---
title: "Differentiable Spectral Normalization for Large-Scale Ising Optimization"
authors:
  - "Thinh Nguyen-Cong"
  - "Thang N. Dinh"
doi: "10.24963/ijcai.2026/258"
url: "https://ijcai.org/proceedings/2026/258"
pdf: "https://ijcai.org/proceedings/2026/0258.pdf"
venue: "Proceedings of the Thirty-Fifth International Joint Conference on Artificial Intelligence"
year: 2026
pages: "2319-2327"
track: "Main Track"
keywords:
  - "Search: Combinatorial search and optimisation"
  - "Constraint Satisfaction and Optimization: Constraint optimization problems"
  - "Machine Learning: Optimization"
  - "Machine Learning: Matrix/tensor methods"
  - "Constraint Satisfaction and Optimization: Mixed discrete and continuous optimization"
---

# Differentiable Spectral Normalization for Large-Scale Ising Optimization

Thinh Nguyen-Cong, Thang N. Dinh

*Proceedings of the Thirty-Fifth International Joint Conference on Artificial Intelligence*, pages 2319-2327. DOI: [10.24963/ijcai.2026/258](https://doi.org/10.24963/ijcai.2026/258)

## Abstract

Spectral relaxation is widely used for large-scale combinatorial optimization due to its computational efficiency. Yet its effectiveness depends critically on the choice of graph normalization, a design decision typically made heuristically. Here, we show that normalization can be treated as a continuous optimization variable rather than a fixed preprocessing choice. Our method, Differentiable Spectral Normalization (DSN), parameterizes the spectral relaxation through a diagonal metric and maximizes the resulting lower bound via projected gradient ascent. Exact gradients are obtained through the Hellmann-Feynman theorem using only the principal eigenpair, maintaining linear complexity per iteration. On benchmark instances ranging from 10^3 to 8.4 x 10^6 nodes, DSN improves solution quality by 3-15% over static spectral methods. Its performance comes within 1-3% of state-of-the-art metaheuristics, such as simulated annealing, at up to 190x lower computational cost on large-scale instances. These results suggest that learning problem-specific relaxation geometry can substantially close the gap between spectral scalability and metaheuristic solution quality.