indeed, if one expects a model to be able to perfectly predict $a$ based on $x$, i.e., there is no noise, then this approach may work well because it will be enough to ensure feasibility with respect to the predicted parameters. However, as soon as there is some uncertainty, i.e., some noise in the relationship, then other techniques are needed to ensure that the resulting decision will be feasible w.r.t. the true parameters $a$. This is a major motivation for the robust optimization methodology in the literature across many years and is our reasoning for considering our robust approach as well. In our paper, we primarily focus on scenarios where feasibility with a very high probability is a primary concern and thus we discarded many other approaches that do not promote a high probability of feasibility. For example, Table 1 shows that the PTO approach, which does something similar to what you suggest, has several orders of magnitude larger probability of infeasibility than our approaches."}, {"review_id": "rdB3OvusNd", "reviewer": "Reviewer_sXBi", "summary": "This paper proposes Smart Predict-then-Optimize with Robust Constraints (SPO-RC), a framework that extends existing predict-then-optimize approaches to handle constraint uncertainty by constructing contextual uncertainty sets through conformal prediction.  In addition, the authors present a convex surrogate, SPO-RC$_+$, to SPO-RC and theoretically and empirically demonstrate the efficacy of their approach.  \n\nOverall, this paper's contribution is substantial, as the authors extend existing predict-then-optimize approaches to constraints in a sound and reasonable manner, which, to the best of my knowledge, has not been done within the contextual optimization literature to date.", "questions": "- Is there a reason within this particular setting that RF models would not be practically as scalable?\n- Have the authors considered naive baselines of just penalizing constraint violation in the objective function and using standard SPO as a stronger baseline?  \n- Do the authors have comprehensive results on out-of-sample performance and feasibility for the alloy benchmark?  \n- Can the authors provide (and include in the appendix) the training times for all methods to give a clear picture of the scalability trade-offs?  \n\n\n\n### Minor Remarks\n- In the camera-ready version, the appendix and main paper should be combined into a single document.  \n-  I have reviewed this paper at a previous conference, and the authors have made some progress towards addressing my concerns by including more experiments, although they are still limited.", "limitations": "See weaknesses.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- **[S1]**: The methodological contributions of this paper are a significant strength.  Predict-and-optimize has become an increasingly growing area of research, and principled methods for problems with constraint uncertainty have remained elusive.  This paper bridges that gap by proposing one of the first approaches to addressing constraint uncertainty, building on SPO+ (a widely used loss function) through the establishment of tractable surrogates, training procedures, and appropriate theory.  \n- **[S2]**: Computational results.  Although the experiments have certain limitations, which will be discussed in the weaknesses section, the authors demonstrate the efficacy of their method on two benchmarks.\n- **[S3]**: Clarity.  The paper is well-structured, and each aspect is well-motivated.  \n\n\n### Weaknesses\n- **[W1]**: The computational experiments are relatively limited.  Both non-toy benchmarks are still extremely small instances, making any empirical claims, in my opinion, weak.  That said, this is a broader trend in the decision-focused learning literature, so this is within the standard, especially given that the authors consider constraint uncertainty.  \n\n- **[W2]**:  Scalability is another weakness of the suggested approach.  In contrast to existing predict-and-optimize approaches that solve a single-level linear or integer optimization problem, this method relies on solving a robust optimization problem, which is generally less efficient.  Given the authors claim that random forests not as scalable as linear models (technically accurate but not often not practically relevant) and avoid comparing to them on the alloy problem, I do encourage them to have a more transparent discussion around scalability limitations and transparent reporting of the time required to train each of the models.  It should be possible to mitigate these issues using approaches similar to those employed to improve efficiency in the objective uncertainty case. However, the paper would benefit from a discussion on this topic to inform readers about the limitations, rather than a single sentence in the contributions.   \n\n- **[W3]**:  Baselines/reporting.  The authors omit results for random forests for the alloy benchmark.  