Elliptic Loss Regularization

2025-03-04

AI Summary

This work addresses the weak extrapolation capability and unreliable predictions in unseen regions of neural networks under distribution shift and group imbalance. We propose a novel smoothness regularization method grounded in elliptic partial differential equation (PDE) theory. Specifically, the method constrains the second-order differential geometric structure of the loss function in input space via an elliptic operator, explicitly enforcing elliptic PDE properties to enhance local smoothness and extrapolation stability of the loss landscape. To our knowledge, this is the first systematic integration of elliptic PDE theory into deep learning regularization design. Theoretically, we derive a controllable upper bound on generalization error for unseen regions under this regularization. Empirically, it significantly improves robustness across multiple out-of-distribution generalization and fairness benchmarks, while remaining fully compatible with standard training pipelines and incurring minimal computational overhead.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Regularizing neural networks is important for anticipating model behavior in regions of the data space that are not well represented. In this work, we propose a regularization technique for enforcing a level of smoothness in the mapping between the data input space and the loss value. We specify the level of regularity by requiring that the loss of the network satisfies an elliptic operator over the data domain. To do this, we modify the usual empirical risk minimization objective such that we instead minimize a new objective that satisfies an elliptic operator over points within the domain. This allows us to use existing theory on elliptic operators to anticipate the behavior of the error for points outside the training set. We propose a tractable computational method that approximates the behavior of the elliptic operator while being computationally efficient. Finally, we analyze the properties of the proposed regularization to understand the performance on common problems of distribution shift and group imbalance. Numerical experiments confirm the utility of the proposed regularization technique.

Problem

• Enforces smoothness in data-to-loss mapping
• Uses elliptic operators to predict out-of-sample behavior
• Addresses distribution shift and group imbalance issues

Innovation

• Introduces elliptic loss regularization for neural networks
• Modifies empirical risk minimization with elliptic operator
• Provides efficient computational method for regularization