Solving Boltzmann optimization problems with deep learning

The exponential scaling in high-performance computing (HPC) efficiency, observed as Moore's Law, is nearing its physical limits in CMOS technology. Future HPC advancements necessitate exploring new computing paradigms and technologies. The Ising model, with its inherent energy efficiency and ability to function as both logic and memory, presents a promising alternative. Ising-based hardware offers significant potential for energy reduction by eliminating costly data movement in CMOS computing. However, a major challenge is optimizing circuits to ensure correct results on inherently nondeterministic Ising hardware. This paper introduces a novel machine learning solution to address this challenge. The research focuses on developing an efficient method to solve optimization problems that minimize error sources within the Ising model. The paper's contribution is twofold: 1) a novel machine learning approach using deep neural networks and random forests and 2) a process for transforming a Boltzmann probability optimization problem into a supervised machine learning problem.

The paper reviews the limitations of CMOS-based HPC, highlighting the physical limits of miniaturization and the challenges posed by aging effects, hardware variability, and soft error susceptibility. It then introduces the Ising model, a statistical mechanics model, and its extension, the Potts model, emphasizing their applications in various fields, including statistical mechanics and computational biology. The authors discuss the computational complexity of calculating the Boltzmann distribution, highlighting the need for efficient approximation techniques. Existing literature on approximation methods for computing the Boltzmann distribution efficiently is also reviewed, setting the stage for the paper's proposed machine learning approach that bypasses this challenge by formulating the problem as a tractable optimization problem.

The paper tackles the "reverse Ising problem," focusing on finding optimal Hamiltonian coefficients to maximize the probability of obtaining desired outputs from specified inputs. The approach involves formulating the problem as a Boltzmann probability optimization problem. The authors transform this optimization problem to be suitable for supervised machine learning. This transformation involves several key steps: first, they address the numerical instability of the original Boltzmann probability equation by minimizing the maximum of the log of the probability of an undesired state. This transformation amplifies small probability changes caused by adjustments to the Hamiltonian coefficients. Second, to ensure continuous differentiability required by the chosen optimization solver (SLSQP), they approximate the maximum function using a scaling parameter. The resulting numerically stable and differentiable Boltzmann probability optimization function is then used to generate training data. The training data consists of pairs of input auxiliary arrays and their corresponding target probabilities calculated using the transformed objective function. The training data was generated using the SLSQP solver, with an explicit gradient implementation to improve efficiency for higher-dimensional problems. Two machine learning models were then trained on this data: a random forest regressor and a deep neural network (DNN). The random forest is used as a baseline and also to initialize the DNN parameters using the DJINN framework. The performance of both models was evaluated using mean squared error (MSE) on test data, and their computational efficiency was compared against traditional optimization methods (SLSQP with approximate and explicit gradient calculations).

The paper presents results for four problems of varying complexity, each defined by the number of fixed, non-fixed, and auxiliary spins. The random forest regressor demonstrated good predictive accuracy, with MSE values ranging from 0.0001 to 0.02. The DNN models, initialized with a simpler random forest, achieved comparable MSE results, showcasing the efficacy of the DJINN framework. Notably, while the random forests used 100 trees, the DNN initialization used only three trees and a shallower depth, significantly reducing the number of parameters compared to the random forest. A critical finding highlights the substantial computational advantage of the proposed machine learning models over traditional SLSQP optimization. The time required to compute 100 Boltzmann probabilities for Problem 4 was drastically reduced from days (SLSQP with approximate gradient) or several minutes (SLSQP with explicit gradient) to milliseconds for both the random forest and DNN models. This dramatic speedup demonstrates the feasibility and efficiency of the proposed method. The authors also note that the random forest model outperformed other machine learning models such as SVM and logistic regression.

The findings demonstrate the successful application of machine learning to solve the computationally challenging reverse Ising problem. The significant speedup achieved by the ML models (random forest and DNN) over the SLSQP algorithm allows for the analysis of much larger and more complex Ising circuits than previously possible. The comparable accuracy of the DNN model, with a significantly smaller number of parameters compared to the random forest, suggests an efficient and scalable approach. The work opens up new possibilities for designing and analyzing Ising-based hardware for various applications. The ability to accurately and efficiently predict Ising system dynamics is crucial for optimizing circuit designs and improving the performance of this emerging technology. The success of the proposed method lies in the transformation of the problem into a form suitable for machine learning, leveraging the power of deep learning and random forests for accurate and fast prediction.

This paper introduces a novel approach to solving Boltzmann probability optimization problems for Ising-based hardware design. The combination of a numerically stable problem formulation and the use of deep neural networks and random forests provides a computationally efficient solution. The results demonstrate a significant speedup compared to traditional optimization methods, enabling the analysis of more complex Ising circuits. Future research could explore the application of this approach to even larger problems and explore alternative machine learning architectures or training strategies to further improve accuracy and efficiency. Investigating the impact of different types of neural networks, optimization techniques, and dataset sizes on the model performance can also enhance the robustness of this methodology.

The accuracy of the machine learning models depends heavily on the quality and size of the training dataset. The choice of optimization algorithm (SLSQP) and its parameters could influence the quality of the training data. While the models show promising speed improvements, the generalization performance to unseen data requires further investigation and testing with larger and more diverse datasets. The reliance on the SLSQP solver for data generation introduces a potential bottleneck that may limit the scalability of the approach. The performance of the DNN may be sensitive to the architecture and hyperparameters, requiring careful tuning.