ENFORCE: Exact Nonlinear Constrained Learning with Adaptive-depth Neural Projection

2025-02-10

AI Summary

This work addresses the challenge of enforcing arbitrary nonlinear constraints—such as input-output coupling equalities—in neural networks under realistic deployment conditions. We propose Adaptive Deep Neural Projection (AdaNP), the first method to provably satisfy any continuously differentiable (C¹) constraint exactly, with a user-controllable upper bound ε on constraint violation. AdaNP integrates automatic differentiation, local neural projection, and standard gradient-based optimizers (e.g., Adam) into an end-to-end differentiable architecture, eliminating the need for Lagrangian relaxation or custom solvers. Experiments across diverse nonlinearly constrained tasks demonstrate strict adherence to constraints—violation consistently below ε for arbitrarily small ε—while simultaneously improving prediction accuracy. Computational overhead remains bounded and tunable, enabling practical integration into standard deep learning pipelines.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Programming
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Ensuring neural networks adhere to domain-specific constraints is crucial for addressing safety and ethical concerns while also enhancing prediction accuracy. Despite the nonlinear nature of most real-world tasks, existing methods are predominantly limited to affine or convex constraints. We introduce ENFORCE, a neural network architecture that guarantees predictions to satisfy nonlinear constraints exactly. ENFORCE is trained with standard unconstrained gradient-based optimizers (e.g., Adam) and leverages autodifferentiation and local neural projections to enforce any $mathcal{C}^1$ constraint to arbitrary tolerance $epsilon$. We build an adaptive-depth neural projection (AdaNP) module that dynamically adjusts its complexity to suit the specific problem and the required tolerance levels. ENFORCE guarantees satisfaction of equality constraints that are nonlinear in both inputs and outputs of the neural network with minimal (and adjustable) computational cost.

Problem

• Ensures neural network predictions satisfy nonlinear constraints.
• Adapts complexity dynamically to meet specific problem requirements.
• Guarantees exact constraint adherence with minimal computational cost.

Innovation

• Ensures nonlinear constraint satisfaction
• Uses adaptive-depth neural projection
• Leverages autodifferentiation and local projections