Global Credit Assignment via Dynamical Criticality

Efficiently training recurrent neural networks on long sequences remains an open challenge. The standard global paradigm, backpropagation through time (BPTT), suffers from vanishing and exploding gradients and memory costs that scale linearly with sequence length. Conversely, biologically inspired local learning rules are memory-efficient but typically introduce severe bias. To bridge this gap, 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 global error propagation, thereby combining online efficiency with gradient descent precision. Theoretically, for a recurrent neural network with $H$ hidden units, COLA requires only an $O(H)$ auxiliary state and constant activation memory, completely independent of sequence length. Empirically, COLA is competitive with BPTT on standard benchmarks and demonstrates superior robustness on stability-sensitive tasks. Finally, we conduct a rigorous analysis of the approximation error to provide a theoretical foundation for reliable online learning. Code is available at \url{https://github.com/Criticality-Cognitive-Computation-Lab/COLA}

Recurrent neural networks are often used to process sequences, such as text, sensor signals, or time-varying images, but training them on long sequences is still difficult. The standard training method, backpropagation through time, works by looking backward through the whole sequence, which can require a lot of memory and can become unstable when the sequence is long. We introduce COLA, a new online learning method that trains recurrent networks step by step, without storing the entire past sequence. COLA is inspired by the idea that both brains and dynamical systems can operate near a critical regime, where signals are stable but still able to travel over long distances in time and across network units. By using this structure, COLA estimates the long-term learning signal with only small local summaries instead of full backward computation. This makes training much more memory-efficient while keeping performance close to standard methods. In our experiments, COLA performs especially well on tasks that require long-term stability, and it also extends to convolutional and spiking recurrent networks.