[Paper Review] Energy-based learning algorithms for analog computing: a comparative study

Benjamin Scellier, Maxence Ernoult+2 more|arXiv (Cornell University)|Dec 22, 2023

Advanced Memory and Neural Computing7 citations

TL;DR

This paper presents a comprehensive comparison of seven energy-based learning (EBL) algorithms—contrastive learning (CL), equilibrium propagation (EP), and coupled learning (CpL) with varying perturbation signs—on deep convolutional Hopfield networks (DCHNs) across five vision tasks. It finds that negative perturbations outperform positive ones, and the centered variant of EP (C-EP) achieves the best performance, especially as task difficulty increases, while also introducing a 13.5× speedup via asynchronous energy minimization and 16-bit precision.

ABSTRACT

Energy-based learning algorithms have recently gained a surge of interest due to their compatibility with analog (post-digital) hardware. Existing algorithms include contrastive learning (CL), equilibrium propagation (EP) and coupled learning (CpL), all consisting in contrasting two states, and differing in the type of perturbation used to obtain the second state from the first one. However, these algorithms have never been explicitly compared on equal footing with same models and datasets, making it difficult to assess their scalability and decide which one to select in practice. In this work, we carry out a comparison of seven learning algorithms, namely CL and different variants of EP and CpL depending on the signs of the perturbations. Specifically, using these learning algorithms, we train deep convolutional Hopfield networks (DCHNs) on five vision tasks (MNIST, F-MNIST, SVHN, CIFAR-10 and CIFAR-100). We find that, while all algorithms yield comparable performance on MNIST, important differences in performance arise as the difficulty of the task increases. Our key findings reveal that negative perturbations are better than positive ones, and highlight the centered variant of EP (which uses two perturbations of opposite sign) as the best-performing algorithm. We also endorse these findings with theoretical arguments. Additionally, we establish new SOTA results with DCHNs on all five datasets, both in performance and speed. In particular, our DCHN simulations are 13.5 times faster with respect to Laborieux et al. (2021), which we achieve thanks to the use of a novel energy minimisation algorithm based on asynchronous updates, combined with reduced precision (16 bits).

Motivation & Objective

• To provide a direct, fair comparison of seven EBL algorithms—CL, EP variants (P-, N-, C-), and CpL variants (P-, N-, C-)—on identical models and datasets.
• To investigate how algorithm performance varies with increasing task difficulty across MNIST, F-MNIST, SVHN, CIFAR-10, and CIFAR-100.
• To identify the most effective EBL algorithm for analog and energy-efficient AI hardware by evaluating performance, stability, and convergence.
• To significantly accelerate EBL training through a novel asynchronous energy minimization algorithm and reduced-precision (16-bit) computation.
• To establish new state-of-the-art results in both accuracy and speed for DCHNs on all five benchmark datasets.

Proposed method

• Trained deep convolutional Hopfield networks (DCHNs) using seven EBL algorithms: CL, P-EP, N-EP, C-EP, P-CpL, N-CpL, and C-CpL, differing in the sign and type of perturbation applied during learning.
• Applied perturbations to contrast the free state (unclamped) and perturbed state (clamped or biased), with weight updates derived from the difference in energy between states.
• Implemented a novel asynchronous energy minimization algorithm that accelerates convergence by updating neurons in a non-simultaneous, randomized order.
• Reduced model precision to 16-bit floating point to enhance training speed without sacrificing accuracy.
• Used identical model architecture and hyperparameters across all algorithms and datasets to ensure fair comparison.
• Conducted simulations on Nvidia A100 GPUs, with training lasting up to six days for full evaluation across five datasets.

Experimental results

Research questions

• RQ1 How do different EBL algorithms (CL, EP, CpL) with varying perturbation signs compare in performance on increasingly complex vision tasks?
• RQ2 Does the use of negative perturbations (N-EP, N-CpL) yield better generalization than positive perturbations (P-EP, P-CpL) or centered perturbations?
• RQ3 Is the centered variant of EP (C-EP) superior to other EBL algorithms in terms of accuracy and convergence stability across diverse datasets?
• RQ4 Can asynchronous energy minimization and 16-bit precision significantly reduce training time while maintaining or improving model performance?
• RQ5 Do the proposed training optimizations enable DCHNs to achieve new state-of-the-art results in both accuracy and speed on standard vision benchmarks?

Key findings

• On MNIST, all EBL algorithms achieve comparable performance, with C-EP and N-EP yielding the lowest test errors (0.44% and 0.42%, respectively).
• As task difficulty increases (e.g., CIFAR-100), performance differences become pronounced: C-EP achieves 37.0% test error, significantly outperforming CL (71.4%) and P-EP (89.4%).
• Negative perturbations (N-EP, N-CpL) consistently outperform positive ones (P-EP, P-CpL), with N-EP achieving 3.51% test error on SVHN compared to 9.64% for CL.
• The centered variant of EP (C-EP) is the top-performing algorithm across all datasets, achieving state-of-the-art results on all five benchmarks.
• The proposed asynchronous energy minimization and 16-bit precision reduce training time by 13.5× compared to prior work (Laborieux et al., 2021), enabling faster convergence.
• On SVHN, with improved weight initialization, C-EP achieves 3.58% test error, outperforming RBP (3.91%) and TBP (3.66%), confirming its robustness on complex data.