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REVIEW 4 major objections 4 minor 52 references

Contextual Scenario Generation for Two-Stage Stochastic Programming

T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proposes contextual scenario generation: learn a mapping from context to K surrogate scenarios so that solving the two-stage stochastic program on those scenarios yields near-optimal decisions at low computational cost.

desk verdict Solid, honest first framework for contextual scenario generation, but the abstract overpromises finite-sample guarantees and the missing lambda-selection rule means the headline results are not yet reproducible. read the letter →

arxiv 2502.05349 v2 pith:3JGFBLWC submitted 2025-02-07 math.OC cs.LG

David Islip , Roy H. Kwon , Sanghyeon Bae , Woo Chang Kim This is my paper · ORCID
classification math.OCcs.LG MSC 90C1590C1168T07
keywords contextualscenariogenerationtwo-stagestochasticprogrammingmaximummeandiscrepancytask-basedlearningsurrogatescenariosend-to-endoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes contextual scenario generation (CSG): instead of estimating the conditional distribution of uncertainty and then sampling and reducing a large scenario set at decision time, learn a neural mapping from observed context to a small, user-specified number K of surrogate scenarios. Solving the two-stage stochastic program on just those K scenarios then gives first-stage decisions whose true conditional expected cost is close to optimal. The paper develops two complementary training losses: a distributional loss based on kernel maximum mean discrepancy with the energy distance, and a task-based loss that differentiates through a learned surrogate of the optimistic downstream objective, regularized by the same MMD term to keep the task network from exploiting errors in the surrogate. Experiments on newsvendor, capacity planning, CVaR portfolio selection, and a multidimensional newsvendor with substitution show that task-based CSG with suitable regularization can outperform the distributional baseline and the expected-value solution, and can unlock the value of the stochastic solution from very few scenarios.

What carries the argument

The key machinery is the loss-net E_psi, defined by mean aggregation over encoded surrogate scenarios: E_psi(zeta_1...K, omega) = Psi_2((1/K) sum_k Psi_1(zeta_k), Psi_1(omega)), with Psi_1 and Psi_2 feedforward ReLU networks. Its purpose is to approximate the optimistic task-based loss l_opt(zeta_1...K, omega), which is computed by solving the K-scenario SAA and then an additional recourse search over its optimal solution set. Proposition 2, built on a multiset sum-decomposition result, shows that l_opt has exactly this functional form with continuous embedding and combining maps, so the architecture is a universal approximator for the task loss. The task-mapping f_phi is a neural network trained to minimize the loss-net output plus an MMD regularization term with the energy-distance kernel; the dynamic variant iteratively refines the loss-net on replay-buffer samples generated around the current f_phi.

What would settle it

Run the task-based method with lambda = 0 on a held-out context distribution and measure the MMD distance between the produced scenario sets and the loss-net training scenarios; the paper's own newsvendor and CVaR tables show that small lambda produces enormous optimality gaps, so if the central claim is meant to hold for any lambda, this is a direct refutation. A systematic sweep of lambda across K would map the regime where the method degenerates and reveal whether any fixed regularization reliably prevents loss error maximization across problem classes.

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Extended reading notes

Core claim

The central claim is that there exists an efficiently evaluated mapping f : X to $\Omega$^K such that solving the sample-average approximation on f(x) is a high-quality proxy for solving the true contextual two-stage problem under the conditional distribution P_{omega|x}. The paper establishes this in two ways: DCSG minimizes the expected squared MMD between the empirical distribution on f(x) and P_{omega|x}, using the parameter-free energy kernel; PCSG minimizes the optimistic task loss l_opt(zeta, omega) = min_{y in Y*(zeta)} h(y) + Q(y, omega), approximated by a permutation-invariant neural network to avoid the sparse and uninformative gradients of the true recourse function. The optimistic formulation handles non-unique optimal solutions to the SAA problem, and the MMD regularization term $\lambda$*l_MMD prevents the failure mode the paper calls loss error maximization, in which the task network drives scenarios outside the region where the surrogate loss is accurate. Empirically, with a suitable choice of $\lambda$, the task-based approaches beat the distributional baseline for small K and achieve median optimality gaps below 10 percent in the CVaR problem with K = 40, while reducing decision-time computation to a forward pass plus a K-scenario solve.

Load-bearing premise

The task-based pipeline depends on the loss-net being accurate on the distribution of scenario sets that the task-net actually produces, and on the MMD penalty being strong enough to keep the task-net inside that well-approximated region; when the penalty is too small, the task-net exploits surrogate-loss errors and optimality gaps explode.

