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Learning at the Speed of Physics: Equilibrium Propagation on Oscillator Ising Machines

Alex Gower

TL;DR

The paper addresses the challenge of training energy-based models efficiently on hardware by leveraging Equilibrium Propagation (EP), a local learning rule that removes the need for a backward pass. It demonstrates a direct mapping of EP onto Oscillator Ising Machines (OIMs), whose continuous phase dynamics perform energy descent and can be tuned with trainable couplings and biases. Empirical results show competitive MNIST and Fashion-MNIST accuracy and robustness to hardware constraints such as phase noise and quantization, with an empirical EP–BPTT correspondence validating gradient-descent properties. The work positions OIMs as a fast, energy-efficient substrate for practical EBMs and highlights the feasibility of implementing energy-based learning directly on physical hardware without major architectural changes.

Abstract

Physical systems that naturally perform energy descent offer a direct route to accelerating machine learning. Oscillator Ising Machines (OIMs) exemplify this idea: their GHz-frequency dynamics mirror both the optimization of energy-based models (EBMs) and gradient descent on loss landscapes, while intrinsic noise corresponds to Langevin dynamics - supporting sampling as well as optimization. Equilibrium Propagation (EP) unifies these processes into descent on a single total energy landscape, enabling local learning rules without global backpropagation. We show that EP on OIMs achieves competitive accuracy ($\sim 97.2 \pm 0.1 \%$ on MNIST, $\sim 88.0 \pm 0.1 \%$ on Fashion-MNIST), while maintaining robustness under realistic hardware constraints such as parameter quantization and phase noise. These results establish OIMs as a fast, energy-efficient substrate for neuromorphic learning, and suggest that EBMs - often bottlenecked by conventional processors - may find practical realization on physical hardware whose dynamics directly perform their optimization.

Learning at the Speed of Physics: Equilibrium Propagation on Oscillator Ising Machines

TL;DR

The paper addresses the challenge of training energy-based models efficiently on hardware by leveraging Equilibrium Propagation (EP), a local learning rule that removes the need for a backward pass. It demonstrates a direct mapping of EP onto Oscillator Ising Machines (OIMs), whose continuous phase dynamics perform energy descent and can be tuned with trainable couplings and biases. Empirical results show competitive MNIST and Fashion-MNIST accuracy and robustness to hardware constraints such as phase noise and quantization, with an empirical EP–BPTT correspondence validating gradient-descent properties. The work positions OIMs as a fast, energy-efficient substrate for practical EBMs and highlights the feasibility of implementing energy-based learning directly on physical hardware without major architectural changes.

Abstract

Physical systems that naturally perform energy descent offer a direct route to accelerating machine learning. Oscillator Ising Machines (OIMs) exemplify this idea: their GHz-frequency dynamics mirror both the optimization of energy-based models (EBMs) and gradient descent on loss landscapes, while intrinsic noise corresponds to Langevin dynamics - supporting sampling as well as optimization. Equilibrium Propagation (EP) unifies these processes into descent on a single total energy landscape, enabling local learning rules without global backpropagation. We show that EP on OIMs achieves competitive accuracy ( on MNIST, on Fashion-MNIST), while maintaining robustness under realistic hardware constraints such as parameter quantization and phase noise. These results establish OIMs as a fast, energy-efficient substrate for neuromorphic learning, and suggest that EBMs - often bottlenecked by conventional processors - may find practical realization on physical hardware whose dynamics directly perform their optimization.
Paper Structure (6 sections, 4 equations, 2 figures)

This paper contains 6 sections, 4 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Oscillator Ising Machines
  3. Equilibrium Propagation
  4. Implementation on Oscillator Ising Machines
  5. Results
  6. Discussion

Figures (2)

  • Figure 1: (Left) Accuracy: 784–500–10 reaches 97.2±0.1% (MNIST) and 88.0±0.1% (FMNIST); 784–120–10 reaches 90.6±1.7% (MNIST/100). Inset: EP--BPTT correspondence, with symmetric EP (dashed) matching BPTT (solid). (Right) Noise robustness (784–120–10): accuracy remains high with optimized $\beta$ (peak 92.0±0.3% at $\xi=0.2$). Inset: fixed-$\beta$ training degrades at large noise $\xi$.
  • Figure 2: (Left) Phase quantization (784–120–10): accuracy holds down to 4-bit (89.8±1.5%), degrades at 2-bit. (Right) Parameter quantization (784–120–10): stable at 10-bit (89.4±1.5%), degrades at 8-bit.