Why sequences break feedforward networks

"X then Y" and "Y then X" are different sequences — in language, "dog bites man" and "man bites dog" are not the same story. But a common way to turn a sequence into a fixed-size vector is to just add up its tokens. Addition doesn't care about order: $X + Y$ and $Y + X$ land on the exact same point. Once that sum is computed, the order is simply gone — there is no way to recover it from the vector alone.

Feed that pooled vector into a plain dense layer with weights $(w_1, w_2)$ and a sigmoid, trained to output $0$ for "X then Y" and $1$ for "Y then X":

• $\hat{y}$ — the network's predicted output.
• $\sigma$ — the sigmoid function, squashing the result into a value between $0$ and $1$.
• $w_1, w_2$ — the two learned weights of the dense layer.
• $p_1, p_2$ — the two components of the pooled vector.
• $p = \text{pool}(\text{sequence})$ — the fixed-size vector produced by summing (pooling) the sequence's tokens together.

1. Pooling erases the order

   Since $p$ is identical for both sequences, $\hat{y}$ is identical for both — for every choice of $w_1, w_2$.

2. No amount of training can fix it

   No amount of training can make this network output two different things for two inputs it has already made indistinguishable.

Drag the point anywhere in this plane. The loss never depends on $w_1$ and $w_2$ separately — only on their sum. Every point along the diagonal where $w_1 + w_2 = 0$ sits in the exact same flat valley, all at the same loss. There's no direction to search in that helps, because the representation this network is working from threw away the one thing that would have let it succeed.

1. Pick a weight — w1 = 2, w2 = -2

   The pooled input is $(1,1)$ for both sequences, so the logit is $2(1) + (-2)(1) = 0$ either way, giving $\hat{y}=\sigma(0)=0.5$ for both.

2. Compute the loss

   • For "X then Y" (target $0$) it's $-\log(1-0.5) = \ln 2$
   • For "Y then X" (target $1$) it's $-\log(0.5) = \ln 2$
3. Average

   $\ln 2 \approx 0.693$ — exactly the loss of a coin flip, because that's genuinely the best this architecture can do.