GeGLU paper
Thread (23 posts)
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Can you fill in "dropout", "zoneout", and " ReLU from §2? I know ReLU But just to have defns in one place
Looking at Section 2 of the GELU paper, here are the definitions of the three concepts mentioned:
ReLU (Rectified Linear Unit)
Where is the indicator function that equals 1 when and 0 otherwise. ReLU deterministically multiplies the input by zero or one based on the input's sign.
Dropout
A stochastic regularization technique that randomly sets some neuron outputs to zero during training. It stochastically multiplies inputs by zero with some probability (the dropout rate). This creates a "pseudoensemble" effect by randomly altering activation decisions through zero multiplication.
Zoneout
A regularization technique for RNNs that stochastically multiplies inputs by one. Unlike dropout which zeros out activations, zoneout preserves the previous hidden state with some probability, effectively keeping the activation unchanged.
The key insight from the GELU paper is that these three techniques all involve multiplying inputs by binary values (0 or 1), but they differ in how they make this decision:
- ReLU: Deterministic based on input sign
- Dropout: Stochastic, input-independent
- Zoneout: Stochastic, input-independent (preserves state)
GELU combines these ideas by using a stochastic mask that depends on the input value , where is the standard normal CDF. The final GELU is the expectation of this stochastic process: .
Do people still use dropout and zone out or is that from a previous era
Dropout is still widely used today, though its application has evolved:
- Still common in: Fully connected layers, computer vision models, and many practical deep learning applications
- Less common in: Modern large language models (LLMs) and transformers, where other regularization techniques are preferred
- Modern variants: DropPath/Stochastic Depth, DropBlock for CNNs, and attention dropout in transformers
Zoneout is much less common now:
- It was primarily designed for RNNs (LSTMs/GRUs)
- Since RNNs have largely been replaced by transformers for most sequence modeling tasks, zoneout usage has declined significantly
- Most modern sequence models use attention mechanisms rather than recurrent architectures
Current regularization landscape:
- Layer normalization and batch normalization are more fundamental
- Weight decay (L2 regularization) remains standard
- Gradient clipping is common for training stability
- Early stopping and learning rate scheduling
- Data augmentation techniques
- Attention dropout in transformers
The GELU paper (2016) was written during the transition period when RNNs were still dominant but transformers were emerging. Today's models rely more on architectural innovations (like attention) and large-scale training rather than traditional stochastic regularization techniques.
So dropout persists but is less central, while zoneout has largely faded with the decline of RNNs.
Can you give a motivating tutorial on the intuition behind the Bernoulli multiplication used to derive GeLU
Thinking Summary
The user is asking for a motivating tutorial on the intuition behind the Bernoulli multiplication used to derive GELU. From the GELU paper embedded above, I can see that GELU is derived from the idea of multiplying the input by a Bernoulli random variable where the probability depends on the input itself. Let me break this down step by step: 1. The key insight is combining properties from dropout, zoneout, and ReLUs 2. ReLU deterministically multiplies input by 0 or 1 based on sign 3. Dropout stochastically multiplies by 0 or 1 randomly 4. GELU makes this stochastic multiplication depend on the input value 5. Specifically, multiply x by m ~ Bernoulli(Φ(x)) where Φ is the CDF of standard normal 6. The expected value of this gives us GELU: E[x·m] = x·Φ(x) I should explain this intuitively and pedagogically.
The Intuition Behind GELU's Bernoulli Multiplication
Let me walk you through the elegant probabilistic thinking that leads to GELU!
Starting Point: What Do Existing Activations Do?
Think about what different neural network components do to an input :
-
ReLU: Multiplies by either 0 or 1, deterministically based on sign
- If : output =
- If : output =
-
Dropout: Multiplies by either 0 or 1, randomly with fixed probability
- With probability : output =
- With probability : output =
The Key Insight: Input-Dependent Stochastic Gating
What if we combined these ideas? Instead of:
- Always dropping negative values (ReLU)
- Randomly dropping values with fixed probability (dropout)
We could randomly drop values with probability that depends on the input itself.
