Activation functions play a crucial role in introducing non-linearities to deep neural networks. We propose a novel approach to designing activation functions by focusing on their gradients and deriving the corresponding functions through integration. Our work introduces the Expanded Integral of the Exponential Linear Unit (xIELU), a trainable piecewise activation function derived by integrating trainable affine transformations applied on the ELU activation function. xIELU combines two key gradient properties: a trainable and linearly increasing gradient for positive inputs, similar to ReLU$^2$, and a trainable negative gradient flow for negative inputs, akin to xSiLU. Conceptually, xIELU can be viewed as extending ReLU$^2$ to effectively handle negative inputs. In experiments with 1.1B parameter Llama models trained on 126B tokens of FineWeb Edu, xIELU achieves lower perplexity compared to both ReLU$^2$ and SwiGLU when matched for the same compute cost and parameter count.

The scaling of neural networks with increasing data and model sizes necessitates more efficient deep learning algorithms. This paper addresses the memory footprint challenge in neural network training by proposing a modification to the handling of activation tensors in pointwise nonlinearity layers. Traditionally, these layers save the entire input tensor for the backward pass, leading to substantial memory use. Our method involves saving the output tensor instead, reducing the memory required when the subsequent layer also saves its input tensor. This approach is particularly beneficial for transformer-based architectures like GPT, BERT, Mistral, and Llama. Application of our method involves taken an inverse function of nonlinearity. To the best of our knowledge, that can not be done analitically and instead we buid an accurate approximations using simpler functions. Experimental results confirm that our method significantly reduces memory usage without affecting training accuracy. The implementation is available at https://github.com/PgLoLo/optiacts.

Activation functions are core components of all deep learning architectures. Currently, the most popular activation functions are smooth ReLU variants like GELU and SiLU. These are self-gated activation functions where the range of the gating function is between zero and one. In this paper, we explore the viability of using arctan as a gating mechanism. A self-gated activation function that uses arctan as its gating function has a monotonically increasing first derivative. To make this activation function competitive, it is necessary to introduce a trainable parameter for every MLP block to expand the range of the gating function beyond zero and one. We find that this technique also improves existing self-gated activation functions. We conduct an empirical evaluation of Expanded ArcTan Linear Unit (xATLU), Expanded GELU (xGELU), and Expanded SiLU (xSiLU) and show that they outperform existing activation functions within a transformer architecture. Additionally, expanded gating ranges show promising results in improving first-order Gated Linear Units (GLU).