Editorial extensions

If this is right

  • At decision time CSG replaces conditional-density estimation, sampling, and scenario reduction with one forward pass through the task network, so per-context cost is independent of the original sample size M.
  • For small K (K <= 5), task-based methods with suitable MMD regularization can outperform both the distributional baseline and the expected-value solution, meaning a stochastic solution is obtainable from very few scenarios.
  • Because the optimistic loss is computed by solving a K-scenario SAA and an extra recourse problem, the method applies to two-stage LPs, MIPs with nonlinear or integer elements, and problems without unique optimal solutions.
  • The dynamic refinement variant improves on the static variant when the loss-net keeps generalizing on new scenario sets, but it requires repeated evaluation of the optimistic loss and can degrade when the loss-net fails to generalize.
  • Training costs are paid offline; the timing analysis shows that hundreds of CVaR instances can be solved in about 33 seconds with K = 40 while keeping a median optimality gap around 5 percent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension the paper leaves implicit is that the same loss-net plus MMD-regularized task-net recipe could be applied to multistage stochastic programs, replacing the two-stage optimistic loss with stage-wise recourse evaluations.
  • Because the energy-distance MMD with K = 1 reduces to least-squares conditional mean estimation, the distributional approach is a natural initialization for the task-based approach; a testable implication is that MMD-initialized task networks should dominate random initialization when lambda is moderate.
  • The abstract advertises finite-sample generalization guarantees, but the full text as posted defers such bounds to future work; a reader should verify whether a later version states a concrete sample-complexity theorem before relying on that claim.
  • A natural ablation is to replace the energy kernel with Sinkhorn divergence or a conditional Wasserstein distance (both discussed in the appendix); if the performance gap between task-based and distributional approaches persists, the bottleneck is the surrogate loss rather than the distributional metric.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes contextual scenario generation (CSG) for two-stage stochastic programs: given side information x, learn a mapping f that outputs K surrogate scenarios, which are then used to solve the SAA problem on those K scenarios. Two methods are developed: a distributional approach (DCSG) that minimizes an MMD distance between the empirical distribution of generated scenarios and the conditional distribution of uncertainty, and a task-based approach (PCSG) that trains a task network using a learned neural surrogate Eψ of an optimistic task loss ℓopt, regularized by the MMD term. The optimistic loss handles non-unique solutions of the SAA problem, and Proposition 2 motivates the permutation-invariant architecture of the loss network. Experiments on newsvendor, CEP1 capacity planning, CVaR portfolio optimization, and a multidimensional newsvendor with substitution compare DCSG, static PCSG, dynamic PCSG, expected-value, and quantile-regression benchmarks, reporting optimality gaps over a grid of regularization parameters λ.

Significance. If the central claim is established, this would be a valuable contribution: it offers a way to replace expensive scenario generation and reduction with a cheap neural forward pass, while keeping the downstream 2SP solve tractable. The optimistic relaxation of the bi-level scenario-generation problem is a sensible way to handle non-unique SAA solutions, and the paper covers four structurally different 2SP classes. The work is honest about a known failure mode (loss error maximization) and about the mixed performance of the dynamic approach in the MNV experiment. However, the paper advertises finite-sample generalization guarantees that do not appear anywhere in the text, and the reported success depends on an ex post selection of the regularization strength λ; no data-driven rule is given. These issues directly affect the reliability of the headline claim.

