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REVIEW 2 major objections 3 minor 14 references

Integrated Learning and Robust Optimization

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Using the same robust decision map to train and deploy — $z^\star_\gamma(\hat y)=\Pi_Z(-\hat y/\gamma)$ — makes decisions robust and aligned at once; the convex surrogate provably matches the target up to a computable threshold.

desk verdict Useful framework with a real gap: the advertised finite-sample guarantees for the surrogate rely on a uniform Fisher-consistency condition that can fail for every fixed gamma. read the letter →

arxiv 2608.08718 v1 pith:4QLQLY4E submitted 2026-08-09 math.OC

Cheng Tan , Yuchen Mao , Shuming Wang , Huan Xu This is my paper · ORCID
classification math.OC MSC 90C1590C0590C2590C47
keywords integratedlearningandoptimizationdecision-focusedrobustcontextualstochasticFisherconsistencysurrogatelossexcessriskboundRademachercomplexity
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

Decisions in transportation, portfolio allocation, and similar settings must be made after seeing context but before costs are realized; the standard pipeline predicts the cost vector and then solves a nominal linear program, which is unstable near switching boundaries and blind to the error that remains in any prediction. This paper claims both defects can be cured by one move: replace the nominal decision map with its regularized robust counterpart $z^\star_\gamma(\hat y)=\arg\min_{z\in Z}\hat y^\top z+\frac{\gamma}{2}\|z\|_2^2$, which is just the Euclidean projection $\Pi_Z(-\hat y/\gamma)$ onto the feasible set, and use that same map to define the training loss and the deployed decision (the ILRO framework). The paper proves that the convex surrogate RSPO+ is Fisher consistent — minimizing it leads to decisions that minimize the true robust objective — exactly when the robustness parameter $\gamma$ stays below a threshold $\bar\gamma(x)$ computable as a linear program, and proves finite-sample excess-risk bounds of order $O_p(N^{-1/2})$ for the robust-loss predictor and $O_p(N^{-1/4})$ (improving to $N^{-1/2}$ under extra growth conditions) for the surrogate. If the claims hold, practitioners get one knob, $\gamma$, that buys hedging against forecast error and decision-quality training at the same time; the experiments place the largest gains precisely in the regimes that motivate decision-focused learning — small samples, high-dimensional decisions, and misspecified models.

What carries the argument

The load-bearing object is the robust decision map $z^\star_\gamma(\hat y)=\arg\min_{z\in Z}\{\hat y^\top z+\tfrac{\gamma}{2}\|z\|_2^2\}$, which completing the square identifies with the Euclidean projection $\Pi_Z(-\hat y/\gamma)$ onto the feasible set. This projection identity does three jobs: it makes the decision map $1/\gamma$-Lipschitz and differentiable almost everywhere, so gradients flow through the actual deployed decision rule and vector contraction transfers loss-side complexity to the prediction class at factor $\sqrt{2}\,r/\gamma$; it yields the convex, everywhere-differentiable surrogate RSPO+ whose gradient is $a(z^\star_\gamma(y)-z^\star_\gamma(\hat y-y/a))$; and it identifies the set of target-risk-optimal predictions as the translated normal cone $-N_Z(z^\star(\bar y))-\gamma z^\star(\bar y)$, collapsing Fisher consistency of the surrogate to one membership condition, $2\bar y\in -N_Z(z^\star(\bar y))-\gamma z^\star(\bar y)$, equivalent to $\gamma\le\bar\gamma(x)$, a linear program when $Z$ is a polyhedron. The companion mechanism for computation is the closed-form Jacobian $\nabla z^\star_\gamma(y_0)=\tfrac{1}{\gamma}(A_S^\top(A_SA_S^\top)^{-1}A_S-I)$ on the active face, which allows first-order descent on the original nonconvex objective. The RSPO loss itself, $\ell_{\mathrm{RSPO}}(\hat y,y)=y^\top z^\star_\gamma(\hat y)-v^\star(y)$, measures the excess realized cost of the robust decision induced by the prediction against the clairvoyant nominal optimum, and is nonnegative by construction.

