A decision-aware weak-to-strong (W2S) framework that leverages both labeled and unlabeled data to improve contextual stochastic optimization and establishes a non-asymptotic upper bound on the excess decision risk of W2S and a complementary lower bound for a strong-only benchmark.
Abstract
Many operational decisions rely on predictive models that estimate uncertain outcomes conditional on observable contexts. Training such models, however, often faces a fundamental data asymmetry: labeled outcomes are scarce or costly to obtain, while contextual covariates are abundant. Motivated by this data asymmetry, we develop a decision-aware weak-to-strong (W2S) framework that leverages both labeled and unlabeled data to improve contextual stochastic optimization. Specifically, we first train a weak model using limited labeled data and then use it to generate predicted outcome distributions on unlabeled contexts. These distributions provide soft supervision for training a strong model. We establish a non-asymptotic upper bound on the excess decision risk of W2S and a complementary lower bound for a strong-only benchmark. Their comparison yields explicit sufficient conditions under which W2S improves downstream decision performance. The key quantity is the correlation dimension between the weak and strong feature representations: when it is small, abundant unlabeled data reduce the effect of teacher errors along non-overlapping directions. A synthetic newsvendor experiment and a comment moderation experiment based on real-world data provide empirical evidence consistent with the theory.
In this paper, we study contextual stochastic optimization (CSO), where decisions are made under uncertainty and the distribution of random parameters can be partially inferred from covariates observed prior to decision-making. In many practical settings, these distributions also depend on the decisions themselves, a phenomenon known as the
decision-dependent effect
. Most existing studies address this issue by imposing structural assumptions on the relationship between decisions and the underlying distributions. However, such assumptions may lead to model misspecification when the true relationship deviates from the assumed form. A prominent alternative is the weighted sample average approximation (wSAA) method proposed by Bertsimas and Kallus (2019), which adapts sample weights based on their similarity to the current decision–context pair. Nevertheless, because these weights are typically computed using complex machine learning models and depend on the decision variables in decision-dependent settings, solving the resulting optimization problem becomes computationally challenging. To overcome this challenge, we extend the wSAA framework from the loss function to its gradient, leading to the notion of the
contextual gradient
. We show that the contextual gradient serves as a meaningful indicator of optimality and leverage this property to develop the
contextual gradient descent (CGD)
algorithm. Our analysis establishes that CGD converges to a neighborhood of the global optimum when the loss function exhibits sufficient strong convexity. Moreover, the derived bounds reveal a key insight: the strength of convexity in the loss function can compensate for the uncertainty introduced by decision-dependent effects. Extensive numerical experiments on both synthetic and real-world datasets demonstrate that CGD consistently outperforms existing methods for contextual optimization under decision-dependent uncertainty.
Wenxuan Liu, Xiangting Liu, Maoqi Liu et al.· Production and operations ma...· 0 citations
Predict-then-optimize systems usually compress uncertainty into a point forecast and then solve a downstream optimization problem as if the forecast were reliable. Distributionally robust optimization (DRO) offers protection against misspecification, but the ambiguity set is often centered at historical samples and uses a fixed radius. We propose \emph{learned predictive ambiguity sets} (LPAS): a deep contextual model outputs a finite nominal scenario distribution, a state-dependent Wasserstein radius, and optionally an anisotropic ground metric. These outputs define a contextual ambiguity set that feeds a DRO decision layer. The radius is trained by a combination of conditional quantile calibration, size regularization, and downstream decision loss, so that robustness is adaptive rather than globally fixed. We derive the finite dual form used by the decision layer, present a staged training algorithm, and evaluate the method on distributionally robust portfolio optimization with 20 S&P 500 constituents from 2018--2026. The proposed method substantially improves over equal-weight, predict-then-optimize, and historical Wasserstein DRO baselines, achieving 26.28% annualized return, Sharpe ratio 1.30, final wealth 1.61, and lower tail loss than a deep fixed-radius DRO baseline while using a smaller average radius. The results show that learned ambiguity radii can recover most of the performance of strong fixed-radius DRO while reducing unnecessary conservatism and improving regime adaptivity.
A novel perturbation test based on a nonsmooth max-difference revenue statistic comparing the best null assortment with the best alternative assortment and asymptotic validity of the proposed p-value under adaptive assortment selection is proposed.
In transfer-learning settings, a model derived from abundant surrogate labels may be deployed in a target population where gold-standard outcomes are unobserved. Evaluating its target performance is essential for determining whether decisions based on the model remain reliable, yet it is difficult when gold labels are scarce, and covariate distributions differ across data sources. We study a three-sample setting with a small gold-labeled source, a larger surrogate-labeled source, and an unlabeled target. Under conditional transportability, we evaluate the surrogate-derived model against the latent gold-standard outcome in the target population. We propose cross-fitted estimators that transport information from the two labeled sources through source-specific density ratios. We also combine outcome-regression augmentation with a kernel correction for estimating the model near a threshold, accounting for uncertainty from all three samples. We establish asymptotically linear inference for TPR and FPR, consistency and pointwise inference for the ROC curve, and asymptotically normal inference for AUC. Simulations assess bias, coverage, and sensitivity to bandwidth and relative sample sizes. A retrospective temporal validation on Chatbot Arena and a semi-synthetic ACS-Income study provide validation in real-world AI applications.
We study a general decision-dependent contextual stochastic program (DD-CSP) in which uncertainty depends on both exogenous contextual information and endogenous decisions. To learn the potentially complex dependence of uncertainty on decisions and contextual information, we employ several nonparametric regression models, including k nearest neighbors (kNN), classification and regression trees (CART), and ReLU neural networks. To account for estimation errors in predicting the uncertainty, we adopt an empirical residuals-based decision-dependent sample average approximation (ER-DD-SAA) framework, which adds empirical residuals to the point predictions from the learned regression models. For each nonparametric regression model, we develop exact mixed-integer programming (MIP) representations that can be seamlessly embedded within the ER-DD-SAA framework. For two-stage ER-DD-SAA problems with kNN, we further propose a tailored decomposition algorithm, named BD-CG, that combines Bender's decomposition with constraint generation. Under suitable assumptions, we prove that the proposed BD-CG converges to a global optimum within a finite number of iterations. From a statistical perspective, we establish the consistency and asymptotic optimality of ER-DD-SAA with all three nonparametric regression models under mild regularity conditions. Numerical experiments on a newsvendor problem with pricing and a two-stage facility location problem demonstrate that the ER-DD-SAA model with nonparametric learning consistently outperforms a parametric benchmark in out-of-sample performance and the proposed reformulations and algorithm substantially improve computational tractability.
An asymmetric variant of minimally consistent concept classes is introduced and used to provide an exact characterization of proper learning with improvements in the real-izable setting, and positive results for more natural Euclidean ball improvement sets are given.