64
$\begingroup$

What decides the choice of function ( Softmax vs Sigmoid ) in a Logistic classifier ?

Suppose there are 4 output classes . Each of the above function gives the probabilities of each class being the correct output . So which one to take for a classifier ?

$\endgroup$
  • 17
    $\begingroup$ The softmax function is nothing but a generalization of the sigmoid, so it's not entirely clear what you mean by "softmax vs. sigmoid." $\endgroup$ – dsaxton Sep 6 '16 at 15:56
  • 2
    $\begingroup$ It is the case with the sigmoid. When we use the sigmoid one class has probability $\exp(\beta^T x) / (\exp(\beta^T x) + 1)$ and the other has probability $1 / (\exp(\beta^T x) + 1)$. $\endgroup$ – dsaxton Sep 6 '16 at 16:05
  • 3
    $\begingroup$ The reddit poster is making a distinction that I think is wrong or at least irrelevant. Whether or not one of the classes has weight one is just a matter of shifting the scores, which has no effect on the probabilities. $\endgroup$ – dsaxton Sep 6 '16 at 19:03
  • 2
    $\begingroup$ Possible duplicate of Binary and multinomial logistic regression $\endgroup$ – Franck Dernoncourt Jan 2 '17 at 14:36
  • 3
    $\begingroup$ "it's not entirely clear what you mean by "softmax vs. sigmoid."" just below the title, there's the body of the question -- very easy to miss, I know. Plus, it's a good title to direct google queries to come here to answer exactly what was asked. $\endgroup$ – michael Oct 31 '17 at 4:54
78
$\begingroup$

The sigmoid function is used for the two-class logistic regression, whereas the softmax function is used for the multiclass logistic regression (a.k.a. MaxEnt, multinomial logistic regression, softmax Regression, Maximum Entropy Classifier).


In the two-class logistic regression, the predicted probablies are as follows, using the sigmoid function:

$$ \begin{align} \Pr(Y_i=0) &= \frac{e^{-\boldsymbol\beta_ \cdot \mathbf{X}_i}} {1 +e^{-\boldsymbol\beta_0 \cdot \mathbf{X}_i}} \, \\ \Pr(Y_i=1) &= 1 - \Pr(Y_i=0) = \frac{1} {1 +e^{-\boldsymbol\beta_ \cdot \mathbf{X}_i}} \end{align} $$

In the multiclass logistic regression, with $K$ classes, the predicted probabilities are as follows, using the softmax function:

$$ \begin{align} \Pr(Y_i=k) &= \frac{e^{\boldsymbol\beta_k \cdot \mathbf{X}_i}} {~\sum_{0 \leq c \leq K}^{}{e^{\boldsymbol\beta_c \cdot \mathbf{X}_i}}} \, \\ \end{align} $$


One can observe that the softmax function is an extension of the sigmoid function to the multiclass case, as explained below. Let's look at the multiclass logistic regression, with $K=2$ classes:

$$ \begin{align} \Pr(Y_i=0) &= \frac{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i}} {~\sum_{0 \leq c \leq K}^{}{e^{\boldsymbol\beta_c \cdot \mathbf{X}_i}}} = \frac{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i}}{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}} = \frac{e^{(\boldsymbol\beta_0 - \boldsymbol\beta_1) \cdot \mathbf{X}_i}}{e^{(\boldsymbol\beta_0 - \boldsymbol\beta_1) \cdot \mathbf{X}_i} + 1} = \frac{e^{-\boldsymbol\beta_ \cdot \mathbf{X}_i}} {1 +e^{-\boldsymbol\beta \cdot \mathbf{X}_i}} \\ \, \\ \Pr(Y_i=1) &= \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}} {~\sum_{0 \leq c \leq K}^{}{e^{\boldsymbol\beta_c \cdot \mathbf{X}_i}}} = \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}}{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}} = \frac{1}{e^{(\boldsymbol\beta_0-\boldsymbol\beta_1) \cdot \mathbf{X}_i} + 1} = \frac{1} {1 +e^{-\boldsymbol\beta_ \cdot \mathbf{X}_i}} \, \\ \end{align} $$

with $\boldsymbol\beta = - (\boldsymbol\beta_0 - \boldsymbol\beta_1)$. We see that we obtain the same probabilities as in the two-class logistic regression using the sigmoid function. Wikipedia expands a bit more on that.

$\endgroup$
  • 1
    $\begingroup$ I am naive in this one, But I see this a lot of time β=−(β0−β1) What could be possible explanation to it? As far as I know in Sigmoids β would be a vector. And they are usually one for given run. Then how come β0 and β1 comes in the picture? $\endgroup$ – Ishan Bhatt Jan 18 '18 at 7:25
  • 1
    $\begingroup$ @IshanBhatt this comment may help. $\endgroup$ – Tom Hale May 4 '18 at 10:14
  • $\begingroup$ strangely enough, i can still regress to multiclasses using just sigmoid :) $\endgroup$ – datdinhquoc Sep 17 at 8:50
15
$\begingroup$

They are, in fact, equivalent, in the sense that one can be transformed into the other.

