GeGLU paper
Thread (23 posts)
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!