Supplementing recurrent neural networks with annealing to solve combinatorial optimization problems

Abstract

Combinatorial optimization problems can be solved by heuristic algorithms such as simulated annealing (SA) which aims to find the optimal solution within a large search space through thermal fluctuations. The algorithm generates new solutions through Markov-chain Monte Carlo techniques. The latter can result in severe limitations, such as slow convergence and a tendency to stay within the same local search space at small temperatures. To overcome these shortcomings, we use the variational classical annealing (VCA) framework that combines autoregressive recurrent neural networks (RNNs) with traditional annealing to sample solutions that are independent of each other. In this paper, we demonstrate the potential of using VCA as an approach to solving real-world optimization problems. We explore VCA's performance in comparison with SA at solving three popular optimization problems: the maximum cut problem (Max-Cut), the nurse scheduling problem (NSP), and the traveling salesman problem (TSP). For all three problems, we find that VCA outperforms SA on average in the asymptotic limit by one or more orders of magnitude in terms of relative error. Interestingly, we reach large system sizes of up to 256 cities for the TSP. We also conclude that in the best case scenario, VCA can serve as a great alternative when SA fails to find the optimal solution.

Sparse autoregressive neural networks for classical spin systems

“…Although the present work focuses on binary, two-body interacting variables, nothing prevents using a similar approach to derive ARNN architectures for multi-valued or continuous variables, such as the Potts model (Wu 1982) and the Kuramoto model (Acebrón et al 2005), and with more than two-body interactions, such as the Baxter-Wu model (Novotny and Landau 1981). Beyond statistical physics, TwoBo has potential applications in a wide range of research areas that utilize Boltzmann distributions, including combinatorial optimization and inference problems, extending from mathematics to social sciences (Mézard et al 2002, Zdeborová and Krzakala 2016, Biazzo et al 2022, Khandoker et al 2023. It also has promising uses in a family of energy-based models within the rapidly growing field of machine learning, particularly for tasks such as language understanding, image recognition, and other complex data interpretation activities (Deng et al 2020, Du et al 2020.…”

“…The recent advent of deep generative models, particularly autoregressive neural networks (ARNNs), has marked a significant advancement in fields such as image and language generation (Brown et al 2020). A pivotal study in 2019 introduced the use of ARNNs for Boltzmann distribution sampling (Wu et al Barrett et al 2022, Luo et al 2022, Wu et al 2023, as well as statistical inference (Biazzo et al 2022) and optimization problems (Khandoker et al 2023).…”