distributionally robust set learning

Directly incorporating machine learning predictions into optimization constraints often yields high constraint violation probabilities. Method: This paper proposes a robust optimization framework that models the model’s loss function as a theoretically certified compact uncertainty set—bypassing conventional distributional assumptions or sampling-based approximations. The framework derives a rigorous upper bound on the constraint violation probability and proves that the resulting uncertainty set radius is up to ten times smaller than those of existing methods. Results: In synthetic experiments, the proposed approach significantly reduces constraint violation probability while compressing the uncertainty set size by an order of magnitude, achieving a favorable trade-off between solution feasibility and conservatism. The core contribution lies in a geometric transformation from the loss function to an uncertainty set, coupled with a probabilistic guarantee mechanism that ensures theoretical soundness and practical efficacy.

This work addresses the limited robustness of traditional empirical risk minimization under distributional shift and its inability to adequately capture tail behavior of the loss distribution. The authors propose a novel stochastic set-valued optimization framework based on hyper-box sets, wherein decision variables are mapped to hyper-boxes and the problem is reformulated as a multi-objective optimization. A key innovation lies in jointly modeling the lower and upper tails of the loss distribution via sub-quantiles and super-quantiles. The resulting formulation is solved using a stochastic multi-gradient algorithm, coupled with a Pareto knee-point selection strategy. This approach significantly enhances model robustness and test-time stability under distributional shifts while maintaining accuracy comparable to that of empirical risk minimization.

This work addresses the performance degradation of set representation learning during inference caused by element-level corruptions, such as outliers or missing entries. To mitigate this issue, the authors propose the SW-DRSO framework, which leverages distributionally robust optimization. By introducing a differentiable barycentric adversarial mechanism, SW-DRSO efficiently transforms the intractable worst-case search over infeasible set perturbations into an optimization problem over simplex weights. This enables the model to minimize the expected loss over potentially corrupted sets during training. Experimental results across four tasks demonstrate that SW-DRSO not only maintains high overall performance but also significantly enhances robustness against element-level corruptions encountered at inference time.

In combinatorial optimization, the empirical risk w.r.t. model parameters is piecewise constant, hindering gradient-based optimization and lacking theoretical generalization guarantees. Method: For contextual stochastic optimization with complex objectives, we propose a perturbation-driven risk smoothing strategy. Our approach integrates statistical learning models with a surrogate combinatorial optimization oracle to construct a context-aware, generalization-controllable decision framework. Contribution/Results: We establish the first unified generalization bound incorporating perturbation bias, statistical error, and optimization error. We introduce the notion of “uniform weak consistency” to characterize the coupled stability between the learning model and the surrogate oracle, proving its universality under mild assumptions. Experiments on stochastic vehicle scheduling demonstrate strong generalization performance. This work provides the first verifiable theoretical generalization framework for contextual stochastic optimization.

This work addresses the lack of provable joint guarantees on generalization and convergence in learned optimization algorithms. Methodologically: (1) it establishes the first PAC-Bayesian generalization bound for unbounded losses, leveraging exponential-family posterior distributions; (2) it formulates optimizer learning as a tractable one-dimensional global optimization problem—convex or non-convex—whose solution is analytically characterizable; and (3) it integrates stochastic optimization design with rigorous theoretical analysis to explicitly trade off convergence rate against generalization error. Empirically, the learned optimizers achieve order-of-magnitude improvements over state-of-the-art methods across four diverse real-world tasks—including neural architecture search, meta-learning, adversarial training, and federated learning—while all gains are underpinned by formal theoretical guarantees. This constitutes the first learning-to-optimize framework endowed with a provably tight PAC-Bayesian generalization bound and jointly certified convergence–generalization performance.

This work addresses the high computational cost and optimization instability of existing Monte Carlo sampling–based variational inference methods for learning neural set functions under weak supervision. The authors propose a sampling-free, continuous relaxation learning framework that reinterprets the evidence lower bound (ELBO) as a continuous relaxation of set functions and introduces a learnable surrogate objective to yield stable and efficient gradients. The approach enjoys approximation guarantees under submodular maximization and reveals a theoretical connection to variational free energy. Experimental results demonstrate that the method significantly outperforms current baselines across multiple real-world tasks, achieving faster convergence and substantially reduced computational overhead.

This work addresses the prediction bias and policy fragility induced by covariate perturbations in data-driven decision-making by proposing the first theoretically grounded robust joint prediction-optimization framework. By integrating robust optimization and designing a computable convex surrogate loss, the method effectively guards against worst-case feature perturbations. Theoretical analysis establishes that its approximation error decays exponentially and that it satisfies Fisher consistency with high probability. Empirical evaluations demonstrate that the proposed framework significantly outperforms existing approaches in out-of-sample performance and training stability.

Traditional robust optimization is often overly conservative due to its exclusive focus on worst-case scenarios, limiting its ability to leverage predictive information for improved scheduling performance. This work proposes the first framework that explicitly incorporates predictions as an independent benchmark in robust scheduling, achieving a principled trade-off between consistency—near-optimality under predicted scenarios—and robustness—guaranteed performance under worst-case uncertainty. By developing a consistency–robustness trade-off mechanism and employing duality theory, upper-envelope reductions, and support-function blocks, the paper systematically analyzes scheduling problems under interval, budgeted, and general uncertainty sets. Smooth $(1+1/\lambda, 1+\lambda)$ trade-offs are established for restricted assignment and related machine models, while the impossibility of constant-factor trade-offs is proven for unrelated machines; constant performance guarantees are provided for identical machines.

This work addresses the limitations of traditional predict-then-optimize approaches, which ignore prediction uncertainty, and existing distributionally robust optimization (DRO) methods that employ fixed-radius ambiguity sets ill-suited for dynamic risk environments. The authors propose a Learnable Prediction Ambiguity Set (LPAS), which jointly learns the center, state-dependent Wasserstein radius, and anisotropic metric of the ambiguity set. This is achieved through end-to-end joint optimization of a deep contextual model and the downstream decision layer, enhanced by conditional quantile calibration and scale regularization to enable state-adaptive robustness. Evaluated on S&P 500 portfolio optimization from 2018 to 2026, LPAS achieves an annualized return of 26.28%, a Sharpe ratio of 1.30, a terminal wealth of 1.61, lower tail risk, and a smaller average ambiguity set radius compared to benchmarks.

This work addresses the challenge of efficiently selecting key uncertainty directions from a finite dictionary to construct computationally tractable robust optimization uncertainty sets. It proposes a data-driven approach that builds atomic uncertainty sets with closed-form support functions by identifying a sparse subset of atoms capable of covering critical evaluation directions—such as gradients, adversarial perturbations, or distributional shifts. The core contribution lies in the design of a monotone and submodular coverage objective, which enables a greedy algorithm with a provable $(1 - 1/e)$ approximation guarantee. The method also provides out-of-sample performance loss bounds and a radius calibration rule, achieving both theoretical approximation guarantees and significant improvements in scalability and out-of-sample robustness.