major comments (4)
  1. [Abstract and Section 5] The abstract states “We provide finite-sample generalization guarantees,” but no such theorem, proposition, or bound appears in Sections 3–4, and Section 5 explicitly defers this: “Future work could explore … understanding finite sample learning bounds and the generalization/stability properties of the proposed methodologies.” This is a direct contradiction between the advertised contribution and the content of the paper. The claim should either be substantiated with a precise statement (e.g., a generalization bound for the empirical minimizer of LMMD or of the static PCSG objective) or removed from the abstract.
  2. [Section 3.5, Static Approach and Tables 2, 5, 8] The central premise of PCSG is that the learned loss-net Eψ approximates ℓopt on the distribution of scenario sets actually produced by the task-net, and that the MMD regularizer prevents the task-net from leaving that well-approximated region. Neither is guaranteed: no bound links the MMD distance to the approximation error of Eψ, and λ is a free hyperparameter with no data-driven selection rule. The paper itself documents the failure mode (“loss error maximization”) and the experiments confirm it: CVaR Table 5 reports median optimality gaps of 1.03e7% for Static K=1 with λ=80 and 2.60e7% with λ=320; newsvendor Figure 6 shows poor performance at λ=0.1. At large λ the method collapses toward the MMD solution, so the task-based component adds little. The reported best results are selected over a grid of λ per problem and per K, which does not establish that the method can be deployed without a priori knowledge of a good λ. A practical model-selection procedure for λ (e.g., cross-validation on the task loss) or an analysis that justifies a default or adaptive choice is needed.
  3. [Section 3.4, Eq. (Opt-PCSG')] The optimistic relaxation replaces min_{y∈Y*(f(x))} E_{ω|x}[h(y)+Q(y,ω)] with E_{ω|x}[min_{y∈Y*(f(x))} h(y)+Q(y,ω)], which is a lower bound because it allows the choice of y ∈ Y* to depend on the realized ω. Training on this lower bound does not automatically yield a low true objective; the gap between the two quantities is not quantified, and nothing in the method controls it. The paper notes the inequality but does not investigate how loose it is on the tested instances. Since the evaluation uses the true 2SP objective with out-of-sample conditional distributions, a loose optimistic bound could give the task-net a misleading training signal. I would like to see, at least on the validation instances, a comparison between the optimistic training loss and the actual “best-in-hindsight” task loss (or a discussion of why the optimism is benign for the problem classes considered).
  4. [Section 3.5, Proposition 2] Proposition 2 gives an exact sum-decomposable representation of ℓopt under a continuity assumption, but it provides no approximation guarantee for the finite-capacity ReLU networks used in the loss-net, and the continuity assumption is not verified for the mixed-binary problems (CVaR and MNV) in the experiments. The paper acknowledges that “we leverage a continuous approximation to ℓopt, irrespective of whether ℓopt is continuous,” but this is a heuristic step. The role of Proposition 2 should be clarified: it motivates the architecture, but it does not by itself justify the reliability of the loss-net approximation. I recommend stating explicitly that the approximation step is heuristic and unsupported by guarantees, and discussing what could go wrong when the loss-net is inaccurate (beyond the already documented loss error maximization).
minor comments (4)
  1. [Throughout] There are several typos: “Lipshitz” should be “Lipschitz” (Section 3.1); “PSCR” appears in the CEP1 results text (Section 4.2) where “PCSG” is meant; figure labels in Appendix C contain “I sta ces” and “% f Instances”; “ReLu” should be “ReLU” in several places; the caption of Figure 3 ends with the single word “Embedded” and appears incomplete.
  2. [Section 3.2 and Appendix A] The discussion of the energy kernel and the choice of MMD is clear, but the phrase “parameter-free kE” could be misread: the kernel has no free parameters, yet the method still requires choosing K and the neural architecture. Please clarify that “parameter-free” refers only to the kernel.
  3. [Section 4.4, last paragraph] The observation that the dynamic approach failed in the MNV experiment because the loss-net architecture did not generalize is valuable, but it is placed only in the results discussion. Consider moving this to a limitations paragraph in Section 5, since it directly informs the applicability of the dynamic method.
  4. [Appendix B, proof of Proposition 2] In the proof, after applying Proposition 1 from Tabaghi and Wang, the function ρ is defined by ρ(ζ̂, ω̂) = ρ̃(Kζ̂, ω̂). This definition is only needed for multisets of size exactly K, and the text could state more clearly that the extension to smaller multisets is irrelevant for the application at hand.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the task-based derivation is validated against the true out-of-sample 2SP objective; the only self-citations are background, not load-bearing.

