A Survey of Contextual Optimization Methods for Decision Making under Uncertainty

1. Introduction

• Contextual optimization uses covariates to estimate an uncertain parameter’s conditional distribution, since decisions based on unconditional distributions can be suboptimal or infeasible.The survey focuses on single- and two-stage contextual optimization.Source: §1 · p. 3
• Decision rule optimization selects the best-performing parameterized covariate-to-action mapping using available data; rules may be linear combinations of functions or deep neural networks.The approach was introduced to operations research by Liyanage and Shanthikumar (2005), popularized by Ban and Rudin (2019), and resembles policy-gradient methods in reinforcement learning (Sutton et al. 1999).Source: §1 · pp. 3–4
• SLO first predicts the conditional distribution of uncertain parameters from covariates, then solves a contextual stochastic optimization problem to obtain an action.To reduce post-decision disappointment from overfitting or misspecification, it can use training-time regularization or adapt the optimization formulation (Smith and Winkler 2006).Source: §1 · p. 4
• ILO searches for a predictive model that steers the optimization problem toward high-performing actions, focusing learning on decision quality rather than prediction precision.The survey covers active strands including smart predict-then-optimize, decision-focused learning, and task-based end-to-end learning, forecasting, or optimization.Source: §1 · p. 4
• The survey extends its ILO review beyond expected value-based models by framing contextual decisions as contextual stochastic optimization problems and linking regret minimization, end-to-end learning, and imitation-based models.It also develops a general ILO taxonomy covering differentiable surrogates and optimizers, unrolling, implicit differentiation, and training procedures.Source: §1 · p. 5

2. Contextual optimization: An overview

• An action z is chosen from a feasible set to minimize cost under unknown parameters y, after observing correlated covariates x.The joint distribution of covariates and uncertain parameters is denoted P.Source: §2 · p. 5

2.1. Contextual problem and policy

• For a risk-neutral decision-maker, the CSO action minimizes expected cost under the conditional distribution P(y|x), and an optimal policy maps covariates to actions.The paper’s main sections focus on uncertain objectives; risk-averse settings and uncertain constraints are discussed in Section 6.Source: §2.1 · pp. 5–6

2.2. Mapping covariate to actions in a data-driven environment

• Because the joint distribution is generally unknown, decision-makers use observed data to approximate the optimal policy; most methods fall into three frameworks.The frameworks are decision rule optimization, sequential learning and optimization, and integrated learning and optimization.Source: §2.2 · p. 6
• Decision rule optimization directly maps covariates to actions through a parameterized policy, but restricts the policy class and optimizes empirical average performance rather than uniform performance across covariates.The empirical distribution also substitutes for the true distribution.Source: §2.2 · p. 7

expected value-based models).

• Sequential learning estimates the conditional distribution by minimizing prediction error, whereas integrated learning trains the predictor to minimize downstream decision costs.Integrated learning addresses the mismatch between prediction divergence and cost: small prediction errors can have large prescription impacts, while higher MSE may still yield near-optimal performance.Source: expected value-based models) · pp. 8–9
• If costs are linear in y, estimating the conditional distribution reduces to mean regression, and mean squared error asymptotically recovers the conditional mean under standard conditions.The survey calls these expected value-based models; more general approaches use conditional-distribution estimators.Source: expected value-based models) · pp. 8–9
• Expected value-based training minimizes average historical costs, making its induced policy optimal for the predicted distribution, but requires solving an optimization model for each data point.The procedure also differentiates through the optimization model, adding computational burden.Source: expected value-based models) · pp. 9–10
• A small prediction error can lead to a suboptimal action, even when another prediction yields the optimal action.Figure 4 illustrates this distinction under c(x,y) := −y⊤x.Source: expected value-based models) · p. 10, Figure 4 · p. 10