The Bernoulli Multiplication Model
Here's the brilliant move: multiply by a Bernoulli random variable where:
where for (the CDF of standard normal).
Why This Makes Sense
-
Larger inputs are more likely to "pass through":
- If , then , so we keep the input ~97.7% of the time
- If , then , so we keep the input only ~2.3% of the time
-
Smooth transition around zero:
- At : , so we have a 50-50 chance
- This creates a smooth probabilistic gate rather than a hard threshold
-
Natural choice of distribution:
- Neural network activations often follow roughly normal distributions (especially with batch normalization)
- Using the normal CDF means we're gating based on "how many standard deviations above average" an input is
From Stochastic to Deterministic: Taking the Expectation
During inference, we want deterministic outputs. So we take the expected value:
This gives us GELU:
Intuitive Interpretation
GELU can be thought of as:
- "Weighted gating": Unlike ReLU which does hard gating based on sign, GELU weights inputs by how much greater they are than other inputs
- "Soft selection": Inputs are scaled by their percentile in the standard normal distribution
- "Probabilistic ReLU": It's like a ReLU where the gating decision is made probabilistically based on the input's magnitude
Practical Example
Consider three inputs:
- : GELU (almost fully passes)
- : GELU (exactly zero, like ReLU)
- : GELU (mostly blocked but some negative signal passes)
This creates a smooth, S-shaped activation that can output negative values (unlike ReLU) but still provides the gating behavior that makes ReLU effective.
What are the implications of passing negative signal through close to zero but not at more negative values?
Thinking Summary
This is a great question about the behavior of GELU for negative inputs. Let me think about the implications of GELU allowing some negative signal through near zero but less as values become more negative. Key points to consider: 1. GELU behavior for negative values 2. Gradient flow implications 3. Comparison with ReLU and ELU 4. Impact on learning dynamics 5. Biological/theoretical motivations
The Implications of GELU's Negative Signal Behavior
This is a really insightful observation! GELU's behavior for negative inputs—allowing some signal through near zero but progressively less as values become more negative—has several important implications.
1. Gradient Flow and the "Dying ReLU" Problem
ReLU's weakness: When a neuron outputs negative values, ReLU completely blocks gradients (derivative = 0). If a neuron gets stuck outputting negative values, it can never recover—the infamous "dying ReLU" problem.
GELU's advantage: For slightly negative inputs (say ), GELU still has non-zero gradients:
- At : GELU with derivative
- At : GELU with derivative
This means neurons can still learn and adjust even when outputting negative values!
2. Information Preservation Near Decision Boundaries
Consider what happens near :
- ReLU: Sharp cutoff—all negative information is lost
- GELU: Gradual filtering—slightly negative signals are attenuated but not eliminated
This is particularly important for:
- Fine distinctions: When the network needs to distinguish between -0.1 and -0.5, GELU preserves this information (outputs ≈ -0.046 vs ≈ -0.154)
- Uncertainty representation: Values near zero might represent uncertain or borderline cases where preserving some signal is valuable
3. Selective Noise Filtering
The progressive attenuation creates an interesting filtering effect:
- Slightly negative (): ~30% of signal passes → useful information preserved
- Moderately negative (): ~2% passes → mostly filtered out
- Very negative (): essentially zero → strong noise rejection
This acts like an adaptive filter that:
- Preserves potentially useful weak negative signals
- Strongly suppresses large negative activations (likely noise or irrelevant features)
4. Smooth Optimization Landscape
The smooth transition creates better optimization properties:
ReLU derivative: { 1 if x > 0, 0 if x ≤ 0 } → discontinuous
GELU derivative: smoothly varies from 0 to 1 → continuous everywhere
Near , GELU's derivative is approximately 0.5, providing consistent gradient flow rather than the abrupt change in ReLU.
