Thread (23 posts)

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: $\text{GeGLU}(x) = \text{GELU}(xW) \otimes (xV)$

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: $xW$ followed by GELU activation
• Value path: $xV$ (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: $\text{GELU}(z) = z \cdot \Phi(z) = z \cdot \frac{1}{2}\left[1 + \text{erf}\left(\frac{z}{\sqrt{2}}\right)\right]$

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

• For large positive values: approaches $z$ (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: $\frac{\partial \text{GeGLU}}{\partial x} = \frac{\partial \text{GELU}(xW)}{\partial x} \otimes (xV) + \text{GELU}(xW) \otimes V$

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.

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

Complete GeGLU Form

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

where:

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

Geometric Impact of Bias Terms

The bias terms significantly affect the geometry:

1. Translation of Activation Surfaces

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

2. Asymmetric Gating

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

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 $(xV + c)$ is now a full affine transformation, enabling:

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

4. Interaction Geometry

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

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.