2.3. Summary

• 2.3. Summary: Training loss determines whether a pipeline uses the decision-rule, sequential, or integrated paradigm, while predictor class and hyperparameters also shape its inductive bias.The paradigms respectively use ERM, minimize estimation error, or minimize downstream policy cost.Source: §2.3 · pp. 10–11
• 2.3. Summary: With limited data, it is unclear in advance which design choices will perform best, so pipelines must be evaluated experimentally.These choices can entail specific methodological challenges, especially for ILO.Source: §2.3 · p. 11

3. Decision rule optimization

• Policy-based methods use a parameterized mapping such as a linear rule or neural network, and require only policy evaluation at decision time.Linear rules are widely used because neural-network policies lack interpretability.Source: §3 · p. 11

3.1. Linear decision rules

• Ban and Rudin (2019) show that ignoring side information can yield asymptotically suboptimal newsvendor decisions, while regularization trades off generalization error and bias.They focus on unconstrained problems because ERM makes it difficult to ensure policy feasibility while maintaining computational tractability; LDRs are not generally asymptotically optimal.Source: §3.1 · p. 12

3.2. RKHS-based decision rules

• RKHS-based policies lift covariates to allow more flexible rules, and Bertsimas and Koduri (2022) establish their asymptotic optimality.For high-dimensional problems, the Gaussian kernel has a slow convergence rate.Source: §3.2 · pp. 12–13

3.3. Non-linear decision rules

• Zhang et al. (2023b) provide consistency results for piecewise-affine decision rules and show they can outperform alternatives on a constrained newsvendor problem.They learn the policy using stochastic majorization-minimization. Neural-network outputs may violate feasibility; softmax can enforce simplex constraints, while application-specific differentiable repair layers can project decisions back to feasibility.Source: §3.3 · pp. 13–14

3.4. Distributionally robust decision rules

• Zhang et al. (2023a) study a distributionally robust contextual newsvendor under a type-1 Wasserstein ambiguity set without assuming explicit structure, using a Shapley extension to avoid degeneracies of ERM for generic Π.Source: §3.4 · p. 14

4.1. Learning conditional distributions

• Discrete conditional distributions are common because they make CSO easier to solve, while continuous distributions often require SAA and make unseen-outcome probabilities difficult to assess.This motivates fixing the support of y to observed values; continuous-distribution CSO often must first be replaced by SAA (Shapiro et al. 2014).Source: §4.1 · p. 15
• Residual-based methods pair any trained regression model with empirical residuals to construct conditional distributions, but reusing training data can underestimate residual error.Kannan et al. (2020) use leave-one-out, or jackknife, residuals to remove this bias.Source: §4.1 · p. 16
• Weight-based SAA assigns historical observations weights, including via kNN, kernel estimators, or random forests, and kNN has asymptotic consistency guarantees for prescriptive performance.Bertsimas and Kallus (2020) give conditions for asymptotic optimality and consistency of prescriptions using kNN, the Nadaraya-Watson kernel estimator, and local methods.Source: §4.1 · pp. 16–17
• When costs are linear, SLO reduces to conditional mean estimation; alternatively, a point prediction can approximate the conditional density, as in last-mile travel-time prediction.Liu et al. (2021) replace the conditional travel-time distribution with a point prediction using covariates such as stop count and trip distance.Source: §4.1 · p. 18

