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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 $x$ :

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

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
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
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.

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@hdevalence.bsky.social

2025-07-03T21:28:09.564Z

What are the implications of passing negative signal through close to zero but not at more negative values?

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Claude 4 Opus (T) via @hdevalence.bsky.social

2025-07-03T21:28:47.000Z

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 $x \in [-2, 0]$ ), GELU still has non-zero gradients:

At $x = -1$ : GELU $(-1) \approx -0.159$ with derivative $\approx 0.08$
At $x = -0.5$ : GELU $(-0.5) \approx -0.154$ with derivative $\approx 0.19$

This means neurons can still learn and adjust even when outputting negative values!

2. Information Preservation Near Decision Boundaries

Consider what happens near $x = 0$ :

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 ( $x \approx -0.5$ ): ~30% of signal passes → useful information preserved
Moderately negative ( $x \approx -2$ ): ~2% passes → mostly filtered out
Very negative ( $x < -3$ ): 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 $x = 0$ , 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 $x \approx -0.72$
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:

$x = 2$ : Strong positive sentiment → output ≈ 1.95
$x = -0.3$ : Slightly negative sentiment → output ≈ -0.11 (some signal preserved)
$x = -3$ : 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!

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