Boltzmann 机学习中 Ising 与 QUBO 变量编码的性能评估

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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.

Comments:	19pages, 10figures
Subjects:	Machine Learning (cs.LG)
Cite as:	arXiv:2510.13210 [cs.LG]
	(or arXiv:2510.13210v2 [cs.LG] for this version)
	https://doi.org/10.48550/arXiv.2510.13210 arXiv-issued DOI via DataCite
Journal reference:	J. Phys. Soc. Jpn. 95, 104006 (2026)
Related DOI:	https://doi.org/10.7566/JPSJ.95.104006 DOI(s) linking to related resources

Submission history

From: Yasushi Hasegawa [view email]
[v1] Wed, 15 Oct 2025 06:57:23 UTC (1,998 KB)
[v2] Fri, 2 Oct 2026 00:31:48 UTC (3,871 KB)