Additionally, essential metrics, such as infeasibility, are not presented for this benchmark.  I do acknowledge that the space is limited, but these could be given in the appendix for a more comprehensive evaluation.  \n\nThe paper’s conceptual advance outweighs the empirical shortcomings, so I lean toward acceptance rather than rejection.", "was_revised": true, "final_justification": "See response to authors in comments below.", "rebuttal": "We first thank you for reading our paper and giving us valuable comments. Let us now answer your questions.\n\n1. **Is there a reason within this particular setting that RF models would not be practically as scalable?**\n\n**A:** Thank you for pointing this out. Our main reason for including Random Forest results in the knapsack example was to illustrate the use of both parametric and non-parametric methods in our robust constraints setting. Similar to Elmachtoub and Grigas (2022), we also wanted to showcase how the SPO-RC+ loss function can perform comparably to a non-parametric method like random forests, when there is enough training data. That being said, the more \"apples to apples'' direct comparison is between the MSE and SPO-RC+ loss functions as they use the same architectures and will scale similarly modulo the challenges of dealing with the a more complicated loss function. We felt that the main point about random forests (and non-parameteric methods more broadly), which scales differently, was made on the first experiment we presented and therefore decided to drop this benchmark from additional experiments. \nWe realize that the sentence in the current version of the paper —\n“(In this experiment only, we compare with random forests to demonstrate a comparison of linear models with a non-parametric method; as random forests is not as scalable, we drop it from the other experiments)” — is an oversimplification of the above discussion and may have caused confusion. In the revised version, we will clarify that scalability was not the main reason for excluding RFs from the later experiments. Rather, we wanted to keep the focus on models that best help evaluate the core contribution of our approach.\n\n2. **Have the authors considered naive baselines of just penalizing constraint violation in the objective function and using standard SPO as a stronger baseline?**\n\n**A:** Thanks for the suggestion. We interpret your suggestion as referring to an exact or inexact penalty method, where the constrained optimization problem is approximately solved by incorporating the constraint into the objective function via a penalty term. This is an interesting suggestion, and something to possibly consider as a baseline, however we did not consider it due to its difficulties with ensuring feasibility when there is uncertainty in the constraint parameters. Indeed, if one expects a model to be able to perfectly predict $a$ based on $x$, i.e., there is no noise, then this approach may work well because it will be enough to ensure feasibility with respect to the predicted parameters. However, as soon as there is some uncertainty, i.e., some noise in the relationship, then other techniques are needed to ensure that the resulting decision will be feasible w.r.t. the true parameters $a$. This is a major motivation for the robust optimization methodology in the literature across many years and is our reasoning for considering our robust approach as well. In our paper, we primarily focus on scenarios where feasibility with a very high probability is a primary concern and thus we discarded many other approaches that do not promote a high probability of feasibility. For example, Table 1 shows that the PTO approach, which does something similar to what you suggest, has several orders of magnitude larger probability of infeasibility than our approaches. Note that we only considered the PTO approach in this one Table to illustrate that it does not promote feasibility and then discarded it as a baseline for this reason. We expect the penalty approach to be similar. \n\n3. **Do the authors have comprehensive results on out-of-sample performance and feasibility for the alloy benchmark?**\n\n**A:** While we did not evaluate the RF model for the alloy production and knapsack benchmarks under an $\\ell\\_1$-based conformity score, most of the out-of-sample performance benchmarks are reported in Figure 3. As for the out-of-sample infeasibility results, they are consistent with those from the knapsack experiments under the $\\ell\\_2$-conformity score.\n\n| Dataset Metric       | PTO     | MSE ($\\mathcal{D}\\_\\text{IR}$) | SPO-RC+ ($\\mathcal{D}\\_\\text{IR}$) |\n|----------------------|---------|-------------------------------|-----------------------------------|\n| Infeasibility (%)    | 30.894  | 0.194                         | 0.188                             |\n\n**Table:** Percentage of infeasible solutions of alloy production instances for different methods on a test set (size 3000), with degc = 8.