4.2. Regularization and distributionally robust optimization

• Nonparametric conditional density methods can be asymptotically consistent yet overly optimistic with many covariates, motivating regularization or distributionally robust optimization.Bertsimas and Kallus (2020) and Notz and Pibernik (2022) are cited for consistency; Bertsimas and Van Parys (2022) discuss the high-dimensional limitation.Source: §4.2 · p. 18
• Distributionally robust optimization minimizes worst-case expected cost over distributions within distance r of an estimated distribution; CVaR portfolio problems admit convex reformulations, with conditions for asymptotic convergence and out-of-sample guarantees.Source: §4.2 · p. 18
• Trimming empirical conditional distributions yields reweighted ambiguity sets linked to partial mass transportation; Kannan et al. (2021) and Kannan et al. (2022) extend rCSO through Wasserstein DRO.Their ambiguity ball has radius r and is centered at the estimated distribution PER(x).Source: §4.2 · p. 19
• Perakis et al. (2023) propose a DRO model for two-stage multi-item joint production and pricing using a partitioned-moment ambiguity set built by clustering residuals.Source: §4.2 · p. 19
• Zhu et al. (2022) inform an expected-value model’s ambiguity set with the estimation metric used to train gˆθ and derive finite-dimensional convex reformulations for linear gθ(x) := θTx.They also promote a “robustness optimization” form.Source: §4.2 · p. 19

5. Integrated learning and optimization

• ILO combines a predictor, an optimizer that maps predictions to decisions, and a task-based loss; predictor parameters are trained to improve the induced policy’s prescriptive performance.The predictor may produce a predicted distribution or a point prediction.Source: §5 · p. 19
• ILO training algorithms use unrolling, implicit differentiation, surrogate differentiable loss functions, or differentiable optimizers.The survey groups its implementation methods into these four categories.Source: §5 · p. 20

5.1. Models

• Expected-value ILO usually trains a point predictor of E[y|x] using expected cost, but its integrated loss is non-convex and can be difficult to differentiate.For a linear-program solution, the gradient may be null or nonexistent because the solution jumps between extreme points as the objective changes.Source: §5.1 · p. 20
• Conditional-distribution ILO models uncertain-parameter distributions using parametric forms (exponential and normal), whereas weighted SAA identifies weights for historical samples; Kallus and Mao (2022) use this approach with an integrated random-forest regressor.Source: §5.1 · pp. 20–21
• Under model misspecification, ILO outperforms decision-rule optimization with neural networks and SLO with maximum likelihood estimation (MLE) on the newsvendor problem.The conditional-distribution approach models uncertain parameters with parametric distributions, including exponential and normal distributions.Source: §5.1 · pp. 20–21
• Minimizing expected regret—comparing a decision based on the mean estimate with the hindsight decision—returns the same optimal parameter vector as expected-value ILO.Both the regret and expected-cost objectives have the same set of minimizers.Source: §5.1 · p. 22
• Action imitation trains a policy to produce decisions close to the optimal hindsight action, but a general distance need not preserve performance under the original task metric.Using cost difference as the distance recovers regret minimization and therefore the same solution as expected-cost ILO.Source: §5.1 · p. 22

5.2. Training by unrolling

• Unrolling approximates the optimization problem with an iterative solver, stores each operation in a computational graph, and enables gradients to be computed by back-propagation.Source: §5.2 · p. 23
• The approach requires extensive amounts of memory because of the large computational graph.Source: §5.2 · p. 23
• The large computational graph exacerbates neural networks’ vanishing and exploding gradient problems (Monga et al. 2021).Source: §5.2 · p. 23

