A Fully First-Order Layer for Differentiable Optimization

Abstract:Differentiable optimization layers enable learning systems to make decisions by solving embedded optimization problems. However, computing gradients via implicit differentiation requires solving a linear system with Hessian terms, which is both compute- and memory-intensive. To address this challenge, we propose a novel algorithm that computes the gradient using only first-order information. The key insight is to rewrite the differentiable optimization as a bilevel optimization problem and leverage recent advances in bilevel methods. Specifically, we introduce an active-set Lagrangian hypergradient oracle that avoids Hessian evaluations and provides finite-time, non-asymptotic approximation guarantees. We show that an approximate hypergradient can be computed using only first-order information in $\tilde{O}(1)$ time, leading to an overall complexity of $\tilde{O}(\delta^{-1}\epsilon^{-3})$ for constrained bilevel optimization, which matches the best known rate for non-smooth non-convex optimization. Furthermore, we release an open-source Python library that can be easily adapted from existing solvers. The source code is available at this https URL.

Machine Learning (cs.LG)

arXiv:2512.02494 [cs.LG]

(or arXiv:2512.02494v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2512.02494 Focus to learn more arXiv-issued DOI via DataCite

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