GeGLU paper
Thread (23 posts)
Can you describe the geometry of the GeGLU activation function?
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:
where 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: followed by GELU activation
- Value path: (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:
This creates a smooth, non-monotonic gating surface with these properties:
- For large positive values: approaches (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:
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.
Is that correct? I thought there were two weight matrices and two bias vector offset terms i.e., xv+b
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
where:
- are two separate weight matrices
- are two separate bias vectors
Geometric Impact of Bias Terms
The bias terms significantly affect the geometry:
1. Translation of Activation Surfaces
- Bias shifts the GELU activation curve horizontally in each dimension
- This moves the "sweet spot" where GELU has maximum curvature
- Bias directly translates the value path output
2. Asymmetric Gating
With bias, the gating becomes:
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 is now a full affine transformation, enabling:
- Non-zero outputs even when
- Richer representational capacity
- Better modeling of data with non-zero means
4. Interaction Geometry
The full form creates:
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.
Why is the multiplication on the left instead of on the right?
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:
The order of operations is:
- Compute → produces hidden vector
- Compute → produces hidden vector
- Element-wise multiply: → produces hidden vector
- Finally multiply by : → produces output
Why This Order Matters
Architectural Reason
The multiplication by happens after the gating because:
- The gating mechanism operates in the hidden space (dimension )
- projects from hidden space back to model dimension ()
- This preserves the standard Transformer FFN structure: expand → activate → project
If We Did First
If we computed , we would:
- Project 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
- : shape
- : shape
- : shape
- : shape (gating in hidden space)
- Final output: shape
The current order maintains the "bottleneck" structure where computation happens in the expanded hidden dimension before projecting back down.
In GELU (xw+b) \otimes (xv+c) why is the multiplication xW on the left rather than the normal vector in the right convention? please explain the whole problem context and all the diffeee t vector spaces and their dimensions
Thinking Summary
The user is asking about the notation conventions in the GeGLU formulation, specifically why the multiplication is written as xW rather than Wx. This is about understanding the different vector spaces, dimensions, and matrix multiplication conventions. Let me explain this thoroughly.
Excellent question! This touches on a fundamental notational convention difference in deep learning. Let me explain the full context.
Notational Conventions
Row Vector Convention (xW)
The paper uses the row vector convention where:
- Input is a row vector:
- Weight matrix :
- Output:
Column Vector Convention (Wx)
Traditional linear algebra often uses:
- Input is a column vector:
- Weight matrix :
- Output:
GeGLU Dimensional Analysis
Let's trace through GeGLU with row vectors:
Input Space
- (e.g., in the paper)
First Linear Transformations
Note: For GeGLU variants, they reduce to of the original to maintain parameter count.
Intermediate Computations
- Gate path:
- GELU applied element-wise:
- Value path:
- Element-wise product:
Final Projection
- Output:
Why Row Vectors?
1. Batch Processing
With row vectors, batching is natural:
- Batch input:
- Computation: works directly
- Each row is an independent example
2. Implementation Efficiency
Most deep learning frameworks (PyTorch, TensorFlow) default to:
# Natural batched operation
output = torch.matmul(input, weight) + bias
# Shape: [batch, d_in] @ [d_in, d_out] = [batch, d_out]
3. Transformer Convention
The Transformer paper and most implementations use row vectors:
- Easier to think about sequence dimension
- Input shape:
[batch, sequence, features]
Complete FFN_GeGLU Flow
Input: x ∈ ℝ^{1×d_model}
↓
┌───┴───┐
↓ ↓
xW+b xV+c (both ∈ ℝ^{1×d_ff})
↓ ↓
GELU() │
↓ ↓
└───⊗───┘ (element-wise multiply)
↓
(·)W₂ (project back to d_model)
↓
Output: ∈ ℝ^{1×d_model}
Dimensional Example
If and (reduced from 3072):
- :
- :
- :
- Intermediate:
- :
- Output:
The row vector convention is simply more convenient for modern deep learning implementations!