Performance Evaluation of Ising and QUBO Variable Encodings in Boltzmann Machine Learning

Abstract:We compare Ising ({-1, +1}) and QUBO ({0, 1}) encodings for Boltzmann machine learning under controlled protocols that fix the sampler, optimizer, and learning-rate design within each comparison. Exploiting the identity that the Fisher information matrix (FIM) equals the covariance of sufficient statistics, we visualize empirical moments from model samples and reveal systematic, representation-dependent differences. QUBO induces larger cross terms between first- and second-order statistics, creating more small-eigenvalue directions in the FIM and lowering spectral entropy. This ill-conditioning explains slower convergence under stochastic gradient descent (SGD). In contrast, full-FIM natural gradient descent (NGD), which rescales updates by the FIM metric, achieves similar convergence across encodings, whereas diagonal-FIM approximation can reintroduce representation-dependent differences. Practically, for SGD-based training, the Ising encoding provides more isotropic curvature and faster convergence; for QUBO, centering/scaling or NGD-style preconditioning mitigates curvature pathologies. These results clarify how representation shapes information geometry and finite-time learning dynamics in Boltzmann machines and yield actionable guidelines for variable encoding and preprocessing.

Machine Learning (cs.LG)

arXiv:2510.13210 [cs.LG]

(or arXiv:2510.13210v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2510.13210 Focus to learn more arXiv-issued DOI via DataCite

J. Phys. Soc. Jpn. 95, 104006 (2026)

https://doi.org/10.7566/JPSJ.95.104006 Focus to learn more DOI(s) linking to related resources

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.