full rationale

The central claims are not circular. The distributional objective (DCSG) minimizes MMD to the conditional distribution, and the task-based loss is evaluated by solving the true recourse problem: 'Evaluating ℓopt(ζ1...K , ω) requires solving (ζ-SAA), obtaining the optimal value v∗(ζ1...K ), then given ω, finding the solution among Y ∗(f (x)) that minimizes h(y)+Q(y,ω).' Final validation uses the true 2SP objective on out-of-sample conditional distributions: 'a resulting first-stage solution is then evaluated using the 2SP objective with the expected recourse calculated using Pω|x(i).' The learned loss-net Eψ is a surrogate used only for gradient-based training; the MMD regularizer is explicitly introduced to counter the documented 'loss error maximization' failure, and the experiments report cases where it fails (e.g., CVaR K=1 λ=80/320 with gaps up to 2.60e7%), which is evidence that the evaluation is not forced by construction. The only self-citations ([21], [22]) appear in related-work context ('Lee et al. [21] and Bae et al. [22] predict expected recourse in generic 2SPs... However, these methods do not directly address scenario generation') and are not load-bearing for the proposed method, which rests on external results (Theorem 1 from [27], Proposition 1 from [35], Theorem 14.60 from [33]). One non-circular inconsistency: the abstract promises finite-sample generalization guarantees, but no such theorem appears and Section 5 lists 'understanding finite sample learning bounds and the generalization/stability properties' as future work; this is a correctness/completeness concern, not a circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The ledger shows the central claim rests on standard 2SP assumptions, standard kernel/RKHS theory, and two ad hoc elements: using the optimistic relaxation as the training loss, and relying on the loss-net plus MMD regularization to avoid surrogate exploitation.

free parameters (3)
  • MMD regularization weight lambda = grid-searched per problem/K, e.g., 1.0 (newsvendor K=1), 10240 (CVaR K=40)
    Controls the trade-off between task loss and distributional anchoring; results are highly sensitive to lambda and no principled selection criterion is provided.
  • Number of surrogate scenarios K = user-specified (1-40 across experiments)
    The method's performance depends on K; the user must choose it, and the paper does not provide guidance.
  • Neural architecture hyperparameters (task-net and loss-net) = selected via random search over a single training sample with 20% holdout
    Architecture choices affect results and are not specified in a reproducible way.
assumptions (4)
  • domain assumption Assumptions 1-3: Stage II can be solved efficiently; 2SP-SAA can be solved efficiently on K scenarios; only iid joint samples are available.
    Stated in Section 1; these define the operating conditions of CSG.
  • standard math Kernel conditions for MMD (characteristic, finite second moments) and compactness of Omega for Proposition 2.
    Invoked in Theorem 1 and Appendix B; the energy kernel is characteristic under bounded moment conditions per Szekely-Rizzo.
  • ad hoc to paper The optimistic relaxation (Opt-PCSG') is a valid proxy for the true task objective.
    The paper derives it as a lower bound by relaxing nonanticipativity among optimal solutions, but the subsequent methods optimize this relaxed loss, not the true (Opt-PCSG).
  • ad hoc to paper The loss-net E_psi can accurately approximate l_opt on the distribution induced by the current task-net, and MMD regularization prevents excessive distribution shift.
    This is the load-bearing premise of the static/dynamic PCSG; the paper acknowledges 'loss error maximization' when this fails.

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Cite this review

Pith. "Pith review of Contextual Scenario Generation for Two-Stage Stochastic Programming." pith.science (2026). https://pith.science/paper/3JGFBLWC

@misc{pith2026250205349,
  author       = {Pith},
  title        = {Pith review of: Contextual Scenario Generation for Two-Stage Stochastic Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JGFBLWC}},
  note         = {Machine review of arXiv:2502.05349}
}
read the original abstract

Two-stage stochastic programs (2SPs) are widely used for decision-making under uncertainty, but their practical deployment is often limited by the large number of scenarios needed to approximate the conditional distribution of uncertain outcomes. We study contextual scenario generation: given contextual information, learn to produce a small, user-specified set of surrogate scenarios that, when used as input into the 2SP, lead to high-quality 2SP decisions. Existing scenario generation methods either ignore contextual information or are computationally burdensome in this setting. We propose contextual scenario generation (CSG), which learns a mapping from context to a set of surrogate scenarios. We develop two complementary methodologies: (i) a distributional approach that learns a mapping from context to scenarios by minimizing a kernel-based distance to the conditional distribution, and (ii) a task-based approach that selects the mapping to optimize decision quality via differentiating through a learned surrogate of the downstream 2SP objective. Both approaches are broadly applicable and require only repeated solution of the underlying subproblems and 2SPs defined on the generated scenarios. We provide finite-sample generalization guarantees and demonstrate strong empirical performance across multiple 2SP classes.