5. Biological Plausibility
This behavior might better model biological neurons:
- Real neurons don't have hard thresholds—they show graded responses
- Slightly sub-threshold inputs can still influence the neuron's state
- Strong inhibition (very negative inputs) effectively silences the neuron
6. Impact on Feature Learning
The selective negative signal passing affects what features the network learns:
Early layers:
- Can learn features that include "slight absence" or "weak opposition"
- Example: In image processing, slight darkening vs. strong shadows
Deep layers:
- Allows for more nuanced representations
- Can represent "not quite X but almost" differently from "definitely not X"
7. Mathematical Properties
The tapering negative response creates interesting dynamics:
- Non-monotonic behavior: GELU has a local minimum around
- Bounded negative output: Unlike ELU which can output arbitrarily negative values, GELU's negative outputs are bounded
Practical Example: Sentiment Analysis
Consider neurons detecting sentiment:
- : Strong positive sentiment → output ≈ 1.95
- : Slightly negative sentiment → output ≈ -0.11 (some signal preserved)
- : Strong negative sentiment → output ≈ -0.004 (essentially blocked)
This allows the network to distinguish between "mildly disappointed" and "extremely angry" in ways ReLU cannot.
The Trade-off
This design represents a careful balance:
- Too much negative signal: Could propagate noise and make training unstable
- No negative signal (ReLU): Loses information and causes gradient death
- GELU's approach: Selectively preserves information based on magnitude
This makes GELU particularly effective in practice, as evidenced by its adoption in transformers like BERT and GPT!
How does the scaling choice play with other model architecture choices such as RMSNorm
Thinking Summary
The user is asking about the GeGLU paper, which is a paper about GLU (Gated Linear Unit) variants in Transformers. They've embedded the paper's content, and there's also a follow-up that cites the definition of GELU from the GELU paper. The last message asks about how the scaling choice σ=1 plays with other model architecture choices such as RMSNorm. Let me analyze what's being asked: 1. The GeGLU paper introduces several GLU variants including GEGLU (Gaussian Error GLU) 2. The GELU paper defines GELU as x * Φ(x) where Φ is the CDF of a standard normal distribution (μ=0, σ=1) 3. The question is about how the choice of σ=1 interacts with architectural choices like RMSNorm This is an interesting technical question about the interplay between activation functions and normalization layers. Let me think about this: - GELU uses σ=1 as a fixed choice (standard normal CDF) - RMSNorm normalizes activations by their root mean square - The question is about how these interact The key insight is that normalization layers like RMSNorm will affect the scale of inputs to the activation function, which could potentially make the fixed σ=1 choice suboptimal in some cases.
The choice of σ=1 in GELU (and consequently in GEGLU) creates an interesting interaction with normalization layers like RMSNorm. Let me break this down:
Scale Sensitivity of GELU
The GELU activation function is defined as:
With σ=1 fixed, the "transition region" where GELU exhibits its characteristic smooth gating behavior occurs roughly in the range [-2, 2]. Outside this range, GELU approximates either 0 (for very negative inputs) or the identity function (for very positive inputs).
Interaction with RMSNorm
RMSNorm normalizes activations by:
Where γ is a learned scale parameter. This normalization ensures that the RMS of the activations is controlled, but the learned γ parameter allows the model to adjust the overall scale.
Key Considerations
-
Effective Temperature: The fixed σ=1 in GELU essentially sets a fixed "temperature" for the activation. With RMSNorm, the learned γ parameter can partially compensate by scaling inputs to match the optimal range for GELU's nonlinearity.
-
Distribution Assumptions: The GELU paper notes that the choice of standard normal (μ=0, σ=1) is motivated by the observation that "neuron inputs tend to follow a normal distribution, especially with Batch Normalization." RMSNorm doesn't guarantee normality but does control scale, which may make the σ=1 assumption less optimal.
-
Depth-Dependent Effects: In deep networks, even with normalization, the effective scale of activations can drift. The fixed σ=1 means GELU can't adapt to these variations, potentially leading to either saturated (nearly linear) or collapsed (nearly zero) regimes in different parts of the network.
Potential Improvements
The GELU paper briefly mentions using learnable μ and σ as hyperparameters:
"We could use the CDF of N(μ, σ²) and have μ and σ be learnable hyperparameters, but throughout this work we simply let μ = 0 and σ = 1."
