A Fully First-Order Layer for Differentiable Optimization

• URL: http://arxiv.org/abs/2512.02494v1
• Date: Tue, 02 Dec 2025 07:36:03 GMT
• Title: A Fully First-Order Layer for Differentiable Optimization
• Authors: Zihao Zhao, Kai-Chia Mo, Shing-Hei Ho, Brandon Amos, Kai Wang,
• Abstract summary: Differentiable optimization layers enable learning systems to make decisions by solving embedded optimization problems.<n>We show that an approximate hyperient can be computed using only first-order information in $too(1)$ time.
• Score: 12.868783495046422
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\tilde{\oo}(1)$ time, leading to an overall complexity of $\tilde{\oo}(δ^{-1}ε^{-3})$ 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. Our code is available here: https://github.com/guaguakai/FFOLayer.

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