Residual connections have been proposed as an architecture-based inductive bias to mitigate the problem of exploding and vanishing gradients and increased task performance in both feed-forward and recurrent networks (RNNs) when trained with the backpropagation algorithm. Yet, little is known about how residual connections in RNNs influence their dynamics and fading memory properties. Here, we introduce weakly coupled residual recurrent networks (WCRNNs) in which residual connections result in well-defined Lyapunov exponents and allow for studying properties of fading memory. We investigate how the residual connections of WCRNNs influence their performance, network dynamics, and memory properties on a set of benchmark tasks. We show that several distinct forms of residual connections yield effective inductive biases that result in increased network expressivity. In particular, those are residual connections that (i) result in network dynamics at the proximity of the edge of chaos, (ii) allow networks to capitalize on characteristic spectral properties of the data, and (iii) result in heterogeneous memory properties. In addition, we demonstrate how our results can be extended to non-linear residuals and introduce a weakly coupled residual initialization scheme that can be used for Elman RNNs.

翻译：残差连接已被提出作为一种基于架构的归纳偏置，用于缓解前馈网络和循环网络（RNN）在使用反向传播算法训练时的梯度爆炸与消失问题，并提升任务性能。然而，关于残差连接如何影响RNN的动态行为及其残差记忆特性，目前仍知之甚少。本文引入了一种弱耦合残差循环网络（WCRNNs），其中残差连接可生成定义明确的李雅普诺夫指数，并允许研究残差记忆的属性。我们在一组基准任务上探究WCRNNs的残差连接如何影响其性能、网络动力学及记忆特性。研究表明，多种不同形式的残差连接能形成有效的归纳偏置，从而增强网络表达能力。具体而言，这些残差连接具有以下特点：（i）使网络动力学接近混沌边缘，（ii）使网络能够利用数据中的特征谱特性，（iii）产生异质性的记忆特性。此外，我们展示了如何将研究结果推广至非线性残差场景，并提出了一种适用于Elman RNN的弱耦合残差初始化方案。