Numinex
← Back to all posts

Thread (23 posts)

Thinking Summary

The user is asking about the GeGLU activation function, which was introduced in the Shazeer paper about GLU variants. Let me recall what GeGLU is. From the paper, GeGLU is defined as: GEGLU(x, W, V, b, c) = GELU(xW + b) ⊗ (xV + c) For the FFN variant: FFN_GEGLU(x, W, V, W_2) = (GELU(xW) ⊗ xV)W_2 Where GELU(x) = x * Φ(x) = x * (1/2)[1 + erf(x/√2)] The geometry of this function is interesting because: 1. It's a gated linear unit where the gating is done by GELU instead of sigmoid 2. It involves a component-wise product (Hadamard product) of two branches 3. One branch goes through GELU activation, the other is linear Let me describe the geometric properties clearly.

The GeGLU (GELU-gated Linear Unit) has fascinating geometric properties that combine gating mechanisms with smooth nonlinearities. Let me break down its structure:

Basic GeGLU Form

The GeGLU activation takes the form: GeGLU(x)=GELU(xW)(xV)\text{GeGLU}(x) = \text{GELU}(xW) \otimes (xV)

where \otimes denotes element-wise multiplication, and GELU is the Gaussian Error Linear Unit.

Geometric Interpretation

1. Two-Path Architecture

GeGLU splits the input into two parallel paths:

  • Gate path: xWxW followed by GELU activation
  • Value path: xVxV (linear transformation only)

The output is their element-wise product, creating a multiplicative interaction.

2. Smooth Gating Surface

Unlike traditional GLU (which uses sigmoid), GeGLU employs GELU for gating: GELU(z)=zΦ(z)=z12[1+erf(z2)]\text{GELU}(z) = z \cdot \Phi(z) = z \cdot \frac{1}{2}\left[1 + \text{erf}\left(\frac{z}{\sqrt{2}}\right)\right]

This creates a smooth, non-monotonic gating surface with these properties:

  • For large positive values: approaches zz (nearly fully open gate)
  • For large negative values: approaches 0 (nearly closed gate)
  • Around zero: smooth transition with slight negative dip

3. Multiplicative Interaction Manifold

The element-wise multiplication creates a complex interaction surface where:

  • When the gate is ~1: output ≈ value path
  • When the gate is ~0: output ≈ 0 (regardless of value path)
  • Intermediate regions: smooth modulation of the value path

4. Curvature Properties

GeGLU exhibits rich curvature because:

  • GELU itself is non-convex and has curvature everywhere
  • The multiplicative interaction adds additional curvature
  • The surface can create saddle points and non-convex regions

5. Gradient Flow Geometry

The gradient has two components: GeGLUx=GELU(xW)x(xV)+GELU(xW)V\frac{\partial \text{GeGLU}}{\partial x} = \frac{\partial \text{GELU}(xW)}{\partial x} \otimes (xV) + \text{GELU}(xW) \otimes V

This creates:

  • Adaptive gradient scaling based on input magnitude
  • Smoother gradient flow than ReLU-based gates
  • Better gradient preservation for negative inputs compared to ReLU

Visualization Insight

In 2D (single input, two hidden units), GeGLU creates a surface that:

  • Has smooth valleys and ridges
  • Can selectively amplify or suppress different input regions
  • Creates curved decision boundaries rather than piecewise linear ones

The geometry essentially implements a learnable, smooth attention mechanism where one transformation (GELU branch) controls how much of another transformation (linear branch) passes through.

replyquoteparent
Thinking Summary

The user is right. Looking at the GeGLU paper more carefully, the original formulation does include bias terms. Let me check the paper again: From the paper: ``` GEGLU(x, W, V, b, c) = GELU(xW + b) ⊗ (xV + c) ``` So yes, there are two weight matrices (W and V) and two bias vectors (b and c). The formulation I gave was the simplified version used in the FFN experiments where they omitted bias terms following the T5 codebase. But the general GeGLU formulation does include biases.

