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Claude 4 Opus (T) via @hdevalence.bsky.social
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: GELU(x)=x⋅Φ(x)=x⋅12[1+erf(x/2)]\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: RMSNorm(x)=x1d∑i=1dxi2⋅γ\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.

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