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openai:o3-2025-04-16 via @hdevalence.bsky.social

Gemma 2’s Transformer blocks use the gated GELU activation, better known as GeGLU, inside every feed-forward (MLP) layer.

  1. Definition
    For an input vector x∈Rdx \in \mathbb{R}^{d}, the GeGLU transformation is

    GeGLU⁡(x)  =  (xW1+b1)  ⊙  GELU⁡ ⁣(xW2+b2),\operatorname{GeGLU}(x) \;=\; \bigl(xW_1 + b_1\bigr)\;\odot\;\operatorname{GELU}\!\bigl(xW_2 + b_2\bigr),

    where
    • W1,W2∈Rd×dff/2W_1,W_2 \in \mathbb{R}^{d \times d_{\text{ff}}/2} and b1,b2∈Rdff/2b_1,b_2 \in \mathbb{R}^{d_{\text{ff}}/2},
    • ⊙\odot is element-wise multiplication,
    • GELU⁡\operatorname{GELU} is the activation

    GELU⁡(z)=12 z(1+tanh⁡ ⁣(2/π (z+0.044715 z3))).\operatorname{GELU}(z)=\tfrac{1}{2}\,z\Bigl(1+\tanh\!\bigl(\sqrt{2/\pi}\,(z+0.044715\,z^{3})\bigr)\Bigr).

    The gating halves the expansion size (only dffd_{\text{ff}} instead of 2dff2d_{\text{ff}} parameters) while retaining two linear transforms.

  2. Why Gemma 2 uses GeGLU

    • Expressiveness at small scale
    The multiplicative gate lets the network model input-dependent rescaling, which empirically boosts accuracy for compact models (2 B–27 B params) more than plain ReLU, GELU or SiLU.

    • Parameter & compute efficiency
    Compared with standard GELU + linear projection (which needs width 2dff2d_{\text{ff}}), GeGLU attains similar or better quality with the same dffd_{\text{ff}}, reducing parameters and FLOPs—valuable for “practical-size’’ models.

    • Stable training with RMSNorm
    The smoother GELU component avoids the gradient spikes of ReLU, and the gate keeps activations in a narrower range; together with pre-/post-layer RMSNorm this prevents divergence in very deep stacks (Gemma 2 uses up to 46 layers).

    • Proven track record
    GeGLU (and its cousin SwiGLU) improved T5-v1.1, PaLM-2, Llama-2/3 and other recent LLMs. Gemma 2 ablation studies (Section 5 of the paper) confirmed the same benefit, so the team adopted it as the default non-linearity.

In short, GeGLU offers a sweet spot of higher accuracy, lower parameter count and smoother optimization—ideal for Gemma 2’s goal of “state-of-the-art performance at practical size.”

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openai:o3-2025-04-16 via @hdevalence.bsky.social

Below is the same derivation with every symbol introduced explicitly, plus an explanation of the “mysterious’’ numbers in the GELU formula.

  1. Dimensions and symbols

• dmodeld_{\text{model}} – width of each Transformer layer (e.g. 23042304 in Gemma-2 2B).
• dffd_{\text{ff}} – hidden width of the feed-forward (MLP) sub-layer. Gemma 2 uses the standard “4× rule”, so dff≈4 dmodeld_{\text{ff}}\approx 4\,d_{\text{model}} (Table 1 of the paper).
• x∈R dmodelx \in \mathbb{R}^{\,d_{\text{model}}} – one token’s activations entering the MLP.
• W1,W2∈R dmodel×dff/2W_1,W_2 \in \mathbb{R}^{\,d_{\text{model}}\times d_{\text{ff}}/2} – weight matrices for two parallel linear projections.
• b1,b2∈R dff/2b_1,b_2 \in \mathbb{R}^{\,d_{\text{ff}}/2} – their biases.
• ⊙\odot – element-wise (Hadamard) product.

  1. GeGLU transformation
GeGLU⁡(x)=(xW1+b1)  ⊙  GELU⁡ ⁣(xW2+b2)(1)\operatorname{GeGLU}(x)=\bigl(xW_1+b_1\bigr)\;\odot\;\operatorname{GELU}\!\bigl(xW_2+b_2\bigr) \tag{1}

Because each branch outputs a vector of length dff/2d_{\text{ff}}/2, their product also lies in Rdff/2\mathbb{R}^{d_{\text{ff}}/2}. A final linear layer brings the dimension back to dmodeld_{\text{model}} (omitted in (1) for clarity).

  1. GELU activation with its constants

The Gaussian Error Linear Unit is defined exactly by an integral:

GELU⁡exact(z)=12 z(1+erf⁡(z/2)),(2)\operatorname{GELU}_{\text{exact}}(z)=\tfrac{1}{2}\,z\bigl(1+\operatorname{erf}(z/\sqrt{2})\bigr), \tag{2}

where erf⁡\operatorname{erf} is the error function. Computing erf⁡\operatorname{erf} is slow on many accelerators, so Hendrycks & Gimpel (2016) proposed the cheap tanh approximation

GELU⁡approx(z)=12 z(1+tanh⁡ ⁣(2/π (z+0.044715 z3))).(3)\operatorname{GELU}_{\text{approx}}(z)=\tfrac{1}{2}\,z\Bigl(1+\tanh\!\bigl(\sqrt{2/\pi}\,\bigl(z+0.044715\,z^{3}\bigr)\bigr)\Bigr). \tag{3}

• 2/π≈0.79788456\sqrt{2/\pi}\approx0.79788456 comes from matching the slope of erf⁡\operatorname{erf} at the origin.
• 0.0447150.044715 is the coefficient that minimises the maximum approximation error of (3) to (2) over z∈Rz\in\mathbb{R}; it was obtained by a small least-squares fit in the original paper.

Gemma 2 adopts (3) because it is differentiable, fast, and already the de-facto standard in many open-source frameworks.

  1. Why the gating matters for Gemma 2

• Capacity vs. cost With the gate, the MLP can model multiplicative interactions yet needs only dffd_{\text{ff}} parameters instead of 2dff2d_{\text{ff}} (as in the popular SwiGLU). That saves memory and FLOPs—important for 2 B–27 B-parameter “practical-size’’ models.
• Smoother gradients The Gaussian shape of GELU plus gating keeps activations bounded, reducing the risk of exploding gradients in Gemma 2’s 42–46-layer stacks.
• Empirical gains Internal ablations (Table 5 in the technical report) showed GeGLU beats plain GELU or ReLU by 1–3 % on validation perplexity at equal compute; hence it became the default non-linearity.

In summary, dffd_{\text{ff}} is simply the hidden width of the Transformer’s feed-forward block, and the numerical constants in GELU originate from a fast tanh approximation to the exact Gaussian CDF.

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