Global Credit Assignment via Dynamical Criticality

Efficiently training recurrent neural networks to capture temporal structure is fundamental to machine intelligence, yet scaling gradient-based learning remains an open challenge. The standard global paradigm, backpropagation through time (BPTT), suffers from memory costs that scale linearly with sequence length and susceptibility to gradient instability. Conversely, biologically inspired local learning rules, while efficient, often introduce severe bias, failing to match the performance of global optimization. Inspired by the criticality observed in biological neural circuits, we introduce Criticality-driven Online Local Alignment (COLA). By leveraging the long-range spatiotemporal correlations inherent to the critical regime, COLA enables a strictly local learning rule to approximate the efficacy of global error propagation, thereby combining the advantages of online plasticity with the precision of gradient descent. Theoretically, for an RNN with $H$ hidden units, the method reduces learning complexity to a $O(H)$ auxiliary state with constant activation memory, independent of sequence length. Empirically, COLA matches BPTT on standard benchmarks and demonstrates superior robustness on stability-sensitive tasks. We support these results with a rigorous analysis on the approximation error, providing a theoretical foundation for reliable, scalable online learning.