Demonstrating Real Advantage of Machine-Learning-Enhanced Monte Carlo for Combinatorial Optimization

TL;DR

The study tackles hard combinatorial optimization in the QUBO/Ising framework by benchmarking a machine-learning–driven Global Annealing (GA) against classical solvers SA and PA on 3D Edwards–Anderson spin-glass instances. GA leverages a shallow MADE autoregressive model to propose global moves while retaining local Metropolis updates, preserving the Gibbs–Boltzmann distribution during annealing. Across $N=10^3$ and $N=14^3$ spins, GA consistently outperforms SA and shows robustness relative to PA, delivering faster convergence to the MEC on harder instances and larger sizes. This work provides the first robust evidence that a ML-assisted optimization method can surpass state-of-the-art classical techniques in a challenging combinatorial setting, with implications for scalable, high-performance optimization in QUBO/Ising problems.

Abstract

Combinatorial optimization problems are central to both practical applications and the development of optimization methods. While classical and quantum algorithms have been refined over decades, machine learning-assisted approaches are comparatively recent and have not yet consistently outperformed simple, state-of-the-art classical methods. Here, we focus on a class of Quadratic Unconstrained Binary Optimization (QUBO) problems, specifically the challenge of finding minimum energy configurations in three-dimensional Ising spin glasses. We use a Global Annealing Monte Carlo algorithm that integrates standard local moves with global moves proposed via machine learning. We show that local moves play a crucial role in achieving optimal performance. Benchmarking against Simulated Annealing and Population Annealing, we demonstrate that Global Annealing not only surpasses the performance of Simulated Annealing but also exhibits greater robustness than Population Annealing, maintaining effectiveness across problem hardness and system size without hyperparameter tuning. These results provide, to our knowledge, the first clear and robust evidence that a machine learning-assisted optimization method can exceed the capabilities of classical state-of-the-art techniques in a combinatorial optimization setting.

Figures (8)

• Figure 1: (a) Schematic description of the three algorithms considered in this work: the well-known Simulated Annealing (SA) and Population Annealing (PA), and the novel machine-learning-assisted Global Annealing (GA). $\theta_l$ and $\theta_g$ represent the number of local and global steps, respectively. (b) Example of how the minimum energy decreases during three different runs of various lengths for a given instance of the problem. The difference $\Delta E$ between the minimum energy found by the algorithms and the exact one (here found using the Gurobi solver gurobi) is plotted as a function of the simulation time for a short (15 seconds), a medium (50 seconds), and a long (150 seconds) run. As the running time increases, the three algorithms manage to progressively find lower energy states.
• Figure 2: Success probability as a function of the mean running time for SA, PA and GA (with and without local moves), together with sigmoidal fits, for easy and hard realizations of the couplings at $N=10^3$.
• Figure 3: Median success probability (solid lines) over 200 instances for $N = 10^3$ instances as a function of the running time for PA (red) and GA (blue), together with the 25th and the 75th percentiles (dotted lines, shaded area) and the best (dash-dotted lines) and worst (color dashed) instances. The black dashed line corresponds to a 90% success probability. To reduce computational cost, instances whose runs achieved a 90% success rate were terminated, and a 100% success rate was assumed thereafter for computing the average quantities. While PA tends to outperform GA on the majority of the runs, there are some instances, like the worst case shown here, in which PA performs much worse than GA. For PA, we have performed 10 MCS for every temperature, as in Ref. wang2015comparing. For GA, we performed 5 global moves per temperature, and 15 local MCS per global move. Both algorithms used a logarithmic spacing of the temperatures during annealing.
• Figure 4: Scatter plot (in log-log scale) of the times it takes on each instance to reach a 90% success rate with PA and $\text{GA}_{15}$, together with the corresponding histograms. Different colors of the datapoints highlight different ratios between the runtimes of the algorithms. The dashed line correspond to equal times. Crosses are points for which PA fails to achieve a 90% success rate within the time limit of about 550 seconds. Data are the same as those used to obtain Fig. \ref{['fig:success_vs_time_L10']}.
• Figure 5: Same plot as in Fig. \ref{['fig:success_vs_time_L10']}, here obtained with 10 instances with $N = 14^3$. The choices of the hyperparameters for both algorithms are the same as in Fig. \ref{['fig:success_vs_time_L10']}. In this case, GA outperforms PA both in the median and in the best/worst case runs.