What would settle it

Replicate the paper's own counterexample at a slope beyond the threshold: fix $x$, take $y\mid x\sim N((1,3/2)^\top,\sigma^2I_2)$ and $Z=\{z\in\mathbb{R}^2:z_1-z_2\le 1,\ z_1\ge -1,\ z_1,z_2\le 0\}$, for which the paper computes $\bar\gamma=3/2$. Train the RSPO+ surrogate with $a=1$ and $\gamma=2$ on an increasingly large sample and measure the average realized cost of the induced decisions against the best RSPO decision. The theory predicts a strictly positive, nonvanishing gap because $2\bar y$ lies outside $Y^\star_{\mathrm{RSPO}}$; if the gap vanishes, the threshold characterization is wrong. The same setup with $\gamma=1\le\bar\gamma$ must show the gap shrink to zero, checking both sides of the claim in one experiment.

Watch

Extended reading notes

Core claim

The paper's central claim is that replacing the nominal decision map $\arg\min_{z\in Z}\hat y^\top z$, which is discontinuous and set-valued, with its regularized robust counterpart $z^\star_\gamma(\hat y)=\arg\min_{z\in Z}\{\hat y^\top z+\tfrac{\gamma}{2}\|z\|_2^2\}=\Pi_Z(-\hat y/\gamma)$, and using this same map to define both the training loss (the RSPO loss $\ell_{\mathrm{RSPO}}(\hat y,y)=y^\top z^\star_\gamma(\hat y)-v^\star(y)$) and the deployed decision, yields a framework that is simultaneously robust and learning–decision aligned. Three mathematical assertions carry the claim. First, the projection identity makes $z^\star_\gamma$ $1/\gamma$-Lipschitz and differentiable almost everywhere, so the training objective carries informative gradients, and the RSPO loss is Lipschitz in the prediction — the basis for a meta excess-risk bound of order $O_p(N^{-1/2})$ with an explicit factor $r/\gamma$. Second, the convex surrogate RSPO+ upper-bounds the RSPO loss, is differentiable everywhere, and, with its free parameter set to $a=1$, has unique population minimizer $2\bar y(x)$ under a central-symmetry condition on $y\mid x$; because the set of target-risk minimizers is the translated normal cone $Y^\star_{\mathrm{RSPO}}=-N_Z(z^\star(\bar y))-\gamma z^\star(\bar y)$, Fisher consistency holds exactly when $0<\gamma\le\bar\gamma(x)$, where $\bar\gamma(x)=\sup\{\gamma\ge 0:2\bar y+\gamma z^\star(\bar y)\in -N_Z(z^\star(\bar y))\}$ is computable as a linear program for polyhedral $Z$. Third, outside that regime the surrogate can converge to a predictor that is asymptotically suboptimal under the target RSPO risk — a nonvanishing bias the paper constructs explicitly in Example 1.

Load-bearing premise

The load-bearing premise is distributional symmetry: conditional on the context, the cost vector must be spread symmetrically about its mean, with positive density around twice that mean, so that the surrogate's best predictor is exactly twice the conditional mean — if the noise is skewed, the surrogate can lock onto a predictor that is secretly biased for the robust objective even with unlimited data, and the theory also needs the nominal problem at the true mean to have one unique optimal solution.