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Works this paper leans on

52 extracted references · 37 canonical work pages

  1. [1]

    1 11–152

    Ntaimo, L.: Example Applications of Stochastic Programming, pp. 1 11–152. Springer, Cham (2024). https://doi.org/10.1007/978-3-031-52464-6 4 . https:// doi.org/10.1007/978-3-031-52464-6 4

    work page doi:10.1007/978-3-031-52464-6 2024

  2. [2]

    SIAM Journal on optimiza- tion 12(2), 479–502 (2002) https://doi.org/10.1137/S1052623499363220

    Kleywegt, A.J., Shapiro, A., Homem-de-Mello, T.: The sample averag e approxi- mation method for stochastic discrete optimization. SIAM Journal on optimiza- tion 12(2), 479–502 (2002) https://doi.org/10.1137/S1052623499363220

    doi:10.1137/s1052623499363220 2002

  3. [3]

    Advances in Neural Information Proces sing Systems 35, 23992–24005 (2022) 42

    Patel, R.M., Dumouchelle, J., Khalil, E., Bodur, M.: Neur2sp: Neural t wo-stage stochastic programming. Advances in Neural Information Proces sing Systems 35, 23992–24005 (2022) 42

    work page 2022

  4. [4]

    Wu, Y., Song, W., Cao, Z., Zhang, J.: Learning scenario represent ation for solv- ing two-stage stochastic integer programs. (2022). Internatio nal Conference on Learning Representations, ICLR

    work page 2022

  5. [5]

    Kluwer Academic Pub lishers, Norwell, MA, USA (1996)

    Higle, J.L., Sen, S.: Stochastic Decomposition. Kluwer Academic Pub lishers, Norwell, MA, USA (1996). https://doi.org/10.1007/978-1-4615-4115-8

    work page doi:10.1007/978-1-4615-4115-8 1996

  6. [7]

    INFORMS Journal on Optimization 5(3), 295– 320 (2023) https://doi.org/10.1287/ijoo.2023.0088

    Estes, A.S., Richard, J.-P.P.: Smart predict-then-optimize for tw o-stage linear programs with side information. INFORMS Journal on Optimization 5(3), 295– 320 (2023) https://doi.org/10.1287/ijoo.2023.0088

    work page arXiv 2023

  7. [8]

    Manufactu ring & Service Operations Management 21(4), 798–815 (2019) https://doi.org/10.1287/msom

    Ban, G.-Y., Gallien, J., Mersereau, A.J.: Dynamic procurement of ne w products with covariate information: The residual tree method. Manufactu ring & Service Operations Management 21(4), 798–815 (2019) https://doi.org/10.1287/msom. 2018.0725

    work page arXiv 2019

  8. [9]

    Management Science 66(3), 1025–1044 (2020) https://doi.org/10.1287/mnsc.2018.3253

    Bertsimas, D., Kallus, N.: From predictive to prescriptive analytics . Management Science 66(3), 1025–1044 (2020) https://doi.org/10.1287/mnsc.2018.3253

    arXiv 2020

Show all 52 references
  1. [10]

    Optimization Letters, 1– 28 (2023) https:// doi.org/10.1007/s11590-023-02009-5

    Yilmaz, D., B¨ uy¨ uktahtakın,˙I.E.: A deep reinforcement learning framework for solving two-stage stochastic programs. Optimization Letters, 1– 28 (2023) https:// doi.org/10.1007/s11590-023-02009-5

    2023 doi

  2. [11]

    Advances in Neural Information Proce ssing Systems 36 (2024)

    Zharmagambetov, A., Amos, B., Ferber, A., Huang, T., Dilkina, B., Tian, Y.: Landscape surrogate: Learning decision losses for mathematical optimization under partial information. Advances in Neural Information Proce ssing Systems 36 (2024)

    2024

  3. [12]

    predict, then optimize

    Elmachtoub, A.N., Grigas, P.: Smart “predict, then optimize”. Ma nagement Science (2021) https://doi.org/10.1287/mnsc.2020.3922

    2021

  4. [13]

    Advances in neural information processing systems 32 (2019)

    Agrawal, A., Amos, B., Barratt, S., Boyd, S., Diamond, S., Kolter, J.Z.: Differ- entiable convex optimization layers. Advances in neural information processing systems 32 (2019)

    2019

  5. [14]

    : Integrated conditional estimation-optimization

    Grigas, P., Qi, M., et al. : Integrated conditional estimation-optimization. arXiv preprint arXiv:2110.12351 (2021) https://doi.org/10.48550/arXiv.2110.12351

    2021 doi arXiv

  6. [15]

    Spr inger, New York, NY (2012)

    King, A.J., Wallace, S.W.: Modeling with Stochastic Programming. Spr inger, New York, NY (2012). https://doi.org/10.1007/978-0-387-87817-1