This suggests that making σ learnable (perhaps per-layer or per-channel) could allow better adaptation to the scale induced by RMSNorm, potentially improving the effectiveness of GEGLU variants in modern architectures.
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!
Can you describe the geometry of the GeGLU activation function?
Thinking Summary
**Describing GeGLU Geometry** I'm focusing on the geometry of the GeGLU activation function. It involves the product of a GELU-transformed gate and a linear value. In a scalar case, I can express it as f(a, b) = GELU(a) * b, which helps visualize it in two dimensions. In terms of gradients, the gating creates an oblique hyperplane, smoothly transitioning between 0 and 1. I think it’s helpful to consider this piecewise and note how the GEGLU behaves like a cubic on the negative side near zero, resulting in a geometric shape that's not entirely symmetrical. **Defining GeGLU Geometry** I'm planning to provide a clear definition and illustrate the shape in 2D, focusing on the gating dimension versus the scaling dimension. I'll also discuss the gradient vector field and explain the iso-lines, which resemble radial "sausages" oriented along the value dimension. Additionally, I want to highlight the differentiation between the gating axis, referred to as the "gate dimension," and the "value dimension." It's important to deliver this in an organized way that clearly conveys the concepts.
Geometry of a single-neuron GeGLU
———————————————————————
(Write , with the gate component and the value component that will ultimately flow forward.)
Scalar form:
-
Domain & codomain
• Domain: the 2-D plane .
• Codomain: the real line (for one neuron) or, in the vector case, the -dimensional subspace spanned by the “value” projection.
Geometrically we are mapping each point of the plane to a signed height above/below it. -
Separating axes
• The -axis controls a smooth gate, ranging continuously from 0 (far negative) through 0.5 (at the origin) toward 1 (large positive).
• The -axis is passed through unchanged and merely gets scaled by that gate.
Thus the plane is foliated into vertical lines (constant ) that are uniformly compressed or expanded. -
Surface shape
Plotting gives a saddle-like surface that is:• Flat along the entire axis at (because the value is zero).
• As increases the surface “fans out’’ in the direction, but its amplitude is modulated by the bell-shaped factor (Fig. below).For fixed the cross-section is exactly the 1-D GELU curve scaled by ; for fixed it is a straight line through the origin with slope .
-
Level sets (iso-contours)
Solving gives
whenever .
These are rational curves that resemble hyperbolas but whose arms pinch smoothly toward the region where the gate shrinks to zero. No contour ever crosses the negative axis because the output there is arbitrarily close to 0. -
Gradient field
• Along the direction gradients are constant w.r.t. : each vertical line has the same slope equal to .
• Gradients vanish as (saturation), peak around , and decay again for very large due to the factor plateauing at .
Vectorised GeGLU inside a Transformer FFN
——————————————————————————————
Let the input vector be linearly projected into two halves:
(gate)
(value)
and produce . (Hadamard product.)
Geometrically:
- The value subspace spanned by columns of forms a -dimensional affine subspace of the model’s representation space.
- The gate vector acts as a smooth, element-wise scaling mask that compresses coordinates of that subspace toward the origin depending on how “positive’’ the companion coordinates in are.
- Consequently, the reachable set of is a warped, axis-aligned “cone’’ inside that subspace whose opening width varies smoothly with . No new directions are introduced; directions are re-weighted.
Comparisons & intuition
——————————————
ReLU gate: sharp half-space clipping, output lies on two flat half-cones separated by a hard plane .
GELU gate: replaces that plane by a sigmoidal transition band, bending the cone walls smoothly. The resulting surface is everywhere differentiable, giving GeGLU its smoother optimization landscape while still preserving a gating geometry (outputs shrink continuously to zero as any gate coordinate drifts negative).
Bottom line
—————
GeGLU carves the plane into smoothly-gated linear pencils: each “value’’ axis is a straight line whose amplitude envelope is the soft bell curve of the GELU applied to its paired “gate’’ axis. In higher dimension, this becomes an axis-aligned, differentiable gating cone embedded in the FFN’s hidden space.