Editorial extensions

If this is right

  • Choosing $\gamma\le\bar\gamma(x)$ makes surrogate training statistically honest: with unlimited data the predictor converges to a target-risk minimizer, and with finite samples the target excess risk decays at rate $O_p(N^{-1/4})$, improving to $O_p(N^{-1/2})$ when the target risk grows no faster than quadratically near the oracle predictor.
  • The excess-risk bounds scale as $r/\gamma$, so a more robust formulation (larger $\gamma$) needs fewer samples for the same statistical guarantee — the paper states the concrete recommendation that, when data are limited, the decision-maker should seek more robust decisions.
  • Because the robust decision map is differentiable almost everywhere, the original nonconvex training objective can be descended by first-order methods; in the portfolio experiments, gradient-based refinement of the surrogate solution lowers out-of-sample decision loss further.
  • Against the post-hoc robust benchmark (identical robust deployment map, nominal training loss), the proposed RSPO+ training attains equal or lower normalized decision loss, with the largest margins at small sample sizes, high decision dimension, and stronger model misspecification — isolating the value of learning–decision alignment itself.
  • Decision gains appear even when relative prediction loss is worse than the benchmarks, implying that aligned training deliberately trades raw forecasting accuracy for decision-relevant accuracy.

Reading between the lines

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

  • A model-based calibration rule the paper leaves implicit: because the provably honest regime is $0<\gamma\le\mathrm{ess\,inf}_x\,\bar\gamma(x)$, a practitioner could restrict hyperparameter tuning to that interval, converting $\gamma$ from a purely heuristic knob into a quantity with a proven upper bound.
  • Remark 2 shows the $\ell_2$ penalty is one instance of a general strongly convex $\phi$, with $1/\mu$ in place of $1/\gamma$; if the Fisher-consistency threshold and the $N^{-1/2}$ rates survive for a non-quadratic $\phi$ such as a Huber-type penalty, the framework extends well beyond quadratic regularization — a direct testable generalization of the paper's own remark.
  • The projection identity suggests a template for other discontinuous decision maps: adding a strongly convex perturbation to a non-differentiable optimization layer smooths it into a trainable surrogate, and this paper's theory predicts the price of that smoothing is exactly the factor $1/\gamma$ in the excess-risk bounds.
  • A natural experiment the paper does not run: hold $\gamma$ fixed at several values and plot excess risk versus $N$ on a log-log scale; the theory predicts slope $-1/2$ for the RSPO predictor and $-1/4$ (or $-1/2$ under quadratic growth) for the surrogate, so the reported rates are directly testable.
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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

2 major / 3 minor

Summary. The paper proposes an integrated learning and robust optimization (ILRO) framework for contextual linear optimization. The robust decision map z*_γ(ŷ) = argmin_{z∈Z} ŷᵀz + (γ/2)‖z‖² is used both as the deployed decision rule (ILRO-Decision) and as the training loss (ILRO-Learning). Since this map is the projection Π_Z(−ŷ/γ), the RSPO loss is Lipschitz and differentiable almost everywhere, unlike the nominal SPO loss. The authors construct a convex surrogate RSPO+ by a limiting representation and linearization, prove its Fisher consistency when γ does not exceed an explicitly computable threshold γ̄(x), derive a meta excess-risk bound for the RSPO predictor (Theorem 3), and provide excess-risk bounds for the RSPO+ predictor under the Fisher-consistent regime (Theorem 4, Corollary 3). Numerical experiments on transportation and portfolio problems compare RSPO+ with least squares, SPO+, and post-hoc robustified SPO+, reporting that RSPO+ achieves the lowest or comparable normalized decision loss.

Significance. If the guarantees hold as stated, the paper makes a substantial contribution to decision-focused learning: it is the first framework to use the same robust decision map in training and deployment, it provides a convex surrogate with an explicit Fisher-consistency threshold computable by a linear program, and it gives finite-sample rates for both the target and surrogate predictors. The mathematical development is careful and largely self-contained: Lemma 1 derives the projection representation, Proposition 4 and Theorem 2 characterize the target minimizer set and the consistency threshold, and Theorem 3 cleanly separates the role of the robust map (r/γ) from the hypothesis-class complexity (Rademacher complexity). The paper also ships explicit finite-dimensional reformulations (Theorem 1), a closed-form Jacobian of the decision map (Proposition 6), and detailed proofs in the electronic companion. The empirical comparison is well designed, with the post-hoc robust benchmark isolating the value of learning–decision alignment.

