**Exponential Decay** is a learning rate schedule where we decay the learning rate with more iterations using an exponential function:

$$ \text{lr} = \text{lr}\_{0}\exp\left(-kt\right) $$

Image Credit: [Suki Lau](https://towardsdatascience.com/learning-rate-schedules-and-adaptive-learning-rate-methods-for-deep-learning-2c8f433990d1)

The **Gaussian Error Linear Unit**, or **GELU**,  is an activation function. The GELU activation function is $x\Phi(x)$, where $\Phi(x)$ the standard Gaussian cumulative distribution function. The GELU nonlinearity weights inputs by their percentile, rather than gates inputs by their sign as in [ReLUs](https://paperswithcode.com/method/relu) ($x\mathbf{1}_{x>0}$). Consequently the GELU can be thought of as a smoother ReLU.

$$\text{GELU}\left(x\right) = x{P}\left(X\leq{x}\right) = x\Phi\left(x\right) = x \cdot \frac{1}{2}\left[1 + \text{erf}(x/\sqrt{2})\right],$$
if $X\sim \mathcal{N}(0,1)$.

One can approximate the GELU with
$0.5x\left(1+\tanh\left[\sqrt{2/\pi}\left(x + 0.044715x^{3}\right)\right]\right)$ or $x\sigma\left(1.702x\right),$
but PyTorch's exact implementation is sufficiently fast such that these approximations may be unnecessary. (See also the [SiLU](https://paperswithcode.com/method/silu) $x\sigma(x)$ which was also coined in the paper that introduced the GELU.)

GELUs are used in [GPT-3](https://paperswithcode.com/method/gpt-3), [BERT](https://paperswithcode.com/method/bert), and most other Transformers.

GELU

Gaussian Error Linear Units (GELUs)

Exponential Decay

**Funnel Transformer** is a type of [Transformer](https://paperswithcode.com/methods/category/transformers) that gradually compresses the sequence of hidden states to a shorter one and hence reduces the computation cost. By re-investing the saved FLOPs from length reduction in constructing a deeper or wider model, the model capacity is further improved. In addition, to perform token-level predictions as required by common pretraining objectives, Funnel-[transformer](https://paperswithcode.com/method/transformer) is able to recover a deep representation for each token from the reduced hidden sequence via a decoder.

The proposed model keeps the same overall skeleton of interleaved S-[Attn](https://paperswithcode.com/method/scaled) and P-[FFN](https://paperswithcode.com/method/dense-connections) sub-modules wrapped by [residual connection](https://paperswithcode.com/method/residual-connection) and [layer normalization](https://paperswithcode.com/method/layer-normalization). But differently, to achieve representation compression and computation reduction, THE model employs an encoder that gradually reduces the sequence length of the hidden states as the layer gets deeper. In addition, for tasks involving per-token predictions like pretraining, a simple decoder is used to reconstruct a full sequence of token-level representations from the compressed encoder output. Compression is achieved via a pooling operation,

Year	2000
Data Source	CC BY-SA - https://paperswithcode.com

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What is: Exponential Decay?

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