Smooth Learning with Hard Constraints via Legendre-Regularized Policies
2026-07-27 • Machine Learning
Machine Learning
AI summaryⓘ
The authors look at how to design policies that decide actions based on the situation, aiming for policies that are both flexible and always produce valid decisions. They introduce a new type of policy, called Legendre-regularized policies, that guarantee feasibility and can be smoothly adjusted using mathematical tools. They prove that these policies behave nicely, can approximate any good policy, and work well in tests involving inventory and resource allocation problems. This approach combines ideas from earlier methods to achieve better decision-making performance.
contextual optimizationpolicy classfeasibility constraintsLegendre regularizationdifferentiabilityoptimizer mapLipschitz continuityuniversal approximationcontextual newsvendorresource allocation
Authors
Zikun Lin, Rui Chen, Yijie Wang
Abstract
We revisit contextual optimization from the perspective of policy class design. A desirable policy class should be expressive enough to learn rich context-decision relationships, should enforce hard feasibility constraints rather than soft penalty terms, and should remain smooth enough for gradient-based training on downstream decision losses. Existing approaches usually emphasize only part of these requirements. We propose Legendre-regularized policies, which parameterize decisions as solutions of regularized optimization problems over the original feasible region. This construction yields policies that are feasible by construction and differentiable with respect to learned latent parameters. We prove that the associated optimizer map is single-valued, maps onto the relative interior of the feasible set, admits an explicit Jacobian, is Lipschitz continuous, and can be made arbitrarily smooth. We also establish a universal approximation result showing that the proposed class can approximate any continuous feasible policy on compact context sets. The framework unifies explicitly regularized optimizers and implicit perturbation-based smooth optimizers. Experiments on contextual newsvendor and resource allocation problems show that our approach improves prescriptive performance relative to the benchmark methods.