major comments (2)
  1. [Section 5.2, Condition 1(iii) and Theorem 4] Theorem 2(ii) establishes only a pointwise lower bound γ̄(x) ≥ 2 dist(ȳ(x), ∂C(x))/‖z*(ȳ(x))‖, which can vanish as ȳ(x) approaches a normal-cone chamber boundary. The text in Section 5.2 states that Condition 1(iii) holds for a fixed γ > 0 'as soon as γ ≤ ess inf_x γ̄(x)', but no assumption in the paper rules out ess inf_x γ̄(x) = 0. This is not vacuous: take the feasible set Z of Example 1 and ȳ(t) = (1,t)ᵀ for t > 0 decreasing to 0. The nominal solution remains (−1,−2), the conditional Gaussian density is positive and centrally symmetric, and the threshold computation gives γ̄(t) = t (the active-cone conditions yield λ₁ = 2t − 2γ and λ₂ = 2t + 2 − 3γ, so the binding constraint is γ ≤ t). If the context distribution assigns positive mass to t_n = 1/n, then for every fixed γ > 0 there is a positive-probability set of x with γ̄(x) < γ, so the unique surrogate minimizer 2ȳ(x) is not a minimizer of the target RSPO risk and Condition 1(iii) fails. Consequently, Theorem 4's bound (20) and Corollary 3's O_p(N^{-1/2}) rate do not follow from the stated assumptions: Step 5 of the proof of Theorem 4 uses Condition 1(iii) to assert R(g₀) = R⋆, and without the uniform positivity of γ̄(x) that equality can fail. This is a load-bearing gap in the paper's finite-sample guarantees for the RSPO+ predictor.
  2. [Section 5.2, Condition 1(ii) and Lemma EC.3] The same uniformity issue arises for the local strong-convexity condition. Lemma EC.3 proves (p₀(x)/γ)-strong convexity of R+(·;x) on B(2ȳ(x), ρ) with p₀(x) = P_{y|x}(y ∈ 2ȳ(x) + γK_ρ), and positivity of p₀(x) follows pointwise from the density-positivity assumption on an open set containing 2ȳ + γZ. The paper then says Condition 1(ii) holds with µ = ess inf_x p₀(x)/γ 'provided ess inf_x p₀(x) > 0', but no assumption in the paper guarantees this essential infimum is positive. The same chamber-boundary construction (or any family of conditional distributions whose concentration region approaches the boundary of the relevant set) makes p₀(x) arbitrarily small on a positive-probability set, so the uniform curvature constant µ required in Theorem 4 may fail to exist. Since Condition 1(ii) is stated as an assumption the theorem is not false as a conditional statement, but the paper's presentation suggests that its distributional conditions provide the curvature; they provide it only pointwise, and the uniform version is an additional, unverified hypothesis.
minor comments (3)
  1. [Section 6] The empirical section reports only boxplots over 20 replications and compares medians; adding confidence intervals or error bars would strengthen the claims about RSPO+ being the lowest or comparable, especially because several figures show overlapping boxes at small sample sizes.
  2. [Abstract and Section 5.2] The abstract states that the finite-sample excess-risk bounds hold for the RSPO+ predictor, but the bound in Theorem 4 requires Condition 1, whose uniform version is not established from the paper's stated distributional assumptions. The abstract should qualify these bounds as conditional on the uniform consistency regime.
  3. [Figure 8] The first two rows of Figure 8 are visually nearly identical and the figure caption does not clearly distinguish the two deg settings within the same panel; labelling each row or differentiating line styles would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: RSPO+ consistency and excess-risk bounds are conditional theorems derived from stated assumptions, with independent external benchmarks.

