What activation function combines properties of the ReLU and sigmoid, providing a smooth and non-linear transformation?

The activation function that combines properties of the ReLU and sigmoid while providing a smooth and non-linear transformation is the GeLU (Gaussian Error Linear Unit). GeLU is particularly advantageous because it introduces stochasticity and smoothness into the activation process, drawing from the principles of the sigmoid function by incorporating probabilistic elements, while also benefiting from the speed of ReLU in terms of computation and gradient propagation.

This function is defined in such a way that it leverages Gaussian distribution characteristics, which allows it to smoothly transition between linear and non-linear transformations. Consequently, GeLU can retain the advantages of both ReLU's robust performance in deep networks and sigmoid's ability to handle non-linear relationships without saturating too quickly. This property makes GeLU highly effective in various neural network architectures, especially in transformer models.

Other options do not exhibit this hybrid between smooth non-linearity and computational efficiency. For example, while ReLU is known for being simple and effective, it lacks the smooth aspects needed for a combined approach. The sigmoid function, on the other hand, suffers from saturation issues that can impede learning in deep networks. ELU (Exponential Linear Unit) is an improvement over ReLU but doesn't integrate the probabilistic elements characteristic of the GeLU.