Probabilistic Approximate Optimization: A New Variational Monte Carlo Algorithm

2025-07-10

AI Summary

This work addresses optimization of large-scale 3D spin glasses and heavy-tailed distribution problems (e.g., the SK-Lévy model). We propose P-AOA, a novel variational Monte Carlo algorithm that employs a parameterized binary stochastic neural network, enabling derivative-free, efficient sampling via independent sample-driven updates of coupling parameters. Our key theoretical contribution is the first rigorous correspondence established between derivative-free parameter updates and full-scale Markovian flow gradients, thereby unifying simulated annealing within a variational framework and enabling joint multi-temperature schedule optimization. P-AOA is co-designed for FPGA-based probabilistic computers, integrating on-chip annealing with parametric sampling to enhance hardware efficiency and convergence speed. Experiments demonstrate that P-AOA outperforms QAOA on a 26-spin SK model and surpasses conventional simulated annealing on both 3D spin glass and SK-Lévy benchmarks.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce a generalized extit{Probabilistic Approximate Optimization Algorithm (PAOA)}, a classical variational Monte Carlo framework that extends and formalizes prior work by Weitz extit{et al.}~cite{Combes_2023}, enabling parameterized and fast sampling on present-day Ising machines and probabilistic computers. PAOA operates by iteratively modifying the couplings of a network of binary stochastic units, guided by cost evaluations from independent samples. We establish a direct correspondence between derivative-free updates and the gradient of the full $2^N imes 2^N$ Markov flow, showing that PAOA admits a principled variational formulation. Simulated annealing emerges as a limiting case under constrained parameterizations, and we implement this regime on an FPGA-based probabilistic computer with on-chip annealing to solve large 3D spin-glass problems. Benchmarking PAOA against QAOA on the canonical 26-spin Sherrington-Kirkpatrick model with matched parameters reveals superior performance for PAOA. We show that PAOA naturally extends simulated annealing by optimizing multiple temperature profiles, leading to improved performance over SA on heavy-tailed problems such as SK-Lévy.

Problem

• Extends variational Monte Carlo for Ising machines
• Optimizes couplings in binary stochastic networks
• Improves performance over QAOA and simulated annealing

Innovation

• Variational Monte Carlo framework for optimization
• Parameterized fast sampling on Ising machines
• Optimizes multiple temperature profiles