full rationale

The central claim—that ILRO trains and deploys through the same robust decision map—is a modeling choice, not a result derived from itself. The RSPO loss and the RSPO+ surrogate are defined from the decision map, and Fisher consistency is then proved by separately characterizing the target minimizers (Proposition 4) and the surrogate minimizers (Proposition 5 and Corollary 1). The threshold gamma_bar(x) in equation (12) is computed from problem data, and Theorem 2 only asserts consistency when 0 < gamma <= gamma_bar(x); Example 1 explicitly demonstrates failure when gamma > gamma_bar(x), which shows the result is falsifiable and not true by construction. The finite-sample excess-risk bounds in Theorems 3 and 4 rest on explicit assumptions (bounded costs, Rademacher complexity, Condition 1, local strong convexity, Fisher-consistent membership) and follow from standard generalization arguments; the Fisher-consistent regime is an assumption, not a fitted parameter renamed as a prediction. The only self-citations (Hu et al. 2025 and Mao et al. 2026, which share author Shuming Wang) are background literature on robust pipelines and are not load-bearing for any theorem in this paper. The reviewer-flagged concern that ess inf_x gamma_bar(x) may be zero, making the Fisher-consistent regime empty for some fixed gamma, is a correctness and assumption-coverage issue rather than a circularity issue: the theorems state conditional claims, and if the assumptions are not guaranteed by the stated conditions, the bounds may not follow, but that is not a self-referential derivation. Accordingly, no circular step was identified.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical or model entities are introduced; the only new objects are mathematical, namely the RSPO loss and the RSPO+ surrogate. The central claim rests on domain restrictions such as density and symmetry of the cost distribution, uniqueness of the nominal solution, polyhedrality of the feasible set, and the exogenous robustness parameter gamma.

free parameters (3)
  • gamma (robustness penalty) = Exogenous in theory; selected by validation from five logarithmically spaced values in [1e-6, 10] in experiments
    The entire ILRO objective and deployed decision change with gamma. Theory requires gamma below a pointwise threshold gamma_bar(x) for Fisher consistency, but the paper provides no data-driven rule to guarantee this, leaving gamma a tuned hyperparameter.
  • lambda (regularization weight) = Selected from five log-spaced values in [1e-3, 10] via validation
    Regularizer in the empirical risk minimization problems (6) and (8); standard tuning, but it affects the reported decision losses.
  • a (surrogate scale) = Fixed at 1 in theory and experiments
    The choice a=1 yields the closed-form surrogate minimizer 2*y_bar(x) under symmetry; other values alter the surrogate and its consistency, as discussed in Remark 4.
assumptions (6)
  • domain assumption Conditional distribution of y given x is absolutely continuous with density positive on an open set containing 2*y_bar + gamma*Z and centrally symmetric about y_bar
    Corollary 1 and Condition 1: without this, the unique surrogate minimizer 2*y_bar is not guaranteed, and Theorem 4's local strong convexity may fail.
  • domain assumption The nominal problem min_{z in Z} y_bar(x)^T z has a unique optimal solution z*(y_bar(x)) for almost every x
    Proposition 4 and Theorem 2 need uniqueness to identify the target-risk minimizer set as a translated normal cone and to ensure the consistency threshold is positive; degeneracy breaks the characterization.
  • domain assumption Feasible set Z is compact convex, and for reformulations and Jacobian results, a bounded polyhedron with nonempty interior
    Lemma 1 requires compactness; Theorem 1 and Proposition 6 require polyhedrality. These are stated but restrict the scope of the framework.
  • domain assumption Oracle predictor g0(x)=2*y_bar(x) lies in the hypothesis class G up to null sets for the N^{-1/2} rates
    Theorem 4 and Corollary 3 assume A^+_G=0, i.e., g0 in G. If the class is misspecified, the target excess risk contains a nonvanishing approximation error.
  • standard math Standard measure-concentration tools: Rademacher bounds, vector contraction inequality, McDiarmid's inequality
    Used throughout Section 5 and Appendix B; these are well-established and not proved in the paper.
  • ad hoc to paper Choice of surrogate scale a=1
    The factor a is not forced by the derivation; a=1 is singled out because it makes 2*y_bar the pointwise minimizer under symmetry (Corollary 1), which is then used for Fisher consistency and risk analysis.

