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The vanishing gradient problem appears when the error signal shrinks as it backpropagates through many layers. The sigmoid and hyperbolic tangent functions have small derivatives, and their product tends toward zero, so the early layers barely learn. ReLU, careful initialisation and normalisation solve the problem.
Choosing an activation function is simple with one base rule: use ReLU in the hidden layers, GELU or SiLU in transformers, and reserve the output for softmax in multiclass classification, sigmoid in binary problems and a linear activation in regression. This comparison gathers formulas, ranges and use cases.
The Mish function is defined as mish(x) = x·tanh(softplus(x)). It is smooth, non-monotonic and self-regularized, properties that let it outperform Swish and ReLU across many tests. The YOLOv4 object detector adopted it in its backbone as the default activation.
The Softplus function is defined as softplus(x) = ln(1 + eˣ): a smooth, always-positive approximation of ReLU whose derivative is exactly the sigmoid. It is differentiable across its whole domain, avoids ReLU's angular kink and its output never quite reaches zero.
The SELU (Scaled Exponential Linear Unit) function is defined as SELU(x) equal to lambda times ELU(x), with lambda near 1.0507 and alpha near 1.6733. Those two constants make activations converge on their own towards zero mean and unit variance layer after layer, building deep networks that normalize themselves without batch normalization.
The ELU (Exponential Linear Unit) activation function returns x for positive inputs and α(eˣ−1) for negative ones. By allowing smooth negative values, it pushes the mean of the activations toward zero, speeds up convergence compared with ReLU, and avoids dead neurons thanks to a gradient that never vanishes completely on the negative side.
The Swish function, also called SiLU, multiplies the input by its sigmoid: swish(x) = x·σ(x). It is smooth, non-monotonic and self-gating, so it often beats ReLU in deep networks. Models like LLaMA and EfficientNet use it as their default activation.
The GELU (Gaussian Error Linear Unit) function multiplies each input by the probability that a standard normal falls below that input. The result is a smooth curve with a continuous derivative that weights inputs by their magnitude, and it has become the default activation inside BERT and GPT.
An activation function is the nonlinear operation each neuron applies to its weighted sum z to produce its output a equals f of z. Without it, stacking layers only chains linear transformations and the whole network collapses into a single one. That nonlinearity is what lets a network learn complex patterns.
The perceptron is the simplest artificial neuron: it takes several inputs, multiplies them by its weights, adds a bias and applies an activation function that decides between two outputs. Frank Rosenblatt introduced it in 1958, and it remains the basic building block of every modern neural network.
The hyperbolic tangent (tanh) is an activation function that squashes any real value into the interval (-1, 1) with zero-centred output, which removes the systematic gradient bias seen with sigmoid. It is the standard activation inside the LSTM and GRU memory cells used in recurrent networks.
The sigmoid function maps any real number to a value between 0 and 1, which makes it the natural activation function for expressing probabilities in a neural network. It is differentiable across its whole domain, though it suffers from saturation and vanishing gradients in deep networks, so today it is mostly reserved for the output layer.
The Softmax function transforms a neural network's output logits into a probability distribution that sums to 1. It is the standard activation for multi-class classification, from image classifiers to the vocabulary layer of language models like GPT.
Leaky ReLU is an activation function derived from ReLU that replaces the zero output for negative inputs with a small slope, typically 0.01. This keeps the gradient from ever reaching zero, which prevents the dying neuron problem and stabilizes training in very deep convolutional, recurrent, and GAN networks.
ReLU (Rectified Linear Unit, f(x) = max(0, x)) has been the dominant activation function in deep learning since AlexNet popularised it in 2012: cheap to compute, resistant to vanishing gradients, and with one well-known weakness, dying ReLU.
The step function, or Heaviside function, is the simplest activation function in a neural network: it maps any input to a binary output, 0 or 1, depending on whether it crosses a fixed threshold. It was the core decision mechanism of Rosenblatt's perceptron in 1958, but its derivative is zero almost everywhere, so modern networks use sigmoid or ReLU instead.
The linear function f(x) = ax + b is the simplest activation a neural network can use: in its identity form, f(x) = x, it is the standard choice for the output layer in regression, because it does not bound the range of possible values. In hidden layers it fails, because composing several linear functions collapses the whole network into one equivalent linear layer.
In a neural network, the input is represented as a column vector x in R^n that the hidden layer transforms through a weight matrix W, a bias vector b, and a non-linear activation function such as ReLU, sigmoid, or tanh. Training adjusts W and b by minimising the loss function via gradient descent and backpropagation.
