Title: Spectral Annealing for Scalable Ising Model Optimization
Primary area: optimization
Keywords: Spectral Annealing, Ising Model, Combinatorial Optimization, Pertubation

Abstract: Large-scale Ising optimization underlies numerous machine learning tasks from correlation clustering to feature selection. Yet existing approaches face a fundamental scalability-quality trade-off: exact methods guarantee optimal solutions but become computationally prohibitive at scale, while metaheuristics achieve scalability by sacrificing solution quality. Spectral relaxations offer a promising middle ground with efficient scaling, but suffer from a critical limitation—they solve exactly one eigenvalue problem, constraining solution quality through lack of exploration.
We introduce \textit{spectral annealing}, a systematic approach that transforms spectral methods from single-shot to exploration-based optimization while preserving computational efficiency.  Our method explores a continuous family of spectral relaxations $\mathbf{N}_\alpha = \mathbf{D}_\alpha^{-1/2} \mathbf{A} \mathbf{D}_\alpha^{-1/2}$ parameterized by $\alpha \in [0,1]$, interpolating from raw adjacency to full normalization. Our key innovation is a \textit{diagonal predictor}, which enables efficient warm-starting by exploiting the algebraic structure of the $\alpha$-parameterized family. We provide rigorous eigen-drift analysis for principled step-size control and GPU implementation with efficient parallelization techniques. Experiments on instances up to 262,000 spins demonstrate favorable time-quality trade-offs. Systematic $\alpha$-exploration provides 2-5× quality improvements over single-parameter spectral methods while maintaining computational efficiency.

TL;DR: Smooth transition of spectral partitioning for scalable Ising model solvers, achieving the efficiency of spectral methods with solution quality of expensive solvers