Is GELU, the ReLU successor ?

Author(s): Poulinakis Kon

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Is GELU the ReLU Successor?

Can we combine regularization and activation functions? In 2016 a paper from authors Dan Hendrycks and Kevin Gimpel came out. Since then, the paper now has been updated 4 times. The authors introduced a new activation function, the Gaussian Error Linear Unit, GELU.

Demystifying GELU

The motivation behind GELU is to bridge stochastic regularizers, such as dropout, with non-linearities, i.e., activation functions.

Dropout regularization stochastically multiplies a neuron’s inputs with 0, randomly rendering them inactive. On the other hand, ReLU activation deterministically multiplies inputs with 0 or 1 dependent upon the input’s value.

GELU merges both functionalities by multiplying inputs by a value from 0 to 1. However, the value of this zero-one mask, while stochastically determined, is also dependent upon the input’s value.

Mathematically, GELU is formulated as :

Φ(x) is the cumulative distribution function (CDF) of the standard normal distribution. The choice of this function stems from the fact that neuron inputs tend to follow a normal distribution, especially when Batch Normalization is used. So, essentially GELU has a higher probability of dropping a neuron (multiplying by 0) while x decreases since P(X ≤ x) becomes smaller. Please take a moment to think about this and let it sink. So the transformation applied by GELU is stochastic, yet it depends upon the input’s value through Φ(x).

Observe how GELU(x) starts from zero for small values of x since the CDF P(X≤x) is almost equal to 0. However, around the value of -2, P(X≤x) starts increasing. Hence we see GELU(x) deviating from zero. For the positive values, since P(X≤x) moves closer to a value of 1, GELU(x) starts approximating ReLU(x). In the figure below, the red line represents the CDF of the Standard Normal Distribution N(0,1) i.e., P(X≤x).

Approximations

GELU can also be approximated through the formulas

if greater feedforward speed is worth the cost of exactness.

Variations

The GELU can also be modified by using different CDFs. For example, if the Logistic Distribution CDF (x) is used, then we would get the Sigmoid Linear Unit (SiLU) x(x). Moreover, we could pick a CDF N(μ, σ) with μ and σ being learnable hyperparameters.

Advantages

The authors in [1], experimented with the use of GELU against ReLU and ELU activation functions in 3 different benchmark datasets covering the tasks of computer vision (CIFAR 10/100 classification), natural language processing (Twitter part of speech tagging), and audio phoneme recognition (TIMIT frame classification).

Throughout their experiments, they observed a consistent improvement in accuracy when using GELU compared to ReLU, and ELU. Analytically :

The table above presents the test error rate in 4 datasets. GELU consistently achieves the lowest test error rate, posing as a promising alternative to ReLU and ELU activations.

An Interesting Fact

The well-known paper “An Image is Worth 16×16 Words: Transformers for Image Recognition at Scale” that made Vision Transformers popular makes use of GELU activation inside the MLP of the encoder transformer block (section 3.1). This suggests that GELU is considered a good option by high-quality researchers.

REFERENCES

[1] Gaussian Error Linear Units (GELUs)

[2] https://en.wikipedia.org/wiki/Normal_distribution

[3] An Image is Worth 16×16 Words: Transformers for Image Recognition at Scale

Thanks for reading, feel free to reach out!

My Links: Medium | LinkedIn | GitHub

Is GELU, the ReLU successor ? was originally published in Towards AI on Medium, where people are continuing the conversation by highlighting and responding to this story.

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