\n\n\n4. **Can the authors provide (and include in the appendix) the training times for all methods to give a clear picture of the scalability trade-offs?**\n\n**A:** For the training times of the alloy production example, the following table summarizes the findings:\n\n| Method                  | MSE ($\\mathcal{D}_\\text{IR}$, 1000) | MSE ($\\mathcal{D}_\\text{IR}$, 5000) | SPO-RC+ ($\\mathcal{D}_\\text{IR}$, 1000) | SPO-RC+ ($\\mathcal{D}_\\text{IR}$, 5000) |\n|-------------------------|--------------------------------------|--------------------------------------|------------------------------------------|------------------------------------------|\n| Knapsack (n=5)          | 6.86                                 | 32.90                                | 30.76                                     | 157.96                                    |\n| Knapsack (n=10)         | 8.67                                 | 35.65                                | 44.36                                     | 214.60                                    |\n| Alloy production (n=5)  | 5.73                                 | 24.01                                | 54.81                                     | 280.58                                    |\n| Alloy production (n=10) | 6.98                                 | 33.24                                | 95.04                                     | 427.95                                    |\n**Table:** Training time comparison across methods and training data sizes.\n\nWe will include the training times in the final version."}, {"review_id": "SFWzo6QrCQ", "reviewer": "Reviewer_E3FA", "summary": "The paper is motivated by providing a smart predict-then-optimize approach for chance-constrained contextual stochastic optimization:\n$$\\min_w\\mathbb{E}(c^\\top w|x) \\text{ s.t. } \\mathbb{P}(h(w,a) \\leq 0|x) \\geq 1-\\alpha,$$\nwhere $c$ and $a$ are random (and unobserved at decision time) but correlated with $x$.  \nA smart predict-then-optimize approach involves building a family of mappings (or conditional distributions) from context $x$ to decision $w$, and training it in an end-to-end manner, i.e., finding the mapping in the class that leads to the minimum expected cost with respect to the (unknown) distribution over $x$, $c$, and $a$.\n\nThe paper introduces a smart predict-then-optimize approach for contextual stochastic optimization with robust constraints:\n$$\\min_{w\\in S}\\mathbb{E}(c^\\top w|x) \\text{ s.t. } h(w,a) \\leq 0, \\text{ for all } a \\in \\mathcal{U}(x),$$\nwhere $\\mathcal{U}(x)$ is the robust feasibility set.  \nThe robust feasibility set is defined (learned) a priori, using an estimate-then-optimize approach (as opposed to a smart predict-then-optimize approach). Here, a conformal prediction technique is used to build this uncertainty set.  \nOnce this is fixed, a smart predict-then-optimize approach is used to learn the objective.  \nThe SPO+ (structured hinge) loss is extended to this case.  \nDecisions to be imitated are obtained by solving the anticipative problem (where the random elements are known).  \nThey may therefore not be feasible for the robust problem (since the constraints for the anticipative problem are weaker as they are not robust).  \nInfeasible solutions cannot be used as targets with the SPO+ loss.  \nAn importance sampling-style reweighting of the feasible solutions in the training set is proposed to compensate for the fact that the infeasible ones have been removed.  \nNumerical experiments evaluate the results.", "questions": "- Why do you present the chance-constrained problem (1) first? Your methodology works only for the robust reformulation (2), which is generally not equivalent to (1) if we want the robust reformulation.\n- How novel is the conformal approach for the robust set? The methodology looks standard.\n- Line 92: \"However, most of these approaches consider uncertainty only in the objective function rather than in the constraints.\" The challenge they represent for SPO approaches has been highlighted by [2]. They deal with a chance-constrained problem in the numerical experiments using an SPO approach that imitates the same single scenario solution. They note that chance constraints are poorly dealt with by using learning only for the objective (numerical experiments on chance-constrained CSP). Do you answer in some way to the challenge they raise? In other words, if we focus on the uncertainty in the constraint, does your SPO approach help in terms of the quality of the solution returned compared to using only the robust approach?\n- Can you give examples of large-scale industrial applications where implementing your approach would be practical?