LNS [1] reveals that in many modern LLMs, nearly half of the deeper layers contribute very little to the final output.
This phenomenon, dubbed the "Curse of Depth", means we're spending enormous computational resources on layers that are essentially just passing information along without performing meaningful transformations.
Pre-Layer Normalization (Pre-LN) architecture, which is standard in most modern LLMs. In a Pre-LN Transformer block, the input to a sub-layer (like Attention or a Feed-Forward Network) is normalized before the main computation. The output is then added back to the original input via a residual connection.
A single Transformer layer's output $x_{l+1}$ can be represented by the following equations:
$$ y = x' + \text{FFN}(\text{LN}(x')) x' = x + \text{Attn}(\text{LN}(x)) $$
The key issue lies in how the variance of these vectors behaves as we go deeper into the network. Because the output of each sub-layer (which has its own variance) is added to the input from the previous layer, the variance accumulates. In paper LNS shows that the variance at layer $l{(\sigma_{x_l}^2)}$ grows sub-exponentially with depth:
$$ \sigma_{x_l}^2 = \sigma_{x_1}^2 \Theta\left(\prod_{k=1}^{l-1}\left(1+\frac{1}{\sigma_{x_k}}\right)\right) $$
This means that with each additional layer, the variance gets a little bigger, and this effect compounds, leading to an exponential explosion of variance in the deeper layers.
Layerwise output variance in a standard LLaMA-130M model with Pre-LN. Notice how the variance (y-axis) grows exponentially as the layer depth increases (represented by the different colored lines).
The answer lies in the gradient, The gradient of a Pre-LN layer (its Jacobian matrix, $J$) is approximately the sum of the identity matrix and the learning term:
$$ J_l = \frac{\partial x_{l+1}}{\partial x_l} \approx I + \frac{\partial \text{SubLayer}(\text{LN}(x_l))}{\partial x_l} $$
Here, $I$ is the identity matrix (which just passes the input through unchanged). The second term is the part that actually learns. [1], [2] both show that the Jacobian of the LayerNorm function itself is inversely proportional to the standard deviation, $\sigma_x$:
$$ \frac{\partial \text{LN}(x)}{\partial x} \approx \frac{1}{\sigma_x}I $$
So, the full Jacobian for the layer is influenced by this inverse relationship:
$$ J_l \approx I + (\text{SubLayer terms}) \times \frac{1}{\sigma_{x_l}} $$
As model goes deeper, the variance $\sigma_{x_l}^2$ explodes, meaning the standard deviation $\sigma_{x_l}^2$ also becomes enormous. Consequently, the term $\frac{1}{\sigma_{x_l}}$ approaches zero. This causes the entire "learning" part of the Jacobian to vanish, leaving:
$$ J_l \approx I + 0 \implies J_l \approx I $$