Sakana AI’s Error Diffusion Trains Dale-Compliant Dual-Stream Networks, Reaching 96.7% MNIST and 61.7% CIFAR-10 Without Backpropagation

Backpropagation dominates deep learning, yet it uses a mechanism the brain likely cannot. Specifically, the backward pass needs exact transposes of forward weight matrices. This is the weight transport problem . Sakana AI’s new paper, Diffusing Blame , confronts this constraint directly. The research team trains networks that obey Dale’s principle while avoiding weight transport entirely.

Error Diffusion (ED) is a local learning rule, first proposed by Kaneko (2000). Each weight update depends on three signals only. These are presynaptic activity, a postsynaptic activation derivative, and a single global error sign. Consequently, ED never transports transposed forward weights or uses random feedback matrices. That locality makes ED naturally compatible with Dale’s principle. However, prior work demonstrated ED only on binary classification and MNIST.

To satisfy that constraint, the research team split each layer into two streams. One stream is excitatory ( p ), and the other is inhibitory ( n ). The forward pass computes excitatory-minus-inhibitory preactivations for each stream:

Here, all four weight matrices stay non-negative element-wise. The biases b p and b n are the exception, since they need not be non-negative. Moreover, the negation signs before W np and W pn are structural, not learned. Therefore cross-stream connections remain inhibitory while all learnable weights stay non-negative. This design needs four weight sub-matrices per layer. As a result, it uses roughly 4× more parameters than a single-stream network. For the same architecture, that is ∼32M versus ∼8M for DFA.

With that architecture in place, the main extension is modulo error routing . This lifts Error Diffusion (ED) beyond binary classification. For hidden unit i , the research team define the routing r(i) = i mod C . Here, C is the output dimension. That unit then learns from the routed error component. In short, each hidden unit is assigned one fixed output channel. Unlike DFA, whose feedback matrices are random, ED uses this structured correspondence.

Building on that routing, the research team adds three fixes for multi-class classification:

With all three innovations, Error Diffusion (ED) reaches 96.7% on MNIST and 61.7% on CIFAR-10. In contrast, seed ED without them collapses to 50.4% and 11.6%. DFA scores higher on both tasks but violates Dale’s principle, using ∼2.84M negative weights. Notably, this is the first time ED has trained convolutional networks. Previously, Fujita (2026) reached ∼55.2% on CIFAR-10 using a flattened MLP. Even so, 61.7% remains far from standard gradient-based methods.

Interestingly, the innovations’ importance flips between tasks. On MNIST, removing layer-specific widths is catastrophic (−71.4 pp), collapsing accuracy toward chance. Batch-centering barely matters there (−0.3 pp). On CIFAR-10, however, the order reverses. Removing batch-centered error becomes the largest drop (−47.9 pp), collapsing four of five seeds. This reversal exposes task-dependent credit-assignment bottlenecks invisible to single-benchmark evaluation.