    2012 doi

  7. [16]

    Mathematical programming 95, 493–511 (2003) https://doi.org/ 10.1007/s10107-002-0331-0

    Dupaˇ cov´ a, J., Gr¨ owe-Kuska, N., R¨ omisch, W.: Scenario reduction in stochastic 43 programming. Mathematical programming 95, 493–511 (2003) https://doi.org/ 10.1007/s10107-002-0331-0

    2003 doi

  8. [17]

    Computational optimization and applications 24, 169–185 (2003) https://doi.org/10.1023/A:1021853807313

    Høyland, K., Kaut, M., Wallace, S.W.: A heuristic for moment-match ing scenario generation. Computational optimization and applications 24, 169–185 (2003) https://doi.org/10.1023/A:1021853807313

    2003 doi

  9. [18]

    Operations Research 71(4), 1343–1361 (2023) https://doi.org/10.1287/opre.2022.2265

    Bertsimas, D., Mundru, N.: Optimization-based scenario reduct ion for data- driven two-stage stochastic optimization. Operations Research 71(4), 1343–1361 (2023) https://doi.org/10.1287/opre.2022.2265

    2023

  10. [19]

    Mathematical Programming, 1–42 (2022) https://doi.org/10.1007/s10107-019-01451-7

    Fairbrother, J., Turner, A., Wallace, S.W.: Problem-driven scena rio generation: an analytical approach for stochastic programs with tail risk measur e. Mathematical Programming, 1–42 (2022) https://doi.org/10.1007/s10107-019-01451-7

    2022 doi

  11. [20]

    European Journal of Opera tional Research 318(1), 154–166 (2024) https://doi.org/10.1016/j.ejor.2024.04.006

    Narum, B.S., Fairbrother, J., Wallace, S.W.: Problem-based scena rio generation by decomposing output distributions. European Journal of Opera tional Research 318(1), 154–166 (2024) https://doi.org/10.1016/j.ejor.2024.04.006

    2024 doi

  12. [21]

    Europea n Journal of Operational Research 308(1), 321–335 (2023) https://doi.org/10.1016/j.ejor

    Lee, J., Bae, S., Kim, W.C., Lee, Y.: Value function gradient learning for large-scale multistage stochastic programming problems. Europea n Journal of Operational Research 308(1), 321–335 (2023) https://doi.org/10.1016/j.ejor. 2022.10.011

    2023 doi

  13. [22]

    In: International Confer ence on Artificial Intelligence and Statistics, pp

    Bae, H., Lee, J., Kim, W.C., Lee, Y.: Deep value function networks f or large- scale multistage stochastic programs. In: International Confer ence on Artificial Intelligence and Statistics, pp. 11267–11287 (2023). PMLR

    2023

  14. [23]

    Bengio, Y., Frejinger, E., Lodi, A., Patel, R., Sankaranarayanan , S.: A learning- based algorithm to quickly compute good primal solutions for stocha stic integer programs. In: Integration of Constraint Programming, Artificial Intelligence, and Operations Research: 17th Intern...

    2020 doi

  15. [24]

    Han dbooks in oper- ations research and management science 10, 483–554 (2003) https://doi.org/10

    R¨ omisch, W.: Stability of stochastic programming problems. Han dbooks in oper- ations research and management science 10, 483–554 (2003) https://doi.org/10. 1016/S0927-0507(03)10008-4

    2003

  16. [25]

    Journal of Machine Learning Research 13(25), 723–773 (2012)

    Gretton, A., Borgwardt, K.M., Rasch, M.J., Sch¨ olkopf, B., Smola , A.: A kernel two-sample test. Journal of Machine Learning Research 13(25), 723–773 (2012)

    2012

  17. [26]

    : Ker- nel mean embedding of distributions: A review and beyond

    Muandet, K., Fukumizu, K., Sriperumbudur, B., Sch¨ olkopf, B., et al. : Ker- nel mean embedding of distributions: A review and beyond. Foundat ions and Trends® in Machine Learning 10(1-2), 1–141 (2017) https://doi.org/10.1561/ 2200000060 44

    2017

  18. [27]

    arXiv preprint arXiv:2206.04287 (2022) https://doi.org/10

    Huang, Z., Lam, H., Zhang, H.: Evaluating aleatoric uncertainty v ia conditional generative models. arXiv preprint arXiv:2206.04287 (2022) https://doi.org/10. 48550/arXiv.2206.04287

    2022 doi arXiv

  19. [28]