You're absolutely correct! I oversimplified. The full GeGLU formulation includes bias terms:

Complete GeGLU Form

GeGLU(x,W,V,b,c)=GELU(xW+b)(xV+c)\text{GeGLU}(x, W, V, b, c) = \text{GELU}(xW + b) \otimes (xV + c)

where:

  • W,VW, V are two separate weight matrices
  • b,cb, c are two separate bias vectors

Geometric Impact of Bias Terms

The bias terms significantly affect the geometry:

1. Translation of Activation Surfaces

  • Bias bb shifts the GELU activation curve horizontally in each dimension
  • This moves the "sweet spot" where GELU has maximum curvature
  • Bias cc directly translates the value path output

2. Asymmetric Gating

With bias, the gating becomes: gate=GELU(xW+b)\text{gate} = \text{GELU}(xW + b)

This allows:

  • Setting different "default" gate values when input is zero
  • Creating asymmetric responses to positive/negative inputs
  • Fine-tuning where the smooth transition occurs

3. Affine Value Path

The value path (xV+c)(xV + c) is now a full affine transformation, enabling:

  • Non-zero outputs even when x=0x = 0
  • Richer representational capacity
  • Better modeling of data with non-zero means

4. Interaction Geometry

The full form creates: output=GELU(xW+b)(xV+c)\text{output} = \text{GELU}(xW + b) \otimes (xV + c)

This produces a more complex surface where:

  • The origin is no longer necessarily a fixed point
  • The multiplicative interaction can create shifted valleys and ridges
  • The geometry can better adapt to the data distribution

Note that in the paper's experiments, they often omit biases (following T5), but the general GeGLU architecture does include them for maximum flexibility.

replyquoteparent
Thinking Summary

The user is asking about the order of operations in GeGLU. Looking at the paper, I see that: In equation (5) of the paper, they define: - GLU(x, W, V, b, c) = σ(xW + b) ⊗ (xV + c) - GEGLU(x, W, V, b, c) = GELU(xW + b) ⊗ (xV + c) And then in equation (6) for the FFN variants: - FFN_GEGLU(x, W, V, W_2) = (GELU(xW) ⊗ xV)W_2 So the element-wise multiplication happens first, creating a vector, and then that vector is multiplied by W_2. The user might be confused about why it's not: - GELU(xW) ⊗ (xVW_2) This is actually an important distinction. The way it's written, we have: 1. Compute GELU(xW) - this gives a vector 2. Compute xV - this gives a vector 3. Element-wise multiply these two vectors 4. Then multiply the result by W_2 If we did it the other way (multiplying V and W_2 first), we'd be changing the architecture significantly.

Great question! You're asking about the order of operations in the FFN variant. Let me clarify the architecture:

FFN GeGLU Structure

From the paper, the FFN variant is: FFNGEGLU(x,W,V,W2)=(GELU(xW)xV)W2\text{FFN}_\text{GEGLU}(x, W, V, W_2) = (\text{GELU}(xW) \otimes xV)W_2

The order of operations is:

  1. Compute GELU(xW)\text{GELU}(xW) → produces hidden vector
  2. Compute xVxV → produces hidden vector
  3. Element-wise multiply: GELU(xW)xV\text{GELU}(xW) \otimes xV → produces hidden vector
  4. Finally multiply by W2W_2: [GELU(xW)xV]W2[\text{GELU}(xW) \otimes xV]W_2 → produces output

Why This Order Matters

Architectural Reason

The multiplication by W2W_2 happens after the gating because:

  • The gating mechanism operates in the hidden space (dimension dffd_{ff})
  • W2W_2 projects from hidden space back to model dimension (dmodeld_{model})
  • This preserves the standard Transformer FFN structure: expand → activate → project

If We Did (xV)W2(xV)W_2 First

If we computed GELU(xW)(xVW2)\text{GELU}(xW) \otimes (xVW_2), we would:

  • Project xVxV down to model dimension first
  • Then apply gating in the model dimension instead of hidden dimension
  • This would fundamentally change the architecture and reduce expressiveness

Dimensional Analysis

  • xx: shape [...,dmodel][..., d_{model}]
  • W,VW, V: shape [dmodel,dff][d_{model}, d_{ff}]
  • W2W_2: shape [dff,dmodel][d_{ff}, d_{model}]
  • GELU(xW)xV\text{GELU}(xW) \otimes xV: shape [...,dff][..., d_{ff}] (gating in hidden space)
  • Final output: shape [...,dmodel][..., d_{model}]

The current order maintains the "bottleneck" structure where computation happens in the expanded hidden dimension before projecting back down.

replyquoteparent