5.3. Training using implicit differentiation

• 100-times speedups compared to Gurobi and CPLEX are reported for OptNet’s batched GPU solver for constrained quadratic programs.Amos and Kolter use KKT conditions and the implicit function theorem to differentiate the optimal solution.Source: §5.3 · p. 23
• On a 254-asset portfolio, ADMM-FP reduced computational time by almost five relative to OptNet, mainly through ADMM in the forward pass (Butler and Kwon 2023a).Their empirical comparison found ADMM-FP faster than unrolling ADMM iterations on the computational graph.Source: §5.3 · p. 23
• cvxpylayers converts disciplined convex programs into conic programs and implicitly differentiates their residual map; PGD projects onto Z each iteration, but when closed-form projection is unavailable in many cases, projection may be costly and PGD may not converge.Source: §5.3 · p. 24
• FP folding analytically differentiates iterative solvers such as ADMM, SQP, and PGD, decoupling forward and backward passes so black-box solvers can be used for the forward pass (Kotary et al. 2023).It can also handle non-convex problems; in portfolio optimization, the authors attribute its superior performance over cvxpylayers to precise Gurobi forward-pass calculations.Source: §5.3 · p. 24
• Alt-Diff uses ADMM to decouple objectives and polyhedral constraints, alternating gradients for primal, dual, and slack variables to reduce Jacobian size when constraints are numerous (Sun et al. 2023b).Its gradients converge to those from KKT differentiation; with iteration truncation, it achieved the same accuracy in less time than cvxpylayers for energy-generation scheduling.Source: §5.3 · p. 25
• IntOpt performs better than QPTL on energy cost-aware scheduling problems; Mandi and Guns (2020) propose it using an interior-point method to differentiate an LP’s homogeneous self-dual formulation.Source: §5.3 · p. 25
• Costa and Iyengar (2023) replace an ILO problem with a surrogate DRO model, using implicit differentiation to learn parameters such as ambiguity-set size instead of cross-validation.Their framework uses a weighted Sharpe-ratio and MSE task loss, with a deviation risk measure controlling return variability under a φ-divergence ambiguity set.Source: §5.3 · p. 25
• Classical implicit differentiation requires a nonsingular solution Jacobian, a condition that can fail in linear programs and smooth quadratic programs (Bolte et al. 2021).Bolte et al. generalize the implicit function theorem to nonsmooth functions using conservative Jacobians, including for conic optimization layers.Source: §5.3 · p. 26
• Bolte et al. (2021) use cvxpylayers to show gradient descent on a quadratic composed with an LP solution map stalls in a non-critical limit cycle as the map switches extreme points.The invertibility condition fails only on a measure-zero line.Source: §5.3 · p. 26
• IFT-based gradient methods can be affected by a non-invertible Jacobian and nonsmoothness, which is difficult to verify a priori.Source: §5.3 · p. 26
• Researchers have pursued surrogate loss functions and perturbation-based models for contextual stochastic optimization to avoid using the IFT.Source: §5.3 · p. 26

5.4. Training using a surrogate differentiable loss function

• Elmachtoub and Grigas (2022) replace the nonconvex, discontinuous SPO loss with SPO+, a convex upper bound with a closed-form subgradient; Loke et al. (2022) combine prediction accuracy and decision quality in DDR.SPO+ is a special case of DDR under specified parameter settings.Source: §5.4 · p. 27
• SPO-relax converges faster with performance similar to SPO+, and yields lower regret and faster convergence than QPTL except for weighted knapsack with low capacity.For MILPs, it computes SPO+ gradients using a continuous relaxation; the approach addresses the computational burden of solving NP-hard optimization problems.Source: §5.4 · p. 27
• Jeong et al. (2022) report up to two orders of magnitude lower expected regret than SPO+ on the training set using a single-level MILP reformulation.Their approach assumes a linear prediction model and a unique optimal decision, and uses symbolic variable elimination to reformulate the bilevel SPO problem.Source: §5.4 · pp. 27–28
• Under specified conditions, SPO, SPO+, and MSE minimizers coincide when the conditional mean lies in H, making SPO+ Fisher consistent; however, SPO+ can be inconsistent for some multiclass problems.The cited consistency analysis assumes distribution knowledge, whereas decision-makers typically have only samples.Source: §5.4 · p. 28
• SLO can converge faster than ILO under certain low-noise assumptions, and stochastically dominates ILO in asymptotic optimality gaps for nonlinear problems when the hypothesis class covers the true distribution.When the model is misspecified, ILO asymptotically outperforms SLO in a general nonlinear setting.Source: §5.4 · p. 29
• Kallus and Mao (2022) propose StochOptForest, which avoids optimization at each candidate split and is asymptotically consistent; Elmachtoub et al. (2020) report that SPO Trees outperform CART.The SPO Trees experiments include a news recommendation problem with real-world data.Source: §5.4 · pp. 28–29

