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Looking at Section 2 of the GELU paper, here are the definitions of the three concepts mentioned:

ReLU (Rectified Linear Unit)

$\text{ReLU}(x) = \max(0, x) = x \cdot \mathbf{1}_{x>0}$

Where $\mathbf{1}_{x>0}$ is the indicator function that equals 1 when $x > 0$ 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 $p$ (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 $m \sim \text{Bernoulli}(\Phi(x))$ that depends on the input value $x$, where $\Phi(x)$ is the standard normal CDF. The final GELU is the expectation of this stochastic process: $\text{GELU}(x) = x\Phi(x)$.

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 $x$:

• ReLU: Multiplies $x$ by either 0 or 1, deterministically based on sign

  • If $x > 0$: output = $x \times 1 = x$
  • If $x \leq 0$: output = $x \times 0 = 0$

• Dropout: Multiplies $x$ by either 0 or 1, randomly with fixed probability

  • With probability $p$: output = $x \times 1$
  • With probability $1-p$: output = $x \times 0$

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 $x$ by a Bernoulli random variable $m$ where:

$m \sim \text{Bernoulli}(\Phi(x))$

where $\Phi(x) = P(Z \leq x)$ for $Z \sim \mathcal{N}(0,1)$ (the CDF of standard normal).

Why This Makes Sense

1. Larger inputs are more likely to "pass through":

   • If $x = 2$, then $\Phi(2) \approx 0.977$, so we keep the input ~97.7% of the time
   • If $x = -2$, then $\Phi(-2) \approx 0.023$, so we keep the input only ~2.3% of the time

2. Smooth transition around zero:

   • At $x = 0$: $\Phi(0) = 0.5$, so we have a 50-50 chance
   • This creates a smooth probabilistic gate rather than a hard threshold

3. 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:

$\mathbb{E}[x \cdot m] = x \cdot \mathbb{E}[m] = x \cdot P(m = 1) = x \cdot \Phi(x)$

This gives us GELU: $\text{GELU}(x) = x \Phi(x)$

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:

• $x = 2$: GELU$(2) = 2 \times 0.977 = 1.954$ (almost fully passes)
• $x = 0$: GELU$(0) = 0 \times 0.5 = 0$ (exactly zero, like ReLU)
• $x = -1$: GELU$(-1) = -1 \times 0.159 = -0.159$ (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.

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: $\text{GELU}(x) = x \cdot \Phi(x) = x \cdot \frac{1}{2}\left[1 + \text{erf}(x/\sqrt{2})\right]$

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: $\text{RMSNorm}(x) = \frac{x}{\sqrt{\frac{1}{d}\sum_{i=1}^{d} x_i^2}} \cdot \gamma$

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

1. 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.

2. 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.

3. 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.

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.

Geometry of a single-neuron GeGLU
———————————————————————
(Write $x\!=\!(x_g,\,x_v)$, with $x_g$ the gate component and $x_v$ the value component that will ultimately flow forward.)

Scalar form:
 $f_{\text{GeGLU}}(x_g, x_v)=\text{GELU}(x_g)\;\cdot\;x_v   =x_g\,\Phi(x_g)\;x_v$

1. Domain & codomain
   • Domain: the 2-D plane $(x_g,x_v)\in\mathbb R^2$.
   • Codomain: the real line $\mathbb R$ (for one neuron) or, in the vector case, the $d_{ff}/2$-dimensional subspace spanned by the “value” projection.
   Geometrically we are mapping each point of the plane to a signed height above/below it.

2. Separating axes
   • The $x_g$-axis controls a smooth gate, ranging continuously from 0 (far negative) through 0.5 (at the origin) toward 1 (large positive).
   • The $x_v$-axis is passed through unchanged and merely gets scaled by that gate.
   Thus the plane is foliated into vertical lines (constant $x_g$) that are uniformly compressed or expanded.

3. Surface shape
   Plotting $z=f_{\text{GeGLU}}(x_g,x_v)$ gives a saddle-like surface that is:

   • Flat along the entire $x_g$ axis at $x_v\!=\!0$ (because the value is zero).
   • As $|x_v|$ increases the surface “fans out’’ in the $\pm x_v$ direction, but its amplitude is modulated by the bell-shaped factor $x_g\Phi(x_g)$ (Fig. below).

   For fixed $x_v$ the cross-section is exactly the 1-D GELU curve scaled by $x_v$; for fixed $x_g$ it is a straight line through the origin with slope $\text{GELU}(x_g)$.

4. Level sets (iso-contours)
   Solving $f(x_g,x_v)=c$ gives
   $x_v = c / \text{GELU}(x_g)$ whenever $\text{GELU}(x_g)\neq0$.
   These are rational curves that resemble hyperbolas but whose arms pinch smoothly toward the $x_g<0$ region where the gate shrinks to zero. No contour ever crosses the negative $x_g$ axis because the output there is arbitrarily close to 0.

5. Gradient field
   $\nabla f = \bigl(x_v\,\text{GELU}'(x_g),\; \text{GELU}(x_g)\bigr)$
   • Along the $x_v$ direction gradients are constant w.r.t. $x_v$: each vertical line has the same slope equal to $\text{GELU}(x_g)$.
   • Gradients vanish as $x_g\!\to\!-\infty$ (saturation), peak around $x_g\!\approx\!1$, and decay again for very large $x_g$ due to the $x_g\Phi(x_g)$ factor plateauing at $x_g$.

Vectorised GeGLU inside a Transformer FFN
——————————————————————————————
Let the input vector $h\in\mathbb R^{d}$ be linearly projected into two halves:

 $g = hW_g\in\mathbb R^{d_{ff}/2}$ (gate)
 $v = hW_v\in\mathbb R^{d_{ff}/2}$ (value)

and produce $y = \text{GELU}(g)\ \otimes\ v$. (Hadamard product.)

Geometrically:

1. The value subspace spanned by columns of $W_v$ forms a $(d_{ff}/2)$-dimensional affine subspace of the model’s representation space.
2. The gate vector $\text{GELU}(g)\in[0,\infty)^{d_{ff}/2}$ 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 $g$ are.
3. Consequently, the reachable set of $y$ is a warped, axis-aligned “cone’’ inside that subspace whose opening width varies smoothly with $g$. 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 $x_g=0$.
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 $(g,v)$ 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.