Score Broadcast and Decorrelation: A General Framework for Broadcast-Based Credit Assignment
Authors: Mustafa Uzun, Mete Erdogan, Cengiz Pehlevan, Alper T. Erdogan
Organizations: KUIS AI Center, Koc University, Turkey · Electrical and Electronics Engineering, Koc University, Turkey · Department of Electrical Engineering, Stanford University, USA · John A. Paulson School of Engineering & Applied Sciences, Harvard University, USA · Kempner Institute, Harvard University, USA · Center for Brain Science, Harvard University, USA
Abstract
We introduce Score Broadcast and Decorrelation (SBD), a principled framework for broadcast-based credit assignment for general families of differentiable losses. Error broadcast is a biologically plausible alternative to backpropagation that sends output information to hidden layers without weight transport. The Error Broadcast and Decorrelation (EBD) framework, recently introduced for the mean-squared-error (MSE) setting, grounded this mechanism in the stochastic orthogonality of optimal estimators, under which the optimal residual is orthogonal to functions of the input. We generalize that foundation by introducing an orthogonality principle between the output score (the gradient of loss with respect to the final-layer output) and hidden-layer activations, which holds whenever the optimal score has conditional mean zero. This single principle unifies broadcast-based credit assignment across the standard differentiable-loss families, including cross-entropy, Bregman divergences, proper scoring rules, and exponential-family negative log-likelihoods. The framework supplies a theoretical grounding for the three-factor learning rule under general losses, with the neuromodulatory factor derived as the broadcast loss score. We derive the cross-entropy case explicitly, characterize the admissible loss class, and introduce a score vector expansion technique that enriches the broadcast signal while preserving the orthogonality framework. Experiments on CIFAR-10 and Tiny ImageNet show that SBD substantially improves over existing broadcast approaches, with score vector expansion delivering further gains. Overall, this work identifies the loss score as the signal to broadcast, supplies the orthogonality theory and theoretical grounding for the three-factor learning rule from neuroscience, and shows how score vector expansion enriches the decorrelation directions of the resulting objective.
Biological neural circuits obey Dale's principle: each neuron's synapses are uniformly excitatory or inhibitory. Artificial networks that respect this constraint must coordinate separate excitatory and inhibitory populations, fundamentally changing how credit is assigned during learning. Several biologically plausible learning rules avoid backpropagation's weight transport requirement, but it has been difficult to achieve strong performance under Dale's principle beyond MNIST. Error Diffusion (ED) was originally proposed in a dual-stream excitatory/inhibitory architecture, where learning is driven by routing global error signals to all layers without transporting transposed forward weights or relying on random feedback matrices. Whether such a rule can scale under Dale's principle across both supervised classification and reinforcement learning remains unknown. Here, we introduce modulo error routing to extend Error Diffusion beyond binary classification, and show that a dual-stream excitatory/inhibitory architecture trained with this method achieves 96.7% on MNIST and establishes a 61.7% baseline on CIFAR-10, demonstrating that representation learning is possible even when strictly enforcing Dale's principle. For the classification setting, we introduce three domain-specific innovations: layer-specific sigmoid widths, batch-centered class error signals, and asymmetric initialization, and ablation analysis reveals that their relative importance reverses between MNIST and CIFAR-10, exposing task-dependent credit-assignment bottlenecks invisible to single-benchmark evaluation. In reinforcement learning, we integrate ED with Proximal Policy Optimization (PPO) and evaluate it on continuous-control tasks in Google Brax and on Craftax, an open-ended exploration task. We show that ED-PPO achieves competitive performance relative to Direct Feedback Alignment, a backpropagation-free baseline.
Long-horizon agentic tasks pose a fundamental credit assignment challenge for outcome-base reinforcement learning: trajectory-level rewards verify final correctness but provide limited guidance on which intermediate reasoning steps or tool interactions contribute to the outcome. The difficulty is especially pronounced in multi-turn search agents, where successful trajectories may contain misleading actions and failed trajectories may contain valuable evidence-gathering steps. We propose PBSD (Privileged Bayesian Self-Distillation), a Bayes-calibrated self-distillation method for fine-grained credit assignment under sparse final rewards. PBSD measures trajectory quality through the posterior-to-prior probability ratio of the verified answer and applies Bayes' rule to convert this hard-to-estimate answer-side ratio into a tractable likelihood ratio between a standard student model and a privileged answer-conditioned teacher model. Autoregressive decomposition of this Bayesian evidence score yields turn-level signals that identify whether each intermediate turn supports or undermines the verified outcome. Consequently, PBSD provides a principled and elegant reweighting scheme that transforms sparse outcome supervision into Bayes-calibrated turn-level credit signals, while remaining fully compatible with standard policy optimization. Experiments demonstrate that PBSD consistently enhances performance across both in-domain and out-of-domain settings, and effectively transfers knowledge from short-context training to long-context inference, suggesting that its fine-grained credit assignment mechanism facilitates more effective policy learning and yields improved generalization.
Backpropagation is the default learning rule for artificial neural networks and is often treated as the settled approach whenever differentiability is available. In this work, we revisit this convention through a theoretical lens of sample efficiency. We introduce a unified vectorized feedback framework for loss-based and reward-based learning on computational graphs, in which synthetic gradients emerge as a natural alternative to backpropagation. We characterize the conditions under which synthetic gradients can achieve a lower gradient-estimation mean squared error than backpropagation. We construct examples illustrating that this sample efficiency advantage can be arbitrarily large. Experiments on contextual bandits and reinforcement learning tasks demonstrate the potential of our theoretical findings.