Consider the fully recurrent network architecture $($without output activation and bias units$)$ defined as

$s(t)=Wx(t)+Ra(t-1)$

$a(t)=f(s(t))$

hat$(y)(t)=Va(t)$

with input vectors $x(t),$ hidden pre$-$activation vectors $s(t),$ hidden activation vectors $a(t),$ activation function $f(*)$ and parameter matrices $R,W,V.$ Let $$ denote the vector of all network parameters shared in time and let $(t)$ denote their usage at time $t.$ Further, let $L(t)=L(y(t),$hat$(y)(t))$ denote the loss function at time $t$ and let $L={\displaystyle \underset{t=1}{\overset{T}{}}}L(t)$ denote the total loss. We use denominator$-$layout convention, i$.$e$.,\frac{\text{d}\text{e}\text{l}\text{L}}{\text{d}\text{e}\text{l}}$ is a column vector.

Which statements about RTRL are true?

a$.$ RTRL computes the gradients $\frac{\text{d}\text{e}\text{l}\text{L}(t)}{\text{d}\text{e}\text{l}}$ during the forward pass already.

b$.$ Schmidhuber's approach divides the input sequence into chunks of size $N($number hidden units$)$ and performs RTRL within these chunks. Then it uses BPTT to consolidate the gradients for these chunks.

c$.$ BPTT considers the recursion $\frac{\text{d}\text{e}\text{l}\text{L}}{\text{d}\text{e}\text{l}\text{s}(t)}=\frac{\text{d}\text{e}\text{l}\text{s}(t+1)}{\text{d}\text{e}\text{l}\text{s}(t)}\frac{\text{d}\text{e}\text{l}\text{L}}{\text{d}\text{e}\text{l}\text{s}(t+1)}+\frac{\text{d}\text{e}\text{l}\text{L}(t)}{\text{d}\text{e}\text{l}\text{s}(t)}$ while RTRL considers the recursion $\frac{\text{d}\text{e}\text{l}\text{s}(t)}{\text{d}\text{e}\text{l}}=\frac{\text{d}\text{e}\text{l}\text{s}(t)}{\text{d}\text{e}\text{l}(t)}+\frac{\text{d}\text{e}\text{l}\text{s}(t-1)}{\text{d}\text{e}\text{l}}\frac{\text{d}\text{e}\text{l}\text{s}(t)}{\text{d}\text{e}\text{l}\text{s}(t-1)}$

d$.$ The term $\frac{\text{d}\text{e}\text{l}\text{s}(t)}{\text{d}\text{e}\text{l}}$ generally has $O(N^4)$ elements, where $N$ denotes the number of hidden units