Sigmoid vs Softmax

Sigmoid and softmax are both functions that turn model scores into values that are easier to interpret, but they solve different kinds of classification problems.

Rule of thumb

Sigmoid = independent yes/no decisions.
Softmax = choose one class out of many.

Quick answer

Use sigmoid for binary classification or multi-label classification.
Use softmax for multi-class classification where the classes are mutually exclusive.

Intuition

Sigmoid takes one score and squashes it to a value between 0 and 1.
Softmax takes a vector of scores and converts it into probabilities that sum to 1.
With sigmoid, each output is treated independently.
With softmax, outputs are competing with each other.

Formulas

Sigmoid

\[\sigma(x) = \frac{1}{1 + e^{-x}}\]

Softmax

\[\text{softmax}(z_i) = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}\]

Visualization

Key differences

Aspect	Sigmoid	Softmax
Input	One score	A vector of scores
Output	One value in `(0, 1)`	A probability distribution
Sum to 1?	No	Yes
Relationship between outputs	Independent	Dependent and competing
Best for	Binary or multi-label classification	Multi-class single-label classification
Example question	"Is this email spam?"	"Is this image a cat, dog, or horse?"

Tiny example

Suppose a model produces these raw scores:

[2.0, 1.0, 0.1]

If you apply sigmoid to each score independently, you get:

[0.881, 0.731, 0.525]

These values do not sum to 1, because each class is evaluated separately.

If you apply softmax, you get:

[0.659, 0.242, 0.099]

These values do sum to 1, so they can be interpreted as a probability distribution across classes.

Why this matters

If one softmax score goes up, the others must go down.
With sigmoid, multiple outputs can be high at the same time.

When to use which

Use sigmoid when

There are only two classes.
Or you have a multi-label problem, where more than one label can be true at once.
Example: a photo can contain dog, car, and tree at the same time.

Use softmax when

You must pick exactly one class from many.
Example: classifying a digit as 0 to 9.

Interview-ready answers

Sigmoid is for independent outputs; softmax is for mutually exclusive outputs.
Sigmoid can be used for multi-label problems; softmax is used for multi-class single-label problems.
Softmax normalizes across classes, while sigmoid treats each class separately.
Sigmoid is commonly paired with binary cross-entropy; softmax with categorical cross-entropy.

Common mistakes

Using softmax for a multi-label problem.
Expecting sigmoid outputs across classes to sum to 1.
Forgetting that softmax depends on the relative size of all scores, not just one score by itself.

Simple Python

            import math

def sigmoid(x: float) -> float:
    return 1 / (1 + math.exp(-x))

def softmax(scores: list[float]) -> list[float]:
    max_score = max(scores)
    exp_scores = [math.exp(score - max_score) for score in scores]
    total = sum(exp_scores)
    return [score / total for score in exp_scores]

One-line memory trick

Sigmoid: "Can each label be true?"
Softmax: "Which one label is most likely?"