Differentiable Spectral Normalization for Large-Scale Ising Optimization

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.