Dynamics and representation structure of local approximations to gradient-based learning in linear recurrent neural networks

Biological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy to satisfy these constraints is to modify gradient descent by neglecting non-local terms to varying degrees, as in random feedback local online (RFLO) learning and truncated backpropagation through time (tBPTT). However, the learning dynamics of these algorithms, and how they compare with BPTT, remain poorly understood. Here, we apply dynamical systems theory to data-aligned linear RNNs to compare stationary solutions, stability properties, and convergence rates, finding a close relationship between BPTT and one-step tBPTT—in contrast with qualitatively distinct behavior for RFLO. We further observe that the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting. Our work provides fundamental insights into how locality constraints shape RNN learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for state-space models.