Diffusing Blame: Task-Dependent Credit Assignment in Biologically Plausible Dual-Stream Networks
Yutaro Yamada, Luca Grillotti, Rujikorn Charakorn, Sebastian Risi, David Ha, Robert Tjarko Lange
cs.LG, cs.NE
2026-06-30
Dual-stream excitatory/inhibitory nets with non-negative weights, trained by Error Diffusion not backprop, reach 96.7% MNIST and 61.7% CIFAR-10, and train competitive PPO policies.
Backpropagation trains deep networks by requiring the backward pass to use exact transposes of the forward weights, the weight transport problem, which has no biological analogue since real neurons do not ship symmetric weight copies around. Several "biologically plausible" rules sidestep weight transport: Feedback Alignment replaces backward weights with fixed random matrices, and Direct Feedback Alignment (DFA) projects output error straight to each hidden layer through fixed random matrices. But all of them still allow arbitrarily signed weights, which breaks a harder biological constraint: Dale's principle, that every neuron's synapses are uniformly excitatory or uniformly inhibitory.
Error Diffusion (ED) is a local rule naturally compatible with Dale's principle. Its updates depend only on presynaptic activity, the postsynaptic activation derivative, and a single global error sign, with no weight transport and no random feedback. The trouble is it had only been validated on binary classification and MNIST. Can a network that strictly enforces Dale's principle stay competitive across both classification and reinforcement learning? That is the question this Sakana AI paper takes on.
The core is a dual-stream architecture: each layer splits into an excitatory (p) and an inhibitory (n) stream. All four weight sub-matrices are element-wise non-negative, and the negation on cross-stream connections is hardcoded structurally, so every learnable parameter stays non-negative while the network can still subtract. The cost is roughly 4x the parameters (32M vs 8M at the same width).
Extending ED from binary to multi-output uses modulo error routing: each hidden unit is assigned a fixed output channel r(i) = i mod C and receives that output dimension's error as its learning signal, with error broadcast through a fixed routing matrix. This is a deterministic, structured unit-to-output correspondence, unlike DFA's random feedback.
For classification the authors add three task-specific tricks: layer-specific sigmoid widths (because the sigmoid derivative directly gates the ED error, and a post-hoc check shows 25x attenuation from output to the first hidden layer; wider sigmoids keep the derivative alive); batch-centered class error (10-way classification uses independent sigmoids rather than softmax, and the 9:1 one-vs-all imbalance persistently suppresses class channels, so they subtract the per-class batch mean to zero-center the error); and asymmetric initialization (excitatory weights 1.5x, inhibitory 0.5x, for an initial E/I ratio near 3:1). In RL, ED replaces backprop between PPO's hidden layers, with ReLU activations.
Classification: full ED hits 96.7±0.1% on MNIST and 61.7±0.7% on CIFAR-10, a large jump over "seed ED" (the dual-stream architecture with none of the three tricks) at 50.4% and 11.6%. DFA scores higher (97.6% MNIST, 69.1% CIFAR-10), but it violates Dale's principle and carries about 2.84M negative weights. The ED-vs-DFA gap widens from 0.9 points on MNIST to 7.4 on CIFAR-10, so the price of staying non-negative grows with task difficulty. The authors are blunt that 62% on CIFAR-10 is still far from competitive with conventional gradient methods.
The cleanest finding is that the ablation ordering flips. On MNIST, removing layer-specific sigmoid widths is catastrophic (96.7% to 25.3%, down 71.4 points); removing batch-centering barely hurts (down 0.3); symmetric initialization has no effect (plus 0.0). On CIFAR-10 the hierarchy reverses: removing batch-centering is now most destructive (61.7% to 13.8%, four of five seeds collapse, down 47.9), uniform width costs 15.1, and symmetric init 5.5. The same component can be load-bearing on one task and negligible on another.
Reinforcement learning (Brax Ant, HalfCheetah, Humanoid, plus Craftax): ED-PPO is strongest on HalfCheetah, where 5494±691 beats BP-PPO's 3520±485 (p<0.001) and matches DFA-PPO (5581±359). On Ant it ties both PPO variants. On Humanoid and Craftax it trails BP-PPO. Craftax flips the comparison once more: DFA-PPO, the standard backprop-free reference, is the weakest method there (19.8), worse than ED-PPO (about 23). The random-feedback pathway that suffices for supervised learning collapses on complex RL.
Two post-hoc findings round it out. Training drives the 3:1 asymmetric initialization toward near-balance (about 1:1), with a depth-dependent inhibitory gradient (layer 1: 1.03, layer 2: 0.90, layer 3: 0.81), loosely matching the cortex's deepening inhibition. The non-negative floor (1e-4) pushes 37.3% of weights to the floor, with cross-stream inhibitory connections pruned hardest (up to 68.8%), which is structured sparsity delivered essentially for free.
For people who work specifically on biologically plausible learning, this extends ED from "MNIST only" into multi-class classification and RL, a real step. For the broader audience, the lasting value is a measurement warning: do not conclude a component matters by looking at one benchmark. Batch-centering that is irrelevant on MNIST becomes the single point of failure on CIFAR-10. That kind of task-dependent bottleneck is invisible in a single-task ablation.
It also hands neuromorphic hardware a more concrete grip. The weights themselves are constrained non-negative (signs come from fixed excitatory or inhibitory pathways), which suits substrates that physically encode non-negative quantities, such as photonics, analog devices, and synapse devices.
The absolute numbers are still weak. 61.7% on CIFAR-10 is nowhere near mainstream gradient methods, and the 0.9 to 7.4 point gap to DFA quantifies the current cost of Dale's principle. ED-PPO has higher variance than BP-PPO across every environment, and on Craftax it sits about 4 reward below BP-PPO (p=0.002), suggesting ED's coarse modular credit assignment is unstable on tasks needing fine-grained temporal attribution.
The 4x parameter overhead may itself be the culprit: with 37.3% of weights pinned to the floor, effective capacity is well below the nominal count, and the authors cannot separate a credit-assignment problem from a capacity problem.
One more caveat: the authors lean on the emergent E/I balance and pathway pruning as a bridge to biology, but the weight-level E/I ratio is an indirect proxy for the cell-count, firing-rate, and synapse-strength balance that defines biological E/I. Treat the analogy as suggestive, not settled.