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Continual Learning in Modern Hopfield Networks with an Application to Diffusion Models

NeurIPS 2026 · Continual learning › Online and replay-based · approach: Plasticity-preserving continual reinforcement learning · 0 citations (OpenAlex)

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Defines intrinsic forgetting as the rise in modern Hopfield energy after a task change, proves high-energy (outlier) memories are forgotten more and benefit more from replay, and uses top-K energy replay to reduce forgetting in Stable Diffusion and a pixel DDPM on split CIFAR-10.

Problem

Continual learning in diffusion/generative models is poorly understood: it is unclear which learned-distribution aspects are forgotten after a task change and which replay samples should be prioritized.

Approach

Uses the modern Hopfield energy (equivalent to the log-sum-exp energy of Gaussian-kernel diffusion scores) as an internal, model-native measure of forgetting; shows analytically in a cluster-plus-outlier MHN under small rotations that outlier memories gain more energy, and that replay gain is proportional to a softmax-based susceptibility times the energy rise, motivating energy-guided top-K repla

Key results

• In the closed-form MHN (N=12, d=50, c=0.35, beta=8), the outlier fixed point shows a larger rotation-induced energy rise than cluster fixed points, as predicted by the theorem.
• On split CIFAR-10, Stable Diffusion v1.5 and a pixel-space DDPM both show post-Task-2 reconstruction MSE at t=0.8 increasing monotonically with Task-1 Hopfield energy.
• Energy-guided top-K replay (K=5000) is reported to reduce forgetting most in the high-energy tail, with aggregated reductions ordered top-K above random above low-energy replay.

Surprising findings

• Low-energy replay helps only low-energy images, while high-energy replay gives large but localized reductions.

Limitations

• Theory covers only finitely many memories and assumes the score function is approximated perfectly; neural-network score error is not analyzed.
• Reconstruction-based forgetting is a heuristic metric and the experiments are limited to CIFAR-10 images and MNIST-style domains.
• Analysis of intrinsic forgetting applies to a rotation-based domain shift, which differs from the actual fine-tuning used in the diffusion experiments.

Experimental setup

Datasets: CIFAR-10 MNIST rotated MNIST

Benchmarks: split CIFAR-10 rotated MNIST

Metrics: reconstruction MSE at fixed timestep Hopfield energy rise

Baselines: random replay no replay energy-guided bottom-K replay

Builds on: modern Hopfield networks diffusion-Hopfield correspondence

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