\n- Suppose you forget about the chance-constrained problem and take as input the (fully reformulated) robust problem. All the uncertainty is now in the objective. What is new in your SPO approach compared to applying directly ElMachtoub and Grigas' approach to that problem? Apart from the infeasible solutions... as feasible targets could perfectly be computed in that setting by taking the best solution (with respect to the cost known a posteriori) that is feasible for the robust problem.\n- My first guess (and I would like you to explain why I am wrong) is that imitating the solution mentioned just above would give better performance than reweighting the training set only on the feasible solutions in the way you do. Indeed, it would use more of the information available. And the paper mentioned above underlined that SPO on the objective is bad at approximating the constraint, so using the realization of the uncertain parameter in the constraint is, in my opinion, not useful in practice.", "limitations": "- The main limitation, in my opinion, is that the uncertainty set $\\mathcal{U}$ is not learned but given. As such, the resulting problem (for this fixed $\\mathcal{U}$) can be recast as a usual contextual stochastic optimization problem. The only difference is that the decision we want to imitate may not be feasible (the reason why reweighting is needed) in the loss.\n- Robust reformulations tend to be computationally complicated, so I guess the approach would not scale. Would it?\n- Only linear objectives are addressed.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths:\n- Contextual stochastic optimization with uncertainty in the constraints is definitely interesting, especially the chance-constrained one (1), but also the second.\n- The general methodology is sound, and the propositions make sense.\n- While the different tools used are not new (SPO+ loss, importance sampling, conformal predictions), their combination for the robust problem is interesting.\n\n### Weaknesses:\n- The fact that the uncertainty set $\\mathcal{U}$ is not learned is a key limitation in practice. Indeed, only the objective is learned in a Smart Predict-then-Optimize way, which is known to be a limitation. As a consequence, the method suffers from the same limits as robust approaches in terms of scalability (the problems are hard to solve practically).\n- One could also argue that, because of this limitation, the improvement with respect to the literature is only incremental. Indeed, once $\\mathcal{U}$ has been fixed, this is a classic SPO setting, and the paper only applies the classic methodology. The only difference is the fact that some anticipative solutions to imitate are not feasible, leading to the importance reweighting.\n- The problems in the numerical experiments are small-scale.\n- Some references are missing. In particular, the SPO+ loss was known 20 years before as the structured Hinge loss in the ML community [1, Chapter 6]. And Line 92, \"However, most of these approaches consider uncertainty only in the objective function rather than in the constraints.\". Such problems and the challenge they represent for SPO approaches have been highlighted by [2]. \n\nSince the difficulty in the chance-constrained optimization is linked to the constraints, and those are dealt with using a classic robust approach, the methodology is not truly a smart predict-then-optimize approach for the chance-constrained optimization problem, but rather for the objective of the problem with robust constraints. This does not prevent publication in my opinion, since the problem is already interesting, but it should be stated more clearly in Section 2 and the introduction (as it is already mentioned in the title).\n\n[1] Nowozin, S., & Lampert, C. H. (2011). Structured learning and prediction in computer vision. Foundations and Trends® in Computer Graphics and Vision, 6(3–4), 185-365.\n\n[2] Parmentier, A. (2022). Learning to approximate industrial problems by operations research classic problems. Operations Research, 70(1), 606-623.", "was_revised": true, "final_justification": "This is a good paper. The contribution is more theoretical than practical.", "rebuttal": "We first thank you for reading our paper and giving us valuable comments. Let us now answer your questions. Because of the word limit, we briefly quoted each weakness/question mentioned in the review.\n\n1. **Since the difficulty in the chance-constrained optimization is linked to the constraints . $\\cdots$ Why do you present the chance-constrained problem (1) first? $\\cdots$**\n\n**A:** Thank you for this helpful observation. We agree with the reviewer’s interpretation of the problem we are addressing—that our methodology focuses on the objective optimization under robust constraints, rather than directly solving the chance-constrained optimization problem. We chose to present the chance-constrained problem (1) first to motivate the need for robustness and to provide a natural progression toward the robust formulation (2). Our goal was to illustrate how the robust approach emerges as a practical and tractable alternative when direct optimization under chance constraints is intractable or impractical. Due to space limitations, we were somewhat brief in clarifying this distinction in the current version. We will make sure to emphasize this point more clearly in the introduction and Section 2 of the camera-ready version.