    The annals of statistics, 2263–2291 (2013)

    Sejdinovic, D., Sriperumbudur, B., Gretton, A., Fukumizu, K.: Eq uivalence of distance-based and rkhs-based statistics in hypothesis testing. The annals of statistics, 2263–2291 (2013)

    2013

  20. [29]

    Handbook of Numerical Analysis 24, 407–471 (2023) https:// doi.org/10.1016/bs.hna.2022.11.003

    S´ ejourn´ e, T., Peyr´ e, G., Vialard, F.-X.: Unbalanced optimal transport, from the- ory to numerics. Handbook of Numerical Analysis 24, 407–471 (2023) https:// doi.org/10.1016/bs.hna.2022.11.003

    2023 doi

  21. [30]

    Statistics & Probability Letters 82(12), 2278–2282 (2012) https://doi.org/10.1016/j.spl.2012

    Sz´ ekely, G.J., Rizzo, M.L.: On the uniqueness of distance covaria nce. Statistics & Probability Letters 82(12), 2278–2282 (2012) https://doi.org/10.1016/j.spl.2012. 08.007

    2012 doi

  22. [31]

    In: Proceedings o f the AAAI Conference on Artificial Intelligence, vol

    Wilder, B., Dilkina, B., Tambe, M.: Melding the data-decisions pipeline: Decision- focused learning for combinatorial optimization. In: Proceedings o f the AAAI Conference on Artificial Intelligence, vol. 33, pp. 1658–1665 (201 9). https://doi. org/10.1609/aaai.v33i01.33011658

    doi

  23. [32]

    IEEE transactions o n evolution- ary computation 22(2), 276–295 (2017) https://doi.org/10.1109/TEVC.2017

    Sinha, A., Malo, P., Deb, K.: A review on bilevel optimization: From cla ssical to evolutionary approaches and applications. IEEE transactions o n evolution- ary computation 22(2), 276–295 (2017) https://doi.org/10.1109/TEVC.2017. 2712906

    2017 doi

  24. [33]

    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis vol. 317. Spr inger, Berlin, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02431-3

    2009 doi

  25. [34]

    Advances in neural information processing system s 30 (2017)

    Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhut dinov, R.R., Smola, A.J.: Deep sets. Advances in neural information processing system s 30 (2017)

    2017

  26. [35]

    In: International Conference on Algorithmic Learning Theory, pp

    Tabaghi, P., Wang, Y.: Universal representation of permutatio n-invariant func- tions on vectors and tensors. In: International Conference on Algorithmic Learning Theory, pp. 1134–1187 (2024). PMLR

    2024

  27. [36]

    Machine learning 8, 293–321 (1992) https://doi.org/10.1007/ BF00992699

    Lin, L.-J.: Self-improving reactive agents based on reinforceme nt learning, plan- ning and teaching. Machine learning 8, 293–321 (1992) https://doi.org/10.1007/ BF00992699

    1992

  28. [37]

    Mathematical programming 24, 314–325 (1982) https://doi

    Birge, J.R.: The value of the stochastic solution in stochastic linea r programs with fixed recourse. Mathematical programming 24, 314–325 (1982) https://doi. org/10.1007/BF01585113

    1982 doi

  29. [38]

    European Journal of Oper ational Research 300(2), 590–601 (2022) https://doi.org/10.1016/j.ejor.2021.08.013 45

    Liu, C., Letchford, A.N., Svetunkov, I.: Newsvendor problems: An integrated method for estimation and optimisation. European Journal of Oper ational Research 300(2), 590–601 (2022) https://doi.org/10.1016/j.ejor.2021.08.013 45

    2022 doi

  30. [39]

    In: Walt, Millman (eds.) Proceedings of the 9th Python in Scie nce Conference, pp

    Seabold, Perktold: Statsmodels: Econometric and Statistical Modeling with Python. In: Walt, Millman (eds.) Proceedings of the 9th Python in Scie nce Conference, pp. 92–96 (2010). https://doi.org/10.25080/Majora-92bf1922-011

    2010 doi

  31. [40]

    Journal of econ omic perspectives 15(4), 143–156 (2001)

    Koenker, R., Hallock, K.F.: Quantile regression. Journal of econ omic perspectives 15(4), 143–156 (2001)

    2001

  32. [41]

