ORACLE: Optimizer-Relative Alignment for Constrained LEarning

2026-10-06

AI Summary

This study addresses the limitation of conventional constraint handling methods, which neglect optimizer-induced gradient reshaping and thereby introduce bias into constraint compatibility assessments. To overcome this, we propose an optimizer-relative constraint learning framework that, for the first time, shifts constraint compatibility evaluation to the post-optimizer stage by constructing an alignment mechanism grounded in native optimizer geometry. This approach integrates joint endpoint linearization, heterogeneous constraint family processing, and update verification techniques to achieve precise alignment of actual update steps within the optimizer's geometric space. Experimental evaluations on PDE benchmarks demonstrate that the proposed framework matches or surpasses native optimizers in 94% of configurations while significantly outperforming existing constraint handling methods.

Abstract

Constraint handling methods typically intervene before the optimizer acts, by modifying the objective or the gradient. Yet momentum, adaptive scaling, and structured preconditioning can substantially reshape that signal before it becomes a parameter update. We formulate optimizer relative constrained learning, where constraint compatibility is assessed on the post optimizer update. Building on this view, we introduce ORACLE, which evaluates the native optimizer's realized step through a joint endpoint linearization of heterogeneous constraint families, constructs the resulting alignment in the optimizer's own geometry, bounds its authority, and commits it only after validation. We evaluate ORACLE across eight Partial Differential Equation benchmarks and four optimizers spanning Euclidean, diagonal adaptive, and structured preconditioned geometries, where it improves or matches native optimizer in 94% of configurations. Cross model analysis shows the same behavior in 92% of configurations, while matched comparisons show improvements over alternative constraint-handling methods acting at the objective, gradient, and post-optimizer levels.

Problem

• constrained learning
• optimizer-relative alignment
• constraint handling
• partial differential equations

Innovation

• Constrained Learning
• Optimizer-Relative Alignment
• Heterogeneous Constraints
• Preconditioned Geometry
• Partial Differential Equations