\n\n2. **How novel is the conformal approach for the robust set? The methodology looks standard.**\n\n**A:** Thank you for your question. As discussed in the literature review, there has been growing interest within the operations research community in using conformal prediction to construct data-driven uncertainty sets. In our work, we adopt conformal prediction as one way to construct context-dependent uncertainty sets based on the available contextual features $x$, but we would like to emphasize that it is not the only method that can be used within our framework.\nTo clarify the contribution of our paper: our goal is not to propose a novel method for constructing uncertainty sets per se, but rather to develop a unified framework that integrates existing techniques—conformal prediction and robust optimization—into our algorithm for decision-focused learning of the objective function. In this sense, we are leveraging well-established results from robust optimization and combining them with recent advances in predictive modeling to address a challenging class of contextual decision-making problems.\n\n3. **Suppose you forget about the chance-constrained problem and take as input the (fully reformulated) robust problem. $\\cdots$  My first guess (and I would like you to explain why I am wrong) is $\\cdots$**\n\n**A:** Thank you for the detailed and thoughtful question.\nThe key difference from the original SPO framework is that we model a context-dependent uncertainty set $\\mathcal{U}(x)$, whereas the original SPO assumes a fixed feasible set across samples. This distinction is critical when constraint parameters are uncertain, because the feasible region varies with $x$. \nTo the best of our understanding, we interpret the suggestion—“feasible targets could perfectly be computed in that setting by taking the best solution (with respect to the cost known a posteriori) that is feasible for the robust problem”—as corresponding to the approach of training on the full dataset without any truncation or reweighting. In our framework, this is exactly what we denote by $\\mathcal{D}\\_o$.\nIn our experiments, we observe that the truncated and reweighted versions ($\\mathcal{D}\\_T$ and $\\mathcal{D}\\_{\\text{IR}})$ do not consistently outperform $\\mathcal{D}\\_o$. This reinforces the point that truncation and reweighting are not always beneficial in practice-which aligns well with the reviewer’s opinion.\nOur primary motivation for introducing truncation and importance reweighting is theoretical: these steps allow us to derive generalization guarantees under sample selection bias. However, as our empirical results show, the practical gains are not guaranteed and may be affected by factors such as overfitting or poor estimation of the density ratio.\nIn practice, we recommend selecting among $\\mathcal{D}\\_o$, $\\mathcal{D}\\_T$ and $\\mathcal{D}\\_{\\text{IR}}$ based on validation performance, providing a flexible and pragmatic approach that adapts to the data at hand.\n\n4. **Line 92: \"However, most of these approaches consider uncertainty only in the objective function rather than in the constraints.\" The challenge they represent for SPO approaches has been highlighted by [2]. $\\cdots$**\n\n**A:** Many thanks for pointing us to [2]. There are some interesting connections to that work that we had not considered before. Let us first comment on your question and then further on the connections to [2].\n\nIn our view, our work takes the constraints (e.g., modeled with chance constraints or robust constraints) as hard constraints in the problem -- they must be satisfied with a very high probability. This justifies the robust approach, and leaves open the possibility of using a decision-focused (or SPO-type) approach for dealing with other parts of the problem, e.g. the objective function, that are risk-neutral. We demonstrate one way to accomplish this by forming the SPO-RC+ loss. In this way, our methodology ensures that the constraints will be satisfied with a high probability while the objective is learned in a decision-focused way. There is no guarantee of course that the chance constraints are dealt with in an ``optimal'' way as the robust approach is just a safe approximation and furthermore there is room in the future to also consider the possibility of learning the constraints in a decision-focused way (or adapting ideas from other works that do this), which we think is a valuable future direction.\n\nSo to answer your question -- in a nutshell our approach is interesting and valuable when there is uncertainty in both the constraints and objective. Uncertainty in only the constraints reduces to robust optimization with a predictor that is not learned in a decision-focused way, and uncertainty only in the objective reduces to SPO. Our approach thus yields better quality solutions for the objective while ensuring feasibility w.r.t. the uncertain constraints with high probability.