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Physical Review Research

Continuous black-box optimization with an Ising machine and random subspace coding

Syun Izawa, Koki Kitai, Shu Tanaka, Ryo Tamura, Koji Tsuda

Published Apr 21, 2022• 34 citations
Publisher

Abstract

A black-box optimization algorithm such as Bayesian optimization finds the extremum of an unknown function by alternating the inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply Ising machines to overcome the difficulty in the continuous black-box optimization. As an Ising machine specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution by Ising machines has to be translated back. Our method has the following three parts: (1) random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector, (2) a quadratic unconstrained binary optimization (QUBO) defined by the acquisition function based on the nonnegative-weighted linear regression model, which is solved by Ising machines, and (3) a penalization scheme to ensure that the solution can be translated back. It is shown with benchmark tests that its performance using the D-Wave Advantage quantum annealer and simulated annealing is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up the possibility of Ising machines and other QUBO solvers, including a quantum approximate optimization algorithm using gated-quantum computers, and may expand its range of application to continuous-valued problems.

DOI
10.1103/physrevresearch.4.023062

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A black-box optimization algorithm such as Bayesian optimization finds the extremum of an unknown function by alternating the inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply Ising machines to overcome the difficulty in the continuous black-box optimization. As an Ising machine specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution by Ising machines has to be translated back. Our method has the following three parts: (1) random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector, (2) a quadratic unconstrained binary optimization (QUBO) defined by the acquisition function based on the nonnegative-weighted linear regression model, which is solved by Ising machines, and (3) a penalization scheme to ensure that the solution can be translated back. It is shown with benchmark tests that its performance using the D-Wave Advantage quantum annealer and simulated annealing is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up the possibility of Ising machines and other QUBO solvers, including a quantum approximate optimization algorithm using gated-quantum computers, and may expand its range of application to continuous-valued problems.
A black-box optimization algorithm such as Bayesian optimization finds extremum of an unknown function by alternating inference of the underlying function and optimization of an acquisition function. In a high-dimensional space, such algorithms perform poorly due to the difficulty of acquisition function optimization. Herein, we apply quantum annealing (QA) to overcome the difficulty in the continuous black-box optimization. As QA specializes in optimization of binary problems, a continuous vector has to be encoded to binary, and the solution of QA has to be translated back. Our method has the following three parts: 1) Random subspace coding based on axis-parallel hyperrectangles from continuous vector to binary vector. 2) A quadratic unconstrained binary optimization (QUBO) defined by acquisition function based on nonnegative-weighted linear regression model which is solved by QA. 3) A penalization scheme to ensure that the QA solution can be translated back. It is shown in benchmark tests that its performance using D-Wave AdvantageTM^{\rm TM}TM quantum annealer is competitive with a state-of-the-art method based on the Gaussian process in high-dimensional problems. Our method may open up a new possibility of quantum annealing and other QUBO solvers including quantum approximate optimization algorithm (QAOA) using a gated-quantum computers, and expand its range of application to continuous-valued problems.
Black-box optimization (BBO) is used in materials design, drug discovery, and hyperparameter tuning in machine learning. The world is experiencing several of these problems. In this review, a factorization machine with quantum annealing or with quadratic-optimization annealing (FMQA) algorithm to realize fast computations of BBO using Ising machines (IMs) is discussed. The FMQA algorithm uses a factorization machine (FM) as a surrogate model for BBO. The FM model can be directly transformed into a quadratic unconstrained binary optimization model that can be solved using IMs. This makes it possible to optimize the acquisition function in BBO, which is a difficult task using conventional methods without IMs. Consequently, it has the advantage of handling large BBO problems. To be able to perform BBO with the FMQA algorithm immediately, we introduce the FMQA algorithm along with Python packages to run it. In addition, we review examples of applications of the FMQA algorithm in various fields, including physics, chemistry, materials science, and social sciences. These successful examples include binary and integer optimization problems, as well as more general optimization problems involving graphs, networks, and strings, using a binary variational autoencoder. We believe that BBO using the FMQA algorithm will become a key technology in IMs, including quantum annealers.
With an increase in the complexity of society, solving multi-objective optimization problems (MOPs) has become crucial. In this study, we introduced a novel method called “quadratic unconstrained binary optimization based on the weighted normal” for solving MOPs using Ising machines, such as quantum annealing and digital annealer (DA), in the field of combinatorial optimization. The proposed method applies the penalty-based boundary intersection method to Ising machines under a setting limited to linear objective functions and maximizes the speed and performance of the DA, which is a quadratic unconstrained binary optimization-specific solver. We demonstrated the effectiveness of the proposed method by solving a real-world problem with a nonconvex shaped Pareto front (component combination problem). The results suggested that the proposed method could handle both convex- and nonconvex-shaped Pareto fronts, expanding the potential applications of Ising machines to solving complex MOPs. This development could significantly enhance decision-making processes, particularly in achieving sustainable development goals.
We introduce kernel-QA, a black-box optimization (BBO) method that constructs surrogate models analytically using low-order polynomial kernels within a quadratic unconstrained binary optimization (QUBO) framework, enabling efficient utilization of Ising machines. The method has been evaluated on artificial landscapes, ranging from uni-modal to multi-modal, with input dimensions extending to 80 for real variables and 640 for binary variables. The results demonstrate that kernel-QA is particularly effective for optimizing black-box functions characterized by local minima and high-dimensional inputs, showcasing its potential as a robust and scalable BBO approach.
In discrete-variable black-box optimization, the number of candidate solutions grows combinatorially, while each evaluation is often expensive. Therefore, it is important to identify promising solutions efficiently within a limited number of trials. Bayesian Optimization of Combinatorial Structures (BOCS), an existing parametric method, works effectively when only a small amount of data is available. However, as the number of observations increases, BOCS tends to repeatedly propose points that have already been evaluated, which leads to search stagnation. A random-point addition strategy has been proposed to address this issue when an evaluated point is proposed, but it cannot sufficiently exploit information from promising data obtained so far. In this study, we propose a hybrid method that uses BOCS as the main search framework and generates alternative unevaluated points using a Gaussian process only when search stagnation is detected. In the Gaussian-process-based component, multiple Lower Confidence Bound (LCB) acquisition functions are adaptively selected to dynamically control the balance between exploitation and exploration. Numerical experiments using fully connected Quadratic Unconstrained Binary Optimization (QUBO) and Higher-order Unconstrained Binary Optimization (HUBO) as black-box functions show that the proposed method finds solutions with better objective values than the conventional random-point addition method in both settings. Additional analyses show that its effectiveness comes from selecting points that promote search progress within Hamming-distance neighborhoods, rather than simply adding low-energy points near promising solutions. Experiments with sparse surrogate models for quantum annealer applications further suggest the importance of retaining near-fully connected representational capacity.
Quantum annealers can solve QUBO problems efficiently but struggle with continuous optimization tasks like regression due to their discrete nature. We introduce Quadratic Continuous Quantum Optimization (QCQO), an anytime algorithm that approximates solutions to unconstrained quadratic programs via a sequence of QUBO instances. Rather than encoding real variables as binary vectors, QCQO implicitly represents them using continuous QUBO weights and iteratively refines the solution by summing sampled vectors. This allows flexible control over the number of binary variables and adapts well to hardware constraints. We prove convergence properties, introduce a step size adaptation scheme, and validate the method on linear regression. Experiments with simulated and real quantum annealers show that QCQO achieves accurate results with fewer qubits, though convergence slows on noisy hardware. Our approach enables quantum annealing to address a wider class of continuous problems.