A Short Course for Modern Large Language ModelsGlossary

Lesson 22 of 23 · Transformer block

Expand, gate, multiply, project back.

Your win: calculate every branch of a small SwiGLU feed-forward network with actual matrices.

15 minutesNeeds: matrix multiplicationOutcome: calculate SwiGLU
Mission link: after attention mixes information across tokens, the feed-forward sublayer transforms each token independently across features.

Two parallel projections form a gated hidden vector

\[ \operatorname{SwiGLU}(\mathbf x) =\left(\operatorname{SiLU}(\mathbf xW_{\text{gate}})\odot(\mathbf xW_{\text{up}})\right)W_{\text{down}}, \] \[ \operatorname{SiLU}(z)=z\,\sigma(z), \qquad \sigma(z)=\frac{1}{1+e^{-z}}. \]
\(d_{\text{model}}\)→two branches of width \(d_{\text{ff}}\)⊙\(d_{\text{ff}}\)→\(d_{\text{model}}\)

The same feed-forward weights are applied separately at every token position. There is no \(T\times T\) token mixing here; attention already performed that role.

Numbers substituted: \(d_{\text{model}}=d_{\text{ff}}=2\)

Let

\[ \mathbf x=\begin{bmatrix}1&-1\end{bmatrix},\quad W_{\text{gate}}=\begin{bmatrix}1&0\\0&1\end{bmatrix},\quad W_{\text{up}}=\begin{bmatrix}1&1\\1&-1\end{bmatrix},\quad W_{\text{down}}=\begin{bmatrix}1&0\\0&-2\end{bmatrix}. \]
  1. Gate projection: \(\mathbf g=\mathbf xW_{\text{gate}}=(1,-1)\).
  2. Apply SiLU: \(\sigma(1)\approx0.731\), \(\sigma(-1)\approx0.269\), so \(\operatorname{SiLU}(\mathbf g)\approx(0.731,-0.269)\).
  3. Up projection: \(\mathbf u=\mathbf xW_{\text{up}}=(0,2)\).
  4. Gate elementwise: \((0.731,-0.269)\odot(0,2)=(0,-0.538)\).
  5. Project down: \((0,-0.538)W_{\text{down}}=(0,1.076)\).

Therefore \(\operatorname{SwiGLU}(1,-1)\approx(0,1.076)\) for these toy weights.

Trace two numbers through both SwiGLU branches

\(x=\)
Gate \(xW_g\)
SiLU(gate)
Up \(xW_u\)
Elementwise product
Down \(W_d\)

The two middle branches have matching widths so their coordinates can multiply one by one.

Why “gate”?

The SiLU branch scales each coordinate of the up branch through elementwise multiplication. A gate near zero suppresses that coordinate; a larger signed gate allows or reverses more of it. The final down projection mixes the gated hidden coordinates back into model width.

Gated feed-forward network with gate, up, elementwise product, and down projection paths
The two expanded branches must have matching \(d_{\text{ff}}\) widths for elementwise multiplication. The down projection restores \(d_{\text{model}}\) so the residual addition is valid.

Code checkpoint · SwiGLU in two lines

The gate and up projections expand to the same intermediate width. Their elementwise product is then projected back to model width.

Show the feed-forward computation
def forward(self, x):
    gate = F.silu(self.gate_proj(x))  # [B, T, Dff]
    up = self.up_proj(x)              # [B, T, Dff]
    gated = gate * up                 # [B, T, Dff]
    return self.down_proj(gated)      # [B, T, D]

Trace it: Which line restores the width required by the second residual addition?

Retrieval check

What must match before \(\operatorname{SiLU}(\mathbf g)\odot\mathbf u\)?

Practice before moving on

  1. Calculate \(\sigma(0)\) and \(\operatorname{SiLU}(0)\).
  2. Calculate \(\operatorname{SiLU}(1)\) to three decimals.
  3. For gate vector \((0.5,-1)\) after SiLU and up vector \((4,3)\), compute their elementwise product.
  4. Recompute the worked example's up projection by matrix multiplication.
  5. If \(d_{\text{model}}=4,d_{\text{ff}}=6\), state the shapes of \(W_{\text{gate}},W_{\text{up}},W_{\text{down}}\) for row-vector notation.
  6. Explain which sublayer mixes tokens and which transforms tokens independently.
Check solutions
  1. \(\sigma(0)=0.5\), so \(\operatorname{SiLU}(0)=0\).
  2. \(1/(1+e^{-1})\approx0.731\).
  3. \((2,-3)\).
  4. \((1,-1)\begin{bmatrix}1&1\\1&-1\end{bmatrix}=(0,2)\).
  5. \(W_{\text{gate}},W_{\text{up}}:4\times6\); \(W_{\text{down}}:6\times4\).
  6. Self-attention mixes information across token positions; the FFN applies the same feature transformation independently to each token.

Primary sources: Shazeer, GLU Variants Improve Transformer; Touvron et al., LLaMA.