\n\nBased on our reading of [2], we think there is potential to apply an approach related to ours to the problem setting of that paper, although there are many open research questions clearly. To do so, two essential steps to apply our approach would be:  (i) replace imitating a single scenario solution with imitating a robust solution, e.g., using a robust problem as the simpler ``well-known'' problem, and (ii) one would have to consider a more involved cost structure in the objective, e.g., cost of delays or travel time, that is treatd in a risk-neutral way. One could possibly then use a standard model with conformal prediction to learn the constraints in a robust way while using a structured prediction type loss to learn the objective. Again, these are very high level ideas and many details remain, but this seems like a promising research direction to us.  \n\n5. **Some references are missing. In particular, the SPO+ loss was known 20 years before as the structured Hinge loss in the ML community [1, Chapter 6]**\n\n**A:** Thanks for pointing to these references, we appreciate it. We will certainly mention the connection to structured prediction, as pointed out in the original Elmachtoub and Grigas (2022) paper, in our camera ready version. The connection to [2] is quite interesting and, as mentioned above, is something that we had not considered before, so we appreciate that. We will certainly also mention this connection in our camera ready version.\n\n6. **Can you give examples of large-scale industrial applications where implementing your approach would be practical?**\n\n**A:** While we have not (yet) collaborated with a real industrial partner to implement our algorithm, we feel that it is promising. Since our approach leverages well established methodologies -- including conformal prediction, robust optimization, and SPO+/structured hinge loss -- it can be smoothly integrated into existing pipelines. Furthermore, since this paper represents basic methodological research, our approach is applicable to a broad range of problems beyond those considered in the experiments, for example it could be applied to other optimization tasks such as battery storage operation [3]\n\n[3] Yeh, C., Christianson, N., Wu, A., Wierman, A., and Yue, Y. (2024). End-to-end conformal\ncalibration for optimization under uncertainty. arXiv preprint arXiv:2409.20534.\n\n7. **The main limitation, in my opinion, is that the uncertainty set $\\mathcal{U}$ is not learned but given.**\n\n**A:** In our framework, the uncertainty set $\\mathcal{U}(x)$ is learned once from data—by first training a constraint-parameter predictor $g(x)$ and applying conformal prediction to construct $\\mathcal{U}(x)$—and is then held fixed while training the objective-cost model $f$; we believe the reviewer’s comment concerns this latter stage, and we view jointly learning $f, g,$ and an adaptive $\\mathcal{U}$ that balances robustness and effectiveness as a promising direction for future work. In the paper’s conclusions and future-directions section, we acknowledge this limitation and flag this extension as a “fully joint learning procedure,” and we intend to pursue it by leveraging recent methods for data-driven uncertainty-set learning.\n\n8. **Robust reformulations tend to be computationally complicated, so I guess the approach would not scale. Would it?**\n\nThank you for your question. We agree with the reviewer that robust reformulations can introduce additional computational complexity. To mitigate this issue, we incorporate recent techniques in our implementation, such as warm-starting, which help reduce the overall computation time during training. These practical adjustments make our approach more scalable, especially in the context of the moderately sized problems considered in our experiments."}]

Author rebuttal:

Meta-review: This paper introduces SPO-RC and its convex surrogate SPO-RC+, extending decision-focused learning to contextual stochastic optimization with uncertain constraints. 
The contribution is regarded as significant and well-grounded: the theory is solid, the methodology integrates robust optimization with predict-then-optimize in a principled way, and the empirical validation, though limited in scale, was judged convincing. After rebuttal and discussion, all reviewers agreed that this is a strong paper and should be accepted.

For the camera-ready, the authors are encouraged to clarify connections to prior work (see reviews), expand the discussion of scalability and computational overhead, and provide additional details such as training times, infeasibility metrics. These improvements will increase the paper’s accessibility and impact.
Decision: Accept (poster)