Credit Assignment via Neural Manifold Noise Correlation

TL;DR

Credit assignment in high-dimensional networks is hampered by the need to estimate the Jacobian and by isotropic perturbations that ignore neural structure. The authors propose Neural Manifold Noise Correlation (NMNC), which restricts perturbations to the learned neural manifold $\mathcal{M}_l$ via an online basis $U_l$ and updates feedback weights using manifold-aligned perturbations, effectively preconditioning gradients. Theoretical and empirical results show that NMNC improves gradient alignment and increases the true-gradient-projected step, yielding large gains in CIFAR-10, ImageNet-scale models, and recurrent nets, along with brain-like representations relative to vanilla noise correlation. These findings suggest that incorporating biologically inspired manifold structure can enhance learning scalability without sacrificing performance, offering a plausible mechanism for credit assignment in real neural circuits.

Abstract

Credit assignment--how changes in individual neurons and synapses affect a network's output--is central to learning in brains and machines. Noise correlation, which estimates gradients by correlating perturbations of activity with changes in output, provides a biologically plausible solution to credit assignment but scales poorly as accurately estimating the Jacobian requires that the number of perturbations scale with network size. Moreover, isotropic noise conflicts with neurobiological observations that neural activity lies on a low-dimensional manifold. To address these drawbacks, we propose neural manifold noise correlation (NMNC), which performs credit assignment using perturbations restricted to the neural manifold. We show theoretically and empirically that the Jacobian row space aligns with the neural manifold in trained networks, and that manifold dimensionality scales slowly with network size. NMNC substantially improves performance and sample efficiency over vanilla noise correlation in convolutional networks trained on CIFAR-10, ImageNet-scale models, and recurrent networks. NMNC also yields representations more similar to the primate visual system than vanilla noise correlation. These findings offer a mechanistic hypothesis for how biological circuits could support credit assignment, and suggest that biologically inspired constraints may enable, rather than limit, effective learning at scale.

Figures (11)

• Figure 1: (Left) Variance of activations explained by principal components across training epochs for each layer of convolutional neural networks trained on CIFAR-10. Epoch 0 refers to the network prior to training, and Epoch 100 is the last epoch. (Right) Same analysis applied to the Jacobian variance. Curves are shown up to 90% cumulative variance explained. mean $\pm$ std, $n=5$ seeds.
• Figure 2: Network size vs. neural manifold dimensionality (TwoNN or #PCs for 90% variance). mean $\pm$ std, $n=5$ seeds.
• Figure 3: Performance and sample efficiency of NMNC and VNC on CIFAR-10. (A) Test accuracy vs. epochs for different learning rules. (B) Test accuracy vs. epochs for varying frequencies of feedback update for NMNC and VNC (No InitJac). Feedback weights are updated every $b$ batch. (C) Best test accuracy vs. noise correlation frequency. Same data as (B). mean $\pm$ std, $n=5$ seeds.
• Figure 4: Alignment between true and estimated gradients in activation space (see \ref{['fig:8']} for alignment in weight space). (A) Cosine similarity angle between the true and estimated gradients for NMNC and VNC across layers. (B) Normalized magnitude of the estimated gradient projected onto the true gradient direction for NMNC and VNC across layers. (C) Same as (A) but for varying frequencies of feedback update. The color scheme is the same as in \ref{['fig:3']}B. mean $\pm$ std, $n=5$ seeds.
• Figure 5: Comparison of NMNC and VNC on ImageNet. Test accuracy of AlexNet on ImageNet when trained with (i) Backprop, (ii) the weight mirror algorithm using vanilla noise correlation (VNC), and (iii) the weight mirror algorithm using neural manifold noise correlation (NMNC). mean $\pm$ std, $n=5$ seeds.