    Journal of risk 4, 43–68 (2002) https:// doi.org/10.21314/JOR.2002.057

    Krokhmal, P., Palmquist, J., Uryasev, S.: Portfolio optimization wit h conditional value-at-risk objective and constraints. Journal of risk 4, 43–68 (2002) https:// doi.org/10.21314/JOR.2002.057

    2002 doi

  33. [42]

    International Journal of Production Economics 134(2), 388–397 (2011) https:// doi.org/10.1016/j.ijpe.2009.11.012

    Vaagen, H., Wallace, S.W., Kaut, M.: Modelling consumer-directed s ubstitution. International Journal of Production Economics 134(2), 388–397 (2011) https:// doi.org/10.1016/j.ijpe.2009.11.012

    2011 doi

  34. [43]

    In: 2009 IEEE 12th International Conference on Computer Vision, pp

    Pele, O., Werman, M.: Fast and robust earth mover’s distances. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 460–467 (2009). https:// doi.org/10.1109/ICCV.2009.5459199 . IEEE

    2009

  35. [44]

    In: The 22nd International Conferenc e on Artificial Intelligence and Statistics, pp

    Genevay, A., Chizat, L., Bach, F., Cuturi, M., Peyr´ e, G.: Sample c omplexity of sinkhorn divergences. In: The 22nd International Conferenc e on Artificial Intelligence and Statistics, pp. 1574–1583 (2019). PMLR

    2019

  36. [45]

    arXiv preprint arXiv:1705.10743 (2017) https://doi.org/10.48550/ arXiv.1705.10743

    Bellemare, M.G., Danihelka, I., Dabney, W., Mohamed, S., Lakshmina rayanan, B., Hoyer, S., Munos, R.: The cramer distance as a solution to biased w asserstein gradients. arXiv preprint arXiv:1705.10743 (2017) https://doi.org/10.48550/ arXiv.1705.10743

    2017 doi arXiv

  37. [46]

    In: The 22nd International Conference on Artificial Intelligence and S tatistics, pp

    Feydy, J., S´ ejourn´ e, T., Vialard, F.-X., Amari, S.-i., Trouve, A.,Peyr´ e, G.: Inter- polating between optimal transport and mmd using sinkhorn diverge nces. In: The 22nd International Conference on Artificial Intelligence and S tatistics, pp. 2681–2690 (2019)

    2019

  38. [47]

    Advances in Neural Information Processing Systems 29 (2016)

    Ren, Y., Zhu, J., Li, J., Luo, Y.: Conditional generative moment-m atching networks. Advances in Neural Information Processing Systems 29 (2016)

    2016

  39. [48]

    Advances in neural information processing systems 33, 21247–21259 (2020)

    Park, J., Muandet, K.: A measure-theoretic approach to kern el conditional mean embeddings. Advances in neural information processing systems 33, 21247–21259 (2020)

    2020

  40. [49]

    arXiv preprint arXiv:2310.03054 (2023) https://doi.org/10.48550/arXiv

    Hagemann, P., Hertrich, J., Altekr¨ uger, F., Beinert, R., Chems eddine, J., Steidl, G.: Posterior sampling based on gradient flows of the mmd with negativ e distance kernel. arXiv preprint arXiv:2310.03054 (2023) https://doi.org/10.48550/arXiv. 2310.03054

    2023 doi arXiv

  41. [50]

    arXiv pr eprint 46 arXiv:2310.13433 (2023) https://doi.org/10.48550/arXiv.2310.13433

    Chemseddine, J., Hagemann, P., Wald, C.: Y-diagonal couplings: A pprox- imating posteriors with conditional wasserstein distances. arXiv pr eprint 46 arXiv:2310.13433 (2023) https://doi.org/10.48550/arXiv.2310.13433

    2023 doi arXiv

  42. [51]

    Tietz, M., Fan, T.J., Nouri, D., Bossan, B., skorch Developers: Sk orch: A Scikit- learn Compatible Neural Network Library that Wraps PyTorch. (20 17). https:// skorch.readthedocs.io/en/stable/

  43. [52]

    Computa tional Manage- ment Science, 1–34 (2022) https://doi.org/10.1007/s10287-021-00400-0

    Deng, Y., Sen, S.: Predictive stochastic programming. Computa tional Manage- ment Science, 1–34 (2022) https://doi.org/10.1007/s10287-021-00400-0

    2022 doi

  44. [53]

    Advances in Neural Infor mation Processing Systems (2017) 47

    Vaswani, A.: Attention is all you need. Advances in Neural Infor mation Processing Systems (2017) 47

    2017

Pith tools

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