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Cite this review

Pith. "Pith review of Integrated Learning and Robust Optimization." pith.science (2026). https://pith.science/paper/4QLQLY4E

@misc{pith2026260808718,
  author       = {Pith},
  title        = {Pith review of: Integrated Learning and Robust Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QLQLY4E}},
  note         = {Machine review of arXiv:2608.08718}
}
read the original abstract

Many operational decisions require solving a linear program whose cost vector is unknown at decision time and must be predicted from contextual information. Because prediction and decision are only weakly aligned, the emerging integrated learning and optimization (ILO) paradigm trains the predictor through the downstream problem, judging a prediction by the decision it induces. However, predictions are inevitably imprecise, so robustness often enters the decision stage. To address this issue, we propose an integrated learning and robust optimization (ILRO) framework, where a robust decision problem is used both to define the training problem (termed the RSPO loss problem), and to produce the deployed decision. Thus, this framework simultaneously achieves both robustness and learning-decision alignment. To tackle its computational challenges, we develop a convex surrogate, RSPO+, and characterize when it is Fisher consistent. Moreover, the RSPO loss possesses informative gradients, allowing us to develop first-order computational methods. We also derive finite-sample excess risk bounds for both RSPO and RSPO+ predictors. Numerical experiments on transportation and portfolio problems, in comparison with multiple benchmarks, show the advantage in decision quality of the proposed framework. The gain is more pronounced for scenarios with limited samples, high-dimensional decisions, and model misspecification.

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Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

  1. [1]

    Advances in Neural Information Processing Systems, volume 32, 9562–9574

    Agrawal A, Amos B, Barratt S, Boyd S, Diamond S, Kolter JZ (2019) Differentiable convex optimization layers. Advances in Neural Information Processing Systems, volume 32, 9562–9574. 38 Amos B, Kolter JZ (2017) OptNet: Differentiable optimization as a layer in neural networks.Proceedings of the 34th International Conference on Machine Learning, volume 70, ...

    work page 2019

  2. [2]

    Since γ >0, the objective is strongly convex, so the minimizer is unique

    BecauseZis nonempty and compact and the objective is continuous, a minimizer exists. Since γ >0, the objective is strongly convex, so the minimizer is unique. The second term in the display is independent ofz; therefore, z⋆ γ(y0) = arg min z∈Z z+ y0 γ 2 2 = ΠZ − y0 γ , which proves the projection representation. Thus, for anyy 1,y 2, the nonexpansiveness ...

    work page 2016

  3. [3]

    The scalar antecedents of the complexity bounds below are classical; the content of this appendix is their adaptation to the multivariate Rademacher complexityRd of Maurer (2016), with the explicit constants required by Table

    work page 2016

  4. [6]

    The common data-generating, valida- tion, testing, and evaluation procedures are stated in that section. E.1 Portfolio Learning Reformulations For the main portfolio experiment, the constraints in Section 6.2 define a bounded polyhedron after the standard epigraph reformulation of∥Σz∥1. Theorem 1 therefore yields finite-dimensional learning formulations f...

    work page 2022

  5. [7]

    Fix an arbitraryϵ >0and chooseg ϵ ∈ Gsuch thatR(gϵ)≤ R⋆(G) +ϵby the definition of the infimum. On the eventE, we have R(bgRSPO)−R ⋆(G) = h R(bgRSPO)− bR(bgRSPO) i + h bR(bgRSPO)− bR(gϵ) i + h bR(gϵ)−R(g ϵ) i + [R(gϵ)−R ⋆(G)] ≤2 sup g∈G R(g)− bR(g) + 0 +ϵ ≤4R1(HℓRSPO ) +b r 2 log(2/δ) N +ϵ, ec16e-companion toC. T an, Y. Mao, S. W ang, H. Xu:Integrated Lear...

    work page 2016

  6. [8]

    Strong duality and dual attainment hold because the primal is a concave quadratic maximization over a nonempty bounded polyhedron