5.4.3. Other surrogates.

• Wilder et al. (2019b) introduce ClusterNet, which uses graph embeddings and soft k-means as a differentiable proxy and outperforms two-stage SLO and other baselines on synthetic data.The experiments cover community detection and facility location.Source: §5.4.3 · p. 29
• Vlastelica et al. (2019) approximate the task loss with a gradient-informative interpolation, while Chung et al. (2022) use task-loss gradients to weight an MSE surrogate.The first approach perturbs predictions to estimate the surrogate gradient; the second uses a first-order Taylor expansion of task loss.Source: §5.4.3 · p. 29
• For medical allocation in Sierra Leone, model (17) solves the optimization once during training and is shown scalable relative to Kallus and Mao (2022)'s task-loss-based feature splitting.Source: §5.4.3 · p. 30
• Lawless and Zhou (2022) combine prediction error with regret, using a squared-error pilot before substituting its induced decision into the loss.Source: §5.4.3 · p. 30
• Lawless and Zhou (2022) give a two-step polynomial-time approximation that performs comparably to SPO+.Source: §5.4.3 · p. 30
• Sun et al. (2023a) learn a cost point estimator to imitate hindsight-optimal solutions by penalizing violations of optimal-basis conditions, deriving generalization-error and suboptimality bounds.Source: §5.4.3 · p. 30

5.5. Training using a surrogate differentiable optimizer

• Perturbing predicted parameters with differentiable-density noise makes the expected solution and associated random-policy cost smooth and differentiable, despite the unperturbed solution gradient being zero almost everywhere.Berthet et al. (2020) exploit this approach, and Monte Carlo sampling of perturbations and solving the perturbed problems can evaluate gradients.Source: §5.5 · pp. 30–31
• Mulamba et al. (2021) use noise-contrastive estimation to address the partition-function challenge, and report improved expected-regret performance over SPO+ and DBB for linear problems.The surrogate replaces likelihood with relative likelihood against sampled suboptimal solutions.Source: §5.5 · p. 32
• Kong et al. (2022) report lower computational time than ILO trained with SQP and cvxpylayers, and lower task loss than sequential MLE models and neural-network policy learning.Their method uses self-normalized importance sampling to approximate the partition function and KL-divergence regularization to avoid overfitting.Source: §5.5 · p. 32

5.5.2. Supervised learning.

• Grigas et al. (2021) show that ICEO-λ has finite-sample guarantees and is asymptotically optimal and consistent when the hypothesis class is well-specified.Their convex, nonnegative decision regularizer assumes discrete support for the uncertain parameter and ensures uniqueness and Lipschitz continuity of the optimizer.Source: §5.5.2 · pp. 32–33
• Cristian et al. (2022) introduce ProjectNet, an end-to-end policy network for uncertain constrained linear programs that uses a differentiable approximation of projection to feasibility.Source: §5.5.2 · p. 33
• Shah et al. (2022) train locally optimized decision losses (LODL) to evaluate predictors on the downstream task, and report better performance than SLO on three resource-allocation problems.They fit convex LODL functions to downstream task losses using generated labels near each training label; the problems are linear top-1 item selection, web advertising, and portfolio optimization.Source: §5.5.2 · p. 33
• Handcrafted surrogate functions perform better only on the web-advertising problem in the reported experiments.Source: §5.5.2 · p. 33

Applications

• Perrault et al. (2020) report higher defender expected utility on synthetic and human-subjects data than sequential models trained with cross-entropy loss.Their ILO model learns attack probabilities over discrete targets to maximize a surrogate for defender expected utility.Source: §5.6 · p. 33
• Wang et al. (2020) report significant performance improvements for non-convex problems compared with the strongly convex case.They replace a large-scale optimization problem with a low-dimensional surrogate by reparameterizing the feasible decision space.Source: §5.6 · p. 33
• Butler and Kwon (2023b) show their ILO portfolio models outperform SLO models using ordinary least squares regression.They provide analytical solutions for unconstrained and equality-constrained mean-variance portfolios; these yield lower variance than exact solutions for the corresponding inequality-constrained problem.Source: §5.6 · p. 34

