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. We apply dynamical systems theory to data-aligned linear RNNs -- whose dynamics can be separated into orthogonal modes -- to compare stationary solutions, stability properties, and convergence rates, finding qualitatively distinct behaviour for RFLO versus BPTT and one-step tBPTT. 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 analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.

翻译：生物和神经形态递归神经网络（RNN）在学习过程中，其可用的信息受到空间和时间局部性约束。常见策略是通过不同程度地忽略非局部项来修改梯度下降，例如随机反馈局部在线学习（RFLO）和截断时间反向传播（tBPTT）。然而，这些算法的学习动力学及其与BPTT的比较仍知之甚少。我们应用动力系统理论于数据对齐的线性RNN（其动力学可分为正交模态），比较稳态解、稳定性性质和收敛速率，发现RFLO与BPTT及单步tBPTT存在定性不同的行为。进一步观察到，RFLO学习的解仅限于初始参数的低秩扰动，这一结果在数据对齐设置之外仍然成立。我们的工作提供了关于局部性约束如何塑造学习动力学的分析性见解，对神经科学学习模型及RNN替代优化方法具有启示意义。