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. What is Error Diffusion? 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. The Dual-Stream Architecture 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: Copy CodeCopiedUse a different Browser p_i = φ_i( +p_{i-1} Wpp − n_{i-1} Wnp + bp ) n_i = φ_i( +n_{i-1} Wnn − p_{i-1} Wpn + bn ) Here, all four weight matrices stay non-negative element-wise. The biases bp and bn are the exception, since they need not be non-negative. Moreover, the negation signs before Wnp and Wpn 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. Modulo Error Routing 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. Three Classification Innovations Building on that routing, the research team adds three fixes for multi-class classification: Layer-specific sigmoid widths use φi(z) = 1/(1 + e−2z/αi). Since the sigmoid derivative directly gates the error signal, attenuation is severe. In fact, post-hoc analysis reveals a 25× decay from the output to the first hidden layer. Wider sigmoids keep derivatives larger, preventing premature saturation. The team sets α = 3.0 for CIFAR-10 convolutional layers and α = 6.0 for fully connected layers. Batch-centered class error subtracts the per-class mini-batch mean. This makes the one-vs-all error zero-mean across the batch for every class. It thereby reduces persistent suppression caused by the 9:1 target imbalance. Asymmetric initialization scales excitatory weights by 1.5× and inhibitory weights by 0.5×. That gives an expected E/I scale ratio of 3:1, while the output layer stays symmetric. Performance 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. Method MNIST CIFAR-10 Dale-compliant Notes Proposed ED 96.7% 61.7% Yes All weights non-negative; first ED on CNNs Seed ED 50.4% 11.6% Yes No innovations; α = 1.0, raw error, symmetric init DFA 97.6% 69.1% No Random feedback; ∼2.84M negative weights The Ablation Reversal 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. Error Diffusion in Reinforcement Learning Beyond classification, the research team integrate ED with Proximal Policy Optimization (PPO). They call the result ED-PPO and test it on Brax locomotion and Craftax. Here, policy-output error is routed to hidden units by output channel. For the scalar value network, the error is broadcast to all units. Importantly, ED-PPO drops the three classification innovations entirely. Across five seeds, ED-PPO beats BP-PPO on HalfCheetah (5494 vs 3520; p < 0.001) and matches DFA-PPO. On Ant, it stays on par with both PPO variants. On Craftax, meanwhile, DFA-PPO is the weakest method (19.8 vs BP-PPO 27.0). Thus random feedback that suffices for classification can fail on open-ended RL. Use Cases and Examples Three settings make this concrete: Neuromorphic and photonic hardware often encodes non-negative synaptic magnitudes physically. ED’s fixed-sign routing maps cleanly onto such substrates, complementing prior photonic DFA work. The non-negative floor drives 37.3% of weights to the floor (10⁻⁴) after training. Inhibitory cross-stream fully connected connections are pruned most, up to 68.8%. This implicit sparsity hints at model compression “for free.” The dedicated inhibitory stream may help continual and open-ended learning. It provides a structural mechanism for dampening large gradient excursions. Comparison How Dale-Compliant Error Diffusion Compares Proposed approach vs. other backpropagation-free and biologically motivated learning rules. “Dale-compliant” means separate excitatory/inhibitory populations with non-negative weights. Method names link to primary sources. Method Backprop-free (no weight transport) How error reaches hidden layers Dale-compliant (E/I, non-negative) Shown on RL Demonstrated reach / notes Error Diffusion — ED / ED-PPO (proposed) Yes Global error sign routed directly to hidden units via modulo routing r(i) = i mod C Yes — dual-stream E/I, non-negative weights Yes (Brax, Craftax) 96.7% MNIST, 61.7% CIFAR-10; RL returns on par with DFA-PPO Backpropagation No — needs transposed forward weights Exact gradient, layer by layer No — arbitrary-sign weights Yes (BP-PPO) Reference baseline; state of the art across tasks Feedback Alignment (FA) Yes Fixed random backward weights, layer by layer No — arbitrary-sign feedback Not shown Learns deep and convolutional nets; limited on harder benchmarks Direct Feedback Alignment (DFA) Yes Output error to
