Which activation function is most susceptible to the vanishing gradient problem, and why?

The sigmoid activation function is most susceptible to the vanishing gradient problem. The sigmoid function outputs values between 0 and 1. During backpropagation, the gradients are multiplied together as they are passed backward through the layers of the neural network. The derivative of the sigmoid function has a maximum value of 0.25. When the input to a sigmoid function is very large or very small, its derivative approaches zero. If several layers in a network use sigmoid activations, the gradients can become increasingly small as they are backpropagated, eventually approaching zero. This means that the weights in the earlier layers receive very small updates during training, effectively preventing them from learning. This is known as the vanishing gradient problem. For example, consider a deep network with many sigmoid layers. During training, if a layer receives a gradient close to zero, the weights in that layer will barely change, and the layer will not learn effectively. This issue makes it difficult to train deep networks with sigmoid activations, leading to poor performance. Other activation functions, such as ReLU, which have a derivative of 1 for positive inputs, are less prone to the vanishing gradient problem because they do not squash gradients as severely as the sigmoid function.