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This research introduces neural slack variables, a novel architecture designed to enforce functional inequality constraints like monotonicity within deep learning models without relying on fragile penalty methods. By coupling primary networks with jointly learned auxiliary components, the approach achieves zero violations in critical industrial applications such as volatility surface modeling and scientific simulations. This advancement addresses a significant gap where existing techniques either fail to guarantee feasibility or impose restrictive inductive biases that limit generalization.
Neural Slack Variables for Shape Constraints
arXiv:2606.13803v1 Announce Type: new Abstract: Enforcing functional inequality constraints such as monotonicity and convexity in neural networks is a fundamental challenge in many industrial and scientific applications. Classical one-sided penalty methods, along with primal-dual methods gated by complementary slackness, provide constraint gradients only at violated locations, resulting in fragile satisfaction. Architectures that guarantee feasibility by construction, on the other hand, remain largely limited to elementary cases and impose additional inductive biases. We introduce neural slack variables, a deep learning native primal-side approach that converts constraint enforcement into a regression problem by coupling the primary network with a jointly learned auxiliary network. The auxiliary network serves as a valid target for the primary network's constraint quantities, inducing feasibility and regularity. Neural slack variables achieve zero measured violations on dense-grid monotonicity and convexity test cases, where penalty and primal-dual baselines leave residual violations, and enable arbitrage-free learning of volatility surfaces, an open industrial challenge in quantitative finance.
Abstract:Enforcing functional inequality constraints such as monotonicity and convexity in neural networks is a fundamental challenge in many industrial and scientific applications. Classical one-sided penalty methods, along with primal-dual methods gated by complementary slackness, provide constraint gradients only at violated locations, resulting in fragile satisfaction. Architectures that guarantee feasibility by construction, on the other hand, remain largely limited to elementary cases and impose additional inductive biases. We introduce neural slack variables, a deep learning native primal-side approach that converts constraint enforcement into a regression problem by coupling the primary network with a jointly learned auxiliary network. The auxiliary network serves as a valid target for the primary network's constraint quantities, inducing feasibility and regularity. Neural slack variables achieve zero measured violations on dense-grid monotonicity and convexity test cases, where penalty and primal-dual baselines leave residual violations, and enable arbitrage-free learning of volatility surfaces, an open industrial challenge in quantitative finance.
Machine Learning (cs.LG)
arXiv:2606.13803 [cs.LG]
(or arXiv:2606.13803v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.13803 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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