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arXiv:2103.10695 · IEEE International Conference on Distributed Computing Systems Workshops (ICDCSW)

QROSS: QUBO Relaxation Parameter optimisation via Learning Solver Surrogates

Tian Huang, Siong Thye Goh, Sabrish Gopalakrishnan, Tao Luo, Qianxiao Li, Hoong Chuin Lau

Published Jul 1, 2021• 5 citations
arXivHTMLPDF

Abstract

A popular method for solving a constrained combinatorial optimisation problem is to first convert it into a quadratic unconstrained binary optimisation (QUBO) problem, and solve it using a standard QUBO solver. However, this relaxation introduces hyper-parameters that balance the objective and penalty terms for the constraints, and their chosen values significantly impact performance. Hence, hyper-parameters tuning is important. Existing generic hyper-parameter tuning methods require multiple expensive calls to a QUBO solver, making them inefficient for performance critical applications when repeated solutions of similar combinatorial optimisation problems are required. In this paper, we propose the QROSS method, in which we build surrogate models of QUBO solvers via learning from solver data on a collection of instances of a problem. In this way, we are able capture the common structure of the instances and their interactions with the solver, and produce promising hyper-parameters with fewer calls to the QUBO solver. We take the Traveling Salesman Problem (TSP) as a case study, where we demonstrate QROSS finds better solutions with fewer calls to QUBO solver, compared with conventional hyper-parameter tuning techniques. Moreover, with simple adaptation methods, QROSS is shown to generalise well to out-of-distribution datasets and different types of QUBO solvers.

