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  • 整体思路以及计算方式
  • 简评
  1. LongConv

Resurrecting Recurrent Neural Networks for Long Sequences

PreviousTime-aware large kernel convolutionsNextCKConv: Continuous Kernel Convolution For Sequential Data

Last updated 2 years ago

论文地址：

  • https://arxiv.org/abs/2303.06349

整体思路以及计算方式

这篇论文主要解决了之前RNN无法在长序列上并行训练，或者说性能一般的问题。

动机：SSM和RNN那么像，为啥SSM work，RNN不work呢？

第一个问题：传统RNN带有非线性激活：

xk=σ(Axk−1+Buk),yk=Cxk+Dukx_k=\sigma\left(A x_{k-1}+B u_k\right), \quad y_k=C x_k+D u_kxk​=σ(Axk−1​+Buk​),yk​=Cxk​+Duk​

所以无法递推得到类似SSM的结果，那解决这点很简单，直接把激活拿掉即可：

xk=Axk−1+Bukx_k=A x_{k-1}+B u_kxk​=Axk−1​+Buk​

假设x−1=0x_{-1}=0x−1​=0，那么：

xk=∑j=0k−1AjBuk−jx_k=\sum_{j=0}^{k-1} A^j B u_{k-j}xk​=j=0∑k−1​AjBuk−j​

最后具体的实现还有一个残差部分，这里罗列一下：

简评

初始化部分可以看看，其他部分，理解SSM的人应该不难自己得到结论。

这样第一个问题就解决，但是如果这样直接训练，效果还是很一般，因为是AjA^jAj的模长可能太大或者太小，作者使用如下方式解决，首先假设AAA为对角阵Λ\LambdaΛ，然后用如下方式初始化：

Λ=diag⁡(exp⁡(−ν+iθ))\Lambda=\operatorname{diag}(\exp (-\nu+i \theta))Λ=diag(exp(−ν+iθ))

这样做的好处是，保证了矩阵的特征值<1<1<1，不会出现模长爆炸的情况，v,θv, \thetav,θ的初始化可以参考论文。

xk=diag⁡(λ)xk−1+γ⊙Bukλj=exp⁡(−exp⁡(νjlog⁡)+iexp⁡(θjlog⁡))γj←(1−∣λj∣2)1/2x_k=\operatorname{diag}(\lambda) x_{k-1}+\gamma \odot B u_k\\ \lambda_j=\exp \left(-\exp \left(\nu_j^{\log }\right)+i \exp \left(\theta_j^{\log }\right)\right)\\ \gamma_j \leftarrow\left(1-\left|\lambda_j\right|^2\right)^{1 / 2}xk​=diag(λ)xk−1​+γ⊙Buk​λj​=exp(−exp(νjlog​)+iexp(θjlog​))γj​←(1−∣λj​∣2)1/2