# ORACLE: Optimizer-Relative Alignment for Constrained LEarning

arXiv: https://arxiv.org/abs/2610.09040 · HTML: https://mlsanity.com/paper/2610.09040/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 59 references that are in the map.
- First posted on arXiv 6 Oct 2026.

## Builds on (most cited first, 19 of 19)

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- [Differentiable Projection for Constrained Deep Learning](https://mlsanity.com/paper/2111.10785/) (2021)
- [Achieving Constraints in Neural Networks: A Stochastic Augmented Lagrangian Approach](https://mlsanity.com/paper/2310.16647/) (2023)
- [Safe Flow Q-Learning: Offline Safe Reinforcement Learning with Reachability-Based Flow Policies](https://mlsanity.com/paper/2603.15136/) (2026)
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- [Gradient-Update Mismatch: Rethinking Conflict-Free Training of Physics-Informed Neural Networks](https://mlsanity.com/paper/2609.01558/) (2026)
- [DePICT: Decision-Preserving Interface for Constrained Downstream Tasks](https://mlsanity.com/paper/2610.03945/) (2026)

## Abstract (arXiv)

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.