DOI
10.1109/icdcsw53096.2021.00013

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A popular method for solving a constrained combinatorial optimisation problem is to first convert it into a quadratic unconstrained binary optimisation (QUBO) problem, and solve it using a standard QUBO solver. However, this relaxation introduces hyper-parameters that balance the objective and penalty terms for the constraints, and their chosen values significantly impact performance. Hence, hyper-parameters tuning is important. Existing generic hyper-parameter tuning methods require multiple expensive calls to a QUBO solver, making them inefficient for performance critical applications when repeated solutions of similar combinatorial optimisation problems are required. In this paper, we propose the QROSS method, in which we build surrogate models of QUBO solvers via learning from solver data on a collection of instances of a problem. In this way, we are able capture the common structure of the instances and their interactions with the solver, and produce promising hyper-parameters with fewer calls to the QUBO solver. We take the Traveling Salesman Problem (TSP) as a case study, where we demonstrate QROSS finds better solutions with fewer calls to QUBO solver, compared with conventional hyper-parameter tuning techniques. Moreover, with simple adaptation methods, QROSS is shown to generalise well to out-of-distribution datasets and different types of QUBO solvers.
In this paper, we propose a hybrid framework to solve large-scale permutation-based combinatorial problems effectively using a high-performance quadratic unconstrained binary optimization (QUBO) solver. To do so, transformations are required to change a constrained optimization model to an unconstrained model that involves parameter tuning. We propose techniques to overcome the challenges in using a QUBO solver that typically comes with limited numbers of bits. First, to smooth the energy landscape, we reduce the magnitudes of the input without compromising optimality. We propose a machine learning approach to tune the parameters for good performance effectively. To handle possible infeasibility, we introduce a polynomial-time projection algorithm. Finally, to solve large-scale problems, we introduce a divide-and-conquer approach that calls the QUBO solver repeatedly on small sub-problems. We tested our approach on provably hard Euclidean Traveling Salesman (E-TSP) instances and Flow Shop Problem (FSP). Optimality gap that is less than 10%10\%10% and 11%11\%11% are obtained respectively compared to the best-known approach.
Many combinatorial optimization problems can be formulated as a problem to determine the order of sequence or to find a corresponding mapping of the objects. We call such problems permutation-based optimization problems. Many such problems can be formulated as a quadratic unconstrained binary optimization (QUBO) or Ising model by introducing a penalty coefficient to the permutation constraint terms. While classical and quantum annealing approaches have been proposed to solve QUBOs to date, they face issues with optimality and feasibility. Here we treat a given QUBO solver as a black box and propose techniques to enhance its performance. First, to ensure an effective search for good quality solutions, a smooth energy landscape is needed; we propose a data scaling approach that reduces the amplitudes of the input without compromising optimality. Second, we need to tune the penalty coefficient. In this paper, we illustrate that for certain problems, it suffices to tune the parameter by performing random sampling on the penalty coefficients to achieve good performance. Finally, to handle possible infeasibility of the solution, we introduce a polynomial-time projection algorithm. We apply these techniques along with a divide-and-conquer strategy to solve some large-scale permutation-based problems and present results for TSP and QAP.
The Travelling Salesman Problem (TSP) is an important combinatorial optimisation problem, and is usually solved on a quantum computer using a Quadratic Unconstrained Binary Optimisation (QUBO) formulation or a Higher Order Binary Optimisation(HOBO) formulation. In these formulations, penalty terms are added to the objective function for outputs that don't map to valid routes. We present a novel formulation which needs fewer binary variables, and where, by design, there are no penalty terms because all outputs from the quantum device are mapped to valid routes. Simulations of a quantum boson sampler were carried out which demonstrate that larger networks can be solved with this penalty-free formulation than with formulations with penalties. Simulations were successfully translated to hardware by running a non-QUBO formulation with penalties on an early experimental prototype ORCA PT-1 boson sampler. Although we worked with a boson sampler, we believe that this novel formulation is relevant to other quantum devices. This work shows that a good embedding for combinatorial optimisation problems can solve larger problems with the same quantum computing resource. The flexibility of boson sampling quantum devices is a powerful asset in solving combinatorial optimisation problem, because it enables formulations where the output string is always mapped to a valid solution, avoiding the need for penalties.
Quantum and quantum-inspired optimisation algorithms are designed to solve problems represented in binary, quadratic and unconstrained form. Combinatorial optimisation problems are therefore often formulated as Quadratic Unconstrained Binary Optimisation Problems (QUBO) to solve them with these algorithms. Moreover, these QUBO solvers are often implemented using specialised hardware to achieve enormous speedups, e.g. Fujitsu's Digital Annealer (DA) and D-Wave's Quantum Annealer. However, these are single-objective solvers, while many real-world problems feature multiple conflicting objectives. Thus, a common practice when using these QUBO solvers is to scalarise such multi-objective problems into a sequence of single-objective problems. Due to design trade-offs of these solvers, formulating each scalarisation may require more time than finding a local optimum. We present the first attempt to extend the algorithm supporting a commercial QUBO solver as a multi-objective solver that is not based on scalarisation. The proposed multi-objective DA algorithm is validated on the bi-objective Quadratic Assignment Problem. We observe that algorithm performance significantly depends on the archiving strategy adopted, and that combining DA with non-scalarisation methods to optimise multiple objectives outperforms the current scalarised version of the DA in terms of final solution quality.
This study introduces a novel train-and-test approach referred to as apprenticeship learning (AL) for generating selection hyper-heuristics to solve the Quadratic Unconstrained Binary Optimisation (QUBO) problem. The primary goal is to automate the design of hyper-heuristics by learning from a state-of-the-art expert and to evaluate whether the apprentice can outperform that expert. The proposed method collects detailed search trace data from the expert and trains the apprentice based on the machine learning models to predict heuristic selection and parameter settings. Multiple data filtering and class balancing techniques are explored to enhance model performance. The empirical results on unseen QUBO instances show that indeed, “student surpasses the teacher”; the hyper-heuristic with the generated heuristic selection not only outperforms the expert but also generalises quite well by solving unseen QUBO instances larger than the ones on which the apprentice was trained. These findings highlight the potential of AL to generalise expert behaviour and improve heuristic search performance.
Quantum and quantum-inspired optimisation algorithms are designed to solve problems represented in binary, quadratic and unconstrained form. Combinatorial optimisation problems are therefore often formulated as Quadratic Unconstrained Binary Optimisation Problems (QUBO) to solve them with these algorithms. Moreover, these QUBO solvers are often implemented using specialised hardware to achieve enormous speedups, e.g. Fujitsu’s Digital Annealer (DA) and D-Wave’s Quantum Annealer. However, these are single-objective solvers, while many real-world problems feature multiple conflicting objectives. Thus, a common practice when using these QUBO solvers is to scalarise such multi-objective problems into a sequence of single-objective problems. Due to design trade-offs of these solvers, formulating each scalarisation may require more time than finding a local optimum.We present the first attempt to extend the algorithm supporting a commercial QUBO solver as a multi-objective solver that is not based on scalarisation. The proposed multi-objective DA algorithm is validated on the bi-objective Quadratic Assignment Problem. We observe that algorithm performance significantly depends on the archiving strategy adopted, and that combining DA with non-scalarisation methods to optimise multiple objectives outperforms the current scalarised version of the DA in terms of final solution quality.