将复杂机器学习问题转化为量子退火可解的二次无约束二值优化形式
Systematic and Efficient Construction of Quadratic Unconstrained Binary Optimization Forms for High-order and Dense Interactions
- 用修正线性单元基函数建模目标函数，实现高阶非线性项的等效二次化
- 在多个测试问题上验证了转化方法的有效性，数值与解析结果一致
- 适用于机器学习超参数优化等黑箱优化场景，提升量子退火应用范围
量子退火（QA）能够高效求解以二次无约束二值优化（QUBO）形式表示的组合优化问题。为扩大QA的应用范围，需通过二次化方法将高阶问题转换为QUBO。然而，涉及机器学习（ML）的复杂问题因强非线性和密集交互关系，现有二次化方法难以适用。为此，本文将目标函数建模为修正线性单元（ReLU）基函数之和，该表达式兼具通用逼近能力与等价的二次多项式形式。研究通过数值和解析方法验证了该方法的可行性。进一步地，结合所提二次化方法与量子退火，设计了一种新型黑箱优化方案：先对机器学习代理回归器进行二次化处理，再输入量子退火求解。
原文摘要 · Abstract (English)
Quantum Annealing (QA) can efficiently solve combinatorial optimization problems whose objective functions are represented by Quadratic Unconstrained Binary Optimization (QUBO) formulations. For broader applicability of QA, quadratization methods are used to transform higher-order problems into QUBOs. However, quadratization methods for complex problems involving Machine Learning (ML) remain largely unknown. In these problems, strong nonlinearity and dense interactions prevent conventional methods from being applied. Therefore, we model target functions by the sum of rectified linear unit bases, which not only have the ability of universal approximation, but also have an equivalent quadratic-polynomial representation. In this study, the proof of concept is verified both numerically and analytically. In addition, by combining QA with the proposed quadratization, we design a new black-box optimization scheme, in which ML surrogate regressors are inputted to QA after the quadratization process.
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