Uncertainty in constraints

• Ignoring covariate information can make SAA solutions infeasible when constraints are uncertain, and ERM poses challenges for constrained contextual stochastic optimization.Rahimian and Pagnoncelli (2022) study conditional chance constraints to ensure feasibility with high probability given observed covariates.Source: §6 · p. 34

Risk aversion

• Risk-averse contextual optimization can use measures such as value-at-risk, but its uncertainty sets must balance risk avoidance with returns.Conditional Robust Optimization learns a covariate-dependent uncertainty set before solving the downstream robust optimization problem; learning that set to minimize downstream cost remains challenging.Source: §6 · p. 35

Toolboxes and benchmarking

• cvxpylayers provides differentiable convex optimization layers, while PyEPO implements methods including SPO+, DBB, and PFYL for problems linear in uncertain parameters.PyEPO also supplies benchmarks and experiments highlighting the advantages of integrated learning.Source: §6 · p. 35
• Comparisons in fixed simulation settings, especially with real-world data, are scarce; a four-dataset newsvendor meta-analysis found no method clearly outperformed all others, and Mandi et al. (2023) benchmarked ILO frameworks on seven problems.Source: §6 · p. 35

Endogenous uncertainty

• Research on decision-dependent uncertainty with covariates remains sparse, although facility location and price-setting examples involve decisions changing demand and an unknown causal relationship.Alley et al. (2023) introduce a price-setting loss function that isolates causal price effects on demand from conditional effects due to other covariates.Source: §6 · p. 36

Data privacy

• Sensitive data from multiple sources may require differential-privacy obfuscation, which can reduce performance; Mieth et al. (2023) use a Wasserstein-ambiguity-set DRO model to value data for power-flow decisions.Source: §6 · p. 36

Interpretability & explainability

• Interpretability remains largely unexplored in contextual optimization: transparent pipelines can use simple models, while explainability can add an algorithm to a complex model.One proposed direction is training tree-based models to approximate complex learning-and-optimization policies.Source: §6 · p. 36

Fairness

• Using protected attributes as context can raise fairness concerns; pricing research seeks to prevent proposed prices from differing significantly across customers or groups (Cohen et al. 2021, 2022).Source: §6 · p. 37

Active research directions

• Gupta and Rusmevichientong (2021) show that Bayes-inspired and regularized policies match optimal in-sample performance in small-data, large-scale regimes, motivating direct learning of debiased in-sample policies.The survey suggests building a general framework to learn such policies by directly minimizing debiased objective functions.Source: §6 · p. 37

Multi-agent decision-making

• Heaton et al. (2022) identify Nash equilibria in contextual games using implicit differentiation of variational inequalities and Jacobian-free backpropagation.Multi-agent perspectives matter when agents in transportation and operations management have different covariates.Source: §6 · p. 37

Costly label acquisition

• Liu et al. (2023a) develop an active learning approach to obtain labels for solving the SPO problem, while active learning for nonlinear contextual optimization is an interesting future direction.Source: §6 · p. 38

Multi-stage contextual optimization

• Although most contextual optimization work addresses single- or two-stage problems, existing approaches build residual-based scenario trees, generalize weighted SAA, or learn end-to-end for inventory replenishment.These approaches are described by Ban et al. (2019), Rios et al. (2015), Bertsimas et al. (2023), and Qi et al. (2023), respectively.Source: §6 · p. 38

Sequential decision-making

• Nikishin et al. (2022a) combine learning and planning for expected returns in tabular and non-tabular MDPs, and their state-action value function has lower approximation error than MLE in tabular MDPs.They use a soft Bellman operator and the IFT for efficient parameter learning.Source: §6 · p. 38

