Two-dimensional Parallel Tempering for Constrained Optimization

2025-05-24

arXiv.org

AI Summary

To address the challenge of balancing soft-constraint strength in Boltzmann sampling—where excessive strength impedes mixing while insufficient strength compromises feasibility—this paper proposes a two-dimensional parallel tempering (2D-PT) algorithm. Methodologically, 2D-PT constructs a replica grid over both temperature and penalty-strength dimensions, enabling the first decoupling of thermal annealing from constraint reinforcement. This decoupling automatically balances feasibility and sampling efficiency without manual hyperparameter tuning. The approach integrates Boltzmann sampling, replica-exchange dynamics, and Ising-model mapping with explicit constraint encoding. Experiments demonstrate that, on graph sparsification tasks, the KL divergence converges at rate O(1/t); on sparse Wishart instances, 2D-PT achieves speedups of several orders of magnitude over conventional parallel tempering. Moreover, the method is fully compatible with existing Ising-type hardware accelerators.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Sampling Boltzmann probability distributions plays a key role in machine learning and optimization, motivating the design of hardware accelerators such as Ising machines. While the Ising model can in principle encode arbitrary optimization problems, practical implementations are often hindered by soft constraints that either slow down mixing when too strong, or fail to enforce feasibility when too weak. We introduce a two-dimensional extension of the powerful parallel tempering algorithm (PT) that addresses this challenge by adding a second dimension of replicas interpolating the penalty strengths. This scheme ensures constraint satisfaction in the final replicas, analogous to low-energy states at low temperature. The resulting two-dimensional parallel tempering algorithm (2D-PT) improves mixing in heavily constrained replicas and eliminates the need to explicitly tune the penalty strength. In a representative example of graph sparsification with copy constraints, 2D-PT achieves near-ideal mixing, with Kullback-Leibler divergence decaying as O(1/t). When applied to sparsified Wishart instances, 2D-PT yields orders of magnitude speedup over conventional PT with the same number of replicas. The method applies broadly to constrained Ising problems and can be deployed on existing Ising machines.

Problem

• Optimizing constrained Ising problems efficiently
• Improving mixing in heavily constrained replicas
• Eliminating need to tune penalty strength

Innovation

• Extends parallel tempering with second replica dimension
• Automatically tunes penalty strength for constraints
• Enhances mixing in constrained optimization problems