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Solving Boltzmann Optimization Problems with Deep Learning

Fiona Knoll, John T. Daly, Jess J. Meyer

TL;DR

The paper tackles optimizing Ising-based Boltzmann systems to realize designated outputs under nondeterministic dynamics by reframing the reverse Ising problem as a learnable Boltzmann-probability optimization. It builds a differentiable surrogate by transforming the max-probability objective with a log-sum-exp approximation and generates training data via an SLSQP solver, producing input-output pairs $(\mathbf{a}, \rho(\mathbf{a}))$. The authors compare random forest regression and DJINN-augmented deep networks as predictors of the Boltzmann probability, showing similar accuracy (MSE around $2\times 10^{-2}$) with far fewer parameters in the DNNs and substantial speedups over traditional optimization. The results indicate that machine-learned predictors can accurately approximate Ising-system dynamics and enable scalable design for larger spin configurations, potentially guiding hardware development for energy-efficient, ground-state-biased computation.

Abstract

Decades of exponential scaling in high performance computing (HPC) efficiency is coming to an end. Transistor based logic in complementary metal-oxide semiconductor (CMOS) technology is approaching physical limits beyond which further miniaturization will be impossible. Future HPC efficiency gains will necessarily rely on new technologies and paradigms of compute. The Ising model shows particular promise as a future framework for highly energy efficient computation. Ising systems are able to operate at energies approaching thermodynamic limits for energy consumption of computation. Ising systems can function as both logic and memory. Thus, they have the potential to significantly reduce energy costs inherent to CMOS computing by eliminating costly data movement. The challenge in creating Ising-based hardware is in optimizing useful circuits that produce correct results on fundamentally nondeterministic hardware. The contribution of this paper is a novel machine learning approach, a combination of deep neural networks and random forests, for efficiently solving optimization problems that minimize sources of error in the Ising model. In addition, we provide a process to express a Boltzmann probability optimization problem as a supervised machine learning problem.

Solving Boltzmann Optimization Problems with Deep Learning

TL;DR

The paper tackles optimizing Ising-based Boltzmann systems to realize designated outputs under nondeterministic dynamics by reframing the reverse Ising problem as a learnable Boltzmann-probability optimization. It builds a differentiable surrogate by transforming the max-probability objective with a log-sum-exp approximation and generates training data via an SLSQP solver, producing input-output pairs . The authors compare random forest regression and DJINN-augmented deep networks as predictors of the Boltzmann probability, showing similar accuracy (MSE around ) with far fewer parameters in the DNNs and substantial speedups over traditional optimization. The results indicate that machine-learned predictors can accurately approximate Ising-system dynamics and enable scalable design for larger spin configurations, potentially guiding hardware development for energy-efficient, ground-state-biased computation.

Abstract

Decades of exponential scaling in high performance computing (HPC) efficiency is coming to an end. Transistor based logic in complementary metal-oxide semiconductor (CMOS) technology is approaching physical limits beyond which further miniaturization will be impossible. Future HPC efficiency gains will necessarily rely on new technologies and paradigms of compute. The Ising model shows particular promise as a future framework for highly energy efficient computation. Ising systems are able to operate at energies approaching thermodynamic limits for energy consumption of computation. Ising systems can function as both logic and memory. Thus, they have the potential to significantly reduce energy costs inherent to CMOS computing by eliminating costly data movement. The challenge in creating Ising-based hardware is in optimizing useful circuits that produce correct results on fundamentally nondeterministic hardware. The contribution of this paper is a novel machine learning approach, a combination of deep neural networks and random forests, for efficiently solving optimization problems that minimize sources of error in the Ising model. In addition, we provide a process to express a Boltzmann probability optimization problem as a supervised machine learning problem.
Paper Structure (9 sections, 29 equations, 2 figures, 4 tables)

This paper contains 9 sections, 29 equations, 2 figures, 4 tables.

Table of Contents

  1. Background of the Boltzmann Optimization Problem
  2. The Reverse Ising Problem
  3. The Boltzmann Probability Objective Function
  4. Size of the System and Complexity of the Problem
  5. Modeling the Boltzmann Probability Optimization Problem
  6. Generating the Training Data
  7. Random Forest Regression Model
  8. Deep Neural Network Model
  9. Performance Evaluations

Figures (2)

  • Figure 1: Subfigure (a) depicts one tree of a random forest created for Problem 3 (from Table \ref{['table_prob']}). Subfigure (b) shows an example set of nodes from the tree in subfigure (a). At each node of the tree, the choice of direction is dependent on the value of a single element of the auxiliary array $\textbf{a}$. Note that the auxiliary array is denoted as $x$ in the figure.
  • Figure 2: Subfigures (a)-(d) provide a visual representation of the results of the random forest regression models (denoted RF in the figure) for the 4 problems listed in Table \ref{['table: rf']}. Subfigures (e)-(h) provide a visual representation of the results of DNN models created using DJINN framework (denoted DJINN in the figure) for the 4 problems listed in Table \ref{['table: nn']}. In each subfigure, the x-axis is the true Boltzmann probability, and the y-axis is the predicted probability. Each point represents the output corresponding to a single auxiliary array $\textbf{a}$.

Theorems & Definitions (1)

  • Example 1