Nonstationary distributions

• A further research direction is to relax stationarity of the joint covariate–uncertain-parameter distribution; Neghab et al. (2022) study a newsvendor with hidden-Markov covariate and demand distributions..Source: §6 · p. 38

Real-world deployment

• Real-world deployment requires addressing decision-aware MDP computational hurdles, including large state-action pairs and high-dimensional policy spaces (Wang et al. 2023).The section also describes a service-call scheduling application formulated as an RMAB to improve maternal and child health in a nonprofit organization.Source: §6 · pp. 38–39
• For RMABs, Mate et al. (2022) cluster beneficiaries’ dynamics and use a Whittle Index, while Wang et al. (2023) use decision-focused learning with soft-top-k arm selection.The soft-top-k selection mitigates the difficulty of differentiating the Whittle Index policy for selecting the top-k arms, using an optimal transport formulation (Xie et al. 2020).Source: §6 · p. 39

• Contextual optimization uses historical covariates and uncertain-parameter values to prescribe decisions after a correlated covariate is revealed.Source: §7 · p. 39
• The survey classifies policy-learning methods as decision rule optimization, sequential learning and optimization, or integrated learning and optimization.Source: §7 · p. 39
• Decision rule optimization parameterizes policies by covariates, whereas the other frameworks estimate conditional distributions or sufficient statistics sequentially or integratively.Source: §7 · p. 39
• The survey establishes uniform notation and terminology, details integrated methods’ training pipelines and modeling choices, and shows these aspects are intertwined.Source: §7 · p. 39
• The survey also identifies active research areas in contextual stochastic optimization problems.Source: §7 · p. 39

Appendix A: Theoretical guarantees

• Consistency requires the sample-based policy to approach the set of optimal decisions for almost all covariates, as defined by Bertsimas and Kallus (2020).The definition expresses this as zero distance from Z*(x).Source: Appendix A · p. 51
• The stated asymptotic conditions imply that policy performance matches the optimal conditional cost for almost all covariates as the sample count grows.The appendix attributes the asymptotic-optimality definition to Bertsimas and Kallus (2020).Source: Appendix A · p. 51
• Fisher consistency requires complete knowledge of the joint distribution P, while calibration asks whether low surrogate excess risk implies small excess true risk.Source: Appendix A · p. 51
• Ho-Nguyen and Kılınç-Karzan (2022) introduce a calibration function that simplifies verification of uniform MSE calibration and show a rate of O(ε^2).Source: Appendix A · p. 51
• Under a conditional-distribution lower-bound assumption, Liu and Grigas (2021) obtain O(ε^2) calibration for polyhedral Z and O(ε) for strongly convex, smooth level-set Z.Source: Appendix A · p. 51
• The appendix defines empirical Rademacher complexity for a hypothesis class under loss ℓ using independent, identically distributed Rademacher random variables.Source: Appendix A · p. 51

• ADMM means alternating direction method of multipliers, and CSO means contextual stochastic optimization; CVaR means conditional value at risk.Source: Appendix B · p. 52
• DRO means distributionally robust optimization, DNN means deep neural network, ERM means empirical risk minimization, and FP means fixed point.Source: Appendix B · p. 52
• ILO means integrated learning and optimization, IFT means implicit function theorem, kNN means k-nearest neighbor, and KKT means Karush–Kuhn–Tucker.Source: Appendix B · p. 52
• KL means Kullback-Leibler, LDR means linear decision rule, MDP means Markov decision process, and ML means machine learning.Source: Appendix B · p. 52
• MLE means maximum likelihood estimation, MILP means mixed-integer linear program, and NW means Nadaraya-Watson.Source: Appendix B · p. 52
• RKHS means reproducing kernel Hilbert space, SAA means sample average approximation, and SLO means sequential learning and optimization.Source: Appendix B · p. 52