An activation function transforms a layer’s output after its weighted sum and bias are computed. In hidden layers, that transformation lets a neural network build more expressive mappings; at the output, it can turn scores into probabilities. Its shape also affects how gradients flow during learning, so choosing an activation is tied to both the layer’s role and its loss function.
What an activation function does
A typical neural-network layer first computes an affine transformation: it combines its inputs with learned weights and adds a bias. The activation function is then applied to that result, commonly element by element in hidden layers. If the affine result is called z, an activation produces g(z).
Without a nonlinear activation between layers, stacking affine transformations still produces an affine transformation overall. Nonlinear activations let a deep network represent more varied relationships. During training, backpropagation also accounts for the activation’s derivative, which influences how much gradient passes through each unit.
How ReLU, sigmoid, and tanh differ
| Function | Definition or output | Typical role and gradient behavior |
|---|---|---|
| ReLU | g(z) = max(0, z) |
A common choice for hidden units. It outputs zero for negative inputs and the input itself for positive inputs. |
| Sigmoid | Maps a real-valued input to a value between 0 and 1. | Useful for representing a binary probability at an output. It saturates at extreme inputs, where its derivative becomes small. |
| Tanh | Maps a real-valued input to a value between -1 and 1. | Centered at zero and more identity-like near zero than logistic sigmoid; it can also saturate at extreme inputs. |
Sigmoid and tanh were widely used in earlier neural networks. Their saturation matters because a small derivative can make gradients too small to support effective learning through those units. ReLU avoids that particular saturation behavior for positive inputs, though the function is still flat for negative inputs.
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When to use sigmoid or softmax at the output
Binary probability output: sigmoid
For a binary prediction, sigmoid can convert a single score into a value interpretable as a probability. Pair it with an appropriate likelihood-based loss. The output activation and objective work together: a suitable likelihood loss can avoid some of the saturation problems that arise from less suitable loss choices.
Multiple discrete classes: softmax
For one choice among multiple discrete classes, softmax converts a vector of scores into values that sum to one, giving a probability distribution across the classes. It exponentiates the scores and normalizes them by their sum. As with sigmoid, choose the loss together with the output interpretation rather than selecting an activation in isolation.
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Why softmax should be computed stably
Directly exponentiating large scores can cause numerical overflow. Softmax is unchanged if the same constant is subtracted from every score, so a stable computation subtracts the largest score first:
softmax(zᵢ) = exp(zᵢ - m) / Σⱼ exp(zⱼ - m), where m = maxⱼ zⱼ.
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This shifts the largest exponent’s input to zero without changing the resulting probabilities, reducing the risk of overflow.
How to choose an activation
- Identify the layer’s job. For hidden transformations, ReLU is a common starting point. For a binary probability output, sigmoid is appropriate; for a distribution over multiple discrete classes, use softmax.
- Check the output range and interpretation. ReLU produces nonnegative values, sigmoid produces values between zero and one, and tanh produces values between negative one and one. Use a probability interpretation only when the output activation and training objective support it.
- Consider gradient behavior. Sigmoid and tanh can saturate at extreme inputs, reducing their derivatives. Tanh is zero-centered and more nearly linear around zero than sigmoid.
- Match the objective to the output. For probabilistic outputs, use a compatible likelihood-based loss rather than treating the activation as an independent choice.
- Use a stable implementation. For softmax, subtract the maximum score before exponentiation, or use a framework implementation that applies an equivalent stable calculation.
Further reading
The chapter “Deep Feedforward Networks” in Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville provides a foundational treatment of activation functions and gradient-based learning.
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