Subsampling Factorization Machine Annealing

TL;DR

An algorithm to solve a black-box optimization problem by improving Factorization Machine Annealing such that SFMA exhibits balanced performance of exploration and exploitation, which is called exploitation-exploration functionality, and indicates the effectiveness of SFMA in a certain class of black-box optimization problems of significant size.

Abstract

Quantum computing and machine learning are state-of-the-art technologies that have been investigated intensively in both academia and industry. The hybrid technology of these two ingredients is expected to be a powerful tool to solve complex problems in many branches of science and engineering such as combinatorial optimization problems and accelerate the creation of next-generation technologies. In this work, we develop an algorithm to solve a black-box optimization problem by improving Factorization Machine Annealing (FMA) such that the training of a machine learning model called Factorization Machine is performed not by a full dataset but by a subdataset that is sampled from a full dataset: Subsampling Factorization Machine Annealing (SFMA). According to such a probabilistic training process, the performance of FMA on exploring a solution space gets enhanced. As a result, SFMA exhibits balanced performance of exploration and exploitation, which we call exploitation-exploration functionality. We conduct numerical benchmarking tests to compare the performance of SFMA with that of FMA. Consequently, SFMA certainly exhibits the exploration-exploitation functionality and outperforms FMA in speed and accuracy. In addition, the performance of SFMA can be further improved by sequentially using two subsampling datasets with different sizes such that the size of the latter dataset is substantially smaller than the former. Such a substantial reduction not only enhances the exploration performance of SFMA but also enables us to run it with correspondingly low computational cost even for a large-scale problem. These results indicate the effectiveness of SFMA in a certain class of black-box optimization problems of significant size: the potential scalability of SFMA in solving large-scale problems with correspondingly low computational cost.

Figures (17)

• Figure 1: Schematic of a procedure to conduct BBO. It is comprised of three steps: (i) data generation, (ii) dataset modeling, and (iii) optimization of a surrogate model.
• Figure 2: (a) Illustration of how to create a sampled dataset $\mathcal{B}_{a}$. It is created by sampling the elements of $\mathcal{D}_{a}$ according to a probability distribution, and the relation between the size of $\mathcal{B}_{a}$ ($|\mathcal{B}_{a}|$) and that of $\mathcal{D}_{a}$ ($|\mathcal{D}_{a}|$): $|\mathcal{B}_{a}| = \left[ R\cdot |\mathcal{D}_{a}| \right]$ ($0<R< 1$). The data point $(\boldsymbol{x}^{(a,i)}, y^{(a,i)})$ is an $i$th ($i=1,\ldots, |\mathcal{B}_{a}|$) element of $\mathcal{B}_{a}$. Owing to such a probabilistic construction of $\mathcal{B}_{a}$, the FM function $f_\text{FM}(\boldsymbol{x}; \boldsymbol{\theta}^{(a)})$ is derived probabilistically (FM parameters $\boldsymbol{\theta}$ are trained probabilistically), and correspondingly, the $a$th best candidate solution $(\boldsymbol{x}^{\dagger(a)}, y^{\dagger(a)}) (= (\boldsymbol{x}^{(a+1)}, y^{(a+1)}))$ is selected from the solution space associated with a deviation. As a result, we become able to explore a wider range of the energy landscape of $f_\text{BB}(\boldsymbol{x})$ than a case of running FMA as indicated by deviations of the best candidate solutions denoted by symbols $\Delta^2\boldsymbol{x}^{(\alpha)}$ ($\alpha=a-3, \ldots, a$) and the exploration performance gets strengthened.
• Figure 3: Results of $\bar{y}^{(a)}_\text{min}$ [panels (a), (c), and (e)] and $R_\text{min,success}^{(a)}$ [panels (b), (d), and (f)]. Plots in panels (a) and (b), (c) and (d), and (e) and (f) display the results for $W_3$ with $N_\text{bit}=12$, $W_4$ with $N_\text{bit}=16$, and $W_2$ with $N_\text{bit}=20$, respectively. The iteration number $N_\text{ite}$ is taken to be $N_\text{ite}=2N_\text{bit}^2+1$. The red (orange) and blue (green) curves represent the results of SFMA and FMA with (without) standardization, respectively. The results of RS are plotted by the tan curves.
• Figure 4: Results of $\bar{y}^{(a)}_\text{min}$ [panels (a), (c), and (e)] and $R_\text{min,success}^{(a)}$ [panels(b), (d), and (f)] of the improved SFMA. Plots in panels (a) and (b), (c) and (d), and (e) and (f) are the results for $W_2$ with $N_\text{bit}=12$ and $N_\text{ite}=4N_\text{bit}^2+1$, $W_7$ with $N_\text{bit}=16$ and $N_\text{ite}=4N_\text{bit}^2+1$, and $W_9$ with $N_\text{bit}=20$ and $N_\text{ite}=6N_\text{bit}^2+1$, respectively. For $N_\text{bit}=12$ and 16, the results of the improved SFMA (ISFMA) are presented by purple curves. On the other hand, for $N_\text{bit}=20$, the two improved algorithms ISFMA$_1$ and ISFMA$_2$ are displayed by purple and pink curves, respectively. To make the comparisons between the performance of the improved SFMA and those of the other algorithms, we also plot the results of the nonstandardized FMA (green), the standardized FMA (blue), the nonstandardized SFMA (orange), the standardized SFMA (red), and RS (tan).
• Figure 5: Results of $\bar{y}^{(a)}_\text{min}$ [panels (a), (c), and (e)] and $R_\text{min,success}^{(a)}$ [panels (b), (d), and (f)] of the standardized SFMA with $N_\text{ite}=2N_\text{bit}^2+1$. Plots in panels (a) and (b), (c) and (d), and (e) and (f) are the results for $W_3$ with $N_\text{bit}=12$, $W_4$ with $N_\text{bit}=16$, and $W_2$ with $N_\text{bit}=20$, respectively. The red and light blue curves represent the results for the annealers selected to be SA and QA, respectively.