    For fixedp i, maximizing overz∈R d gives sup z∈Rd Li(z,p i) =−b ⊤pi + 1 2aγ ∥ci +A ⊤pi∥2 2, where the supremum is attained atz= 1 aγ (ci +A ⊤pi). Strong duality and dual attainment hold because the primal is a concave quadratic maximization over a nonempty bounded polyhedron. Indeed, for convex quadratic programs with affine constraints, strong duality re...

    work page 2004

  7. [9]

    Step 4: conclusion.By Steps 1-3, for everyxthe unique minimizer of the pointwiseSPO + risk witha= 1belongstoY ⋆ SPO.IntegratingoverP x viatheinterchangeabilityprinciple,asinSection4.2, yields Fisher consistency of theSPO+ loss witha= 1with respect to theSPOloss.□ Proof of Proposition 6.LetS=S y0 for notational simplicity and writez 0 =z ⋆ γ(y0). Since the...

    work page 2002

  8. [12]

    NX i=1 σiφ(xi)⊤ F # . By Jensen’s inequality, Eσ,X

    In each example, the excess risk bound follows from the complexity bound by a direct application of Theorem 3, so only the complexity bounds require proof. D.1 Bounded Affine Class We begin with affine predictors, which cover the linear models used in predict-then-optimize pipelines and yield an explicit dimension-dependent bound. The scalar analogue of t...

    work page 2006

Show all 14 references
  1. [13]

    Advances in Neural Information Processing Systems, volume 33, 9508–9519

    Berthet Q, Blondel M, Teboul O, Cuturi M, Vert JP, Bach F (2020) Learning with differentiable perturbed optimizers. Advances in Neural Information Processing Systems, volume 33, 9508–9519. Bertsimas D, Kallus N (2020) From predictive to prescriptive analytics.Management Scienc...

    2020

  2. [14]

    Taking expectation over the sample establishes (EC.15).□ Remark EC.1 (The two classes are not nested).Neither class contains the other

    N . Taking expectation over the sample establishes (EC.15).□ Remark EC.1 (The two classes are not nested).Neither class contains the other. A bounded affine class generates infinitely many prediction patterns already on even a single sam- ple point: for the identity context ma...

    2018

  3. [21]

    Kannan R, Bayraksan G, Luedtke JR (2025) Technical note—data-driven sample average approximation with covari- ate information.Operations Research73(6):3245–3259

    39 Kannan R, Bayraksan G, Luedtke JR (2024) Residuals-based distributionally robust optimization with covariate information.Mathematical Programming207(1):369–425. Kannan R, Bayraksan G, Luedtke JR (2025) Technical note—data-driven sample average approximation with covari- ate...

    2024 doi

  4. [71]

    predict, then optimize

    Bertsimas D, Sim M (2004) The price of robustness.Operations Research52(1):35–53. Boyd S, Vandenberghe L (2004)Convex Optimization(Cambridge University Press). Chenreddy AR, Delage E (2024) End-to-end conditional robust optimization.Proceedings of the Fortieth Conference on Un...

    2004 arXiv

  5. [1376]

    Integrated Learning and Robust Optimization

    Wilder B, Dilkina B, Tambe M (2019) Melding the data-decisions pipeline: Decision-focused learning for combinatorial optimization.Proceedings of the AAAI Conference on Artificial Intelligence33(1):1658–1665. Yeh C, Christianson N, Wu A, Wierman A, Yue Y (2025) End-to-end confo...

    2019

  6. [1701]

    Massart P (2000) Some applications of concentration inequalities to statistics.Annales de la Faculté des sciences de Toulouse: Mathématiques9(2):245–303

    MaoY,Saldanha-daGamaF,WangS,WangS(2026)Predictiveproduction-and-serviceplanning:Ambiguityaversion with performance guarantees.Management ScienceForthcoming. Massart P (2000) Some applications of concentration inequalities to statistics.Annales de la Faculté des sciences de Tou...

    2026

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