Suppose that your data is represented by a vector $\boldsymbol{x}$, of arbitrary dimension, and you built a binary classifier for it, using an affine transformation followed by a softmax:

\begin{equation} \begin{pmatrix} z_0 \\ z_1 \end{pmatrix} = \begin{pmatrix} \boldsymbol{w}_0^T \\ \boldsymbol{w}_1^T \end{pmatrix}\boldsymbol{x} + \begin{pmatrix} b_0 \\ b_1 \end{pmatrix}, \end{equation} \begin{equation} P(C_i | \boldsymbol{x}) = \text{softmax}(z_i)=\frac{e^{z_i}}{e^{z_0}+e^{z_1}}, \, \, i \in \{0,1\}. \end{equation}

Let's transform it into an equivalent binary classifier that uses a sigmoid instead of the softmax. First of all, we have to decide which is the probability that we want the sigmoid to output (which can be for class $C_0$ or $C_1$). This choice is absolutely arbitrary and so I choose class $C_0$. Then, my classifier will be of the form:

\begin{equation} z' = \boldsymbol{w}'^T \boldsymbol{x} + b', \end{equation} \begin{equation} P(C_0 | \boldsymbol{x}) = \sigma(z')=\frac{1}{1+e^{-z'}}, \end{equation} \begin{equation} P(C_1 | \boldsymbol{x}) = 1-\sigma(z'). \end{equation}

The classifiers are equivalent if the probabilities are the same, so we must impose:

\begin{equation} \sigma(z') = \text{softmax}(z_0) \end{equation}

Replacing $z_0$, $z_1$ and $z'$ by their expressions in terms of $\boldsymbol{w}_0,\boldsymbol{w}_1, \boldsymbol{w}', b_0, b_1, b'$ and $\boldsymbol{x}$ and doing some straightforward algebraic manipulation, you may verify that the equality above holds if and only if $\boldsymbol{w}'$ and $b'$ are given by:

\begin{equation} \boldsymbol{w}' = \boldsymbol{w}_0-\boldsymbol{w}_1, \end{equation} \begin{equation} b' = b_0-b_1. \end{equation}

$\endgroup$
  • $\begingroup$ @null Ok, I if you ask that, then you did not understand my explanation. Let me address your specific problem: if you tell me you are feeding your data to a sigmoid, then it must be a one-dimensional number, $x$. When feeding it to a sigmoid, you get the probability of $x$ being in one of your two classes, for instance $C_0$: $P(C_0|x)=σ(x)$. Then, the probability of $x$ being in $C_1$ is: $P(C_1|x)=1−P(C_0|x)=σ(x)$. Now let's replace your sigmoid by a softmax. (To be continued). $\endgroup$ – D... Jun 25 '17 at 23:49
  • $\begingroup$ (Continuation). In order to apply a softmax to a classification problem with two classes, you need your one dimensional data to be transformed into a two dimensional vector. Therefore, we need to define our $w_0$ and $w_1$. Let's choose $w_0=1$. Since $w_1$ must satisfy $w′=w_0−w_1$, we have $1=1−w_1$, so $w_1=0$. Now, we have $z_0=w_0x=x$ and $z_1=w_1x=0$. Using this, you can immediately verify that $σ(x)=\text{softmax}(z_0)$. $\endgroup$ – D... Jun 25 '17 at 23:49
  • $\begingroup$ Moreover, any combination of $w_0$ and $w_1$ that satisfies $w'=w_0-w_1$ (that is, $1=w_1-w_0$) would lead to the exact same result. This shows that the softmax has one redundant parameter. Although this may seem stupid, it is in fact an interesting property, since it allows normalization of the parameters $w_i$, which promotes numerical stability of the learning algorithm and inference. But this is just an extra comment, it is not important to answer your question :) $\endgroup$ – D... Jun 25 '17 at 23:50
  • $\begingroup$ Thanks a lot. I got it. In your first comment the probability $P(C_1|x)$ should probably be $1-\sigma(x)$. I now understand what is the idea behind the transformation. $\endgroup$ – null Jun 27 '17 at 11:05
  • $\begingroup$ Glad that you understood it ;) Yes, it's a typo, it obviously should be $P(C_1|x)=1 - \sigma(x)$. Thanks for pointing it out! $\endgroup$ – D... Jun 27 '17 at 11:19
8
$\begingroup$

I've noticed people often get directed to this question when searching whether to use sigmoid vs softmax in neural networks. If you are one of those people building a neural network classifier, here is how to decide whether to apply sigmoid or softmax to the raw output values from your network:

  • If you have a multi-label classification problem = there is more than one "right answer" = the outputs are NOT mutually exclusive, then use a sigmoid function on each raw output independently. The sigmoid will allow you to have high probability for all of your classes, some of them, or none of them. Example: classifying diseases in a chest x-ray image. The image might contain pneumonia, emphysema, and/or cancer, or none of those findings.
  • If you have a multi-class classification problem = there is only one "right answer" = the outputs are mutually exclusive, then use a softmax function. The softmax will enforce that the sum of the probabilities of your output classes are equal to one, so in order to increase the probability of a particular class, your model must correspondingly decrease the probability of at least one of the other classes. Example: classifying images from the MNIST data set of handwritten digits. A single picture of a digit has only one true identity - the picture cannot be a 7 and an 8 at the same time.

Reference: for a more detailed explanation of when to use sigmoid vs. softmax in neural network design, including example calculations, please see this article: "Classification: Sigmoid vs. Softmax."

$\endgroup$
-1
$\begingroup$

Adding to all the previous answers - I would like to mention the fact that any multi-class classification problem can be reduced to multiple binary classification problems using "one-vs-all" method, i.e. having C sigmoids (when C is the number of classes) and interpreting every sigmoid to be the probability of being in that specific class or not, and taking the max probability.

So for example, in the MNIST digits example, you could either use a softmax, or ten sigmoids. In fact this is what Andrew Ng does in his Coursera ML course. You can check out here how Andrew Ng used 10 sigmoids for multiclass classification (adapted from Matlab to python by me), and here is my softmax adaptation in python.

Also, it's worth noting that while the functions are equivalent (for the purpose of multiclass classification) they differ a bit in their implementation (especially with regards to their derivatives, and how to represent y).

A big advantage of using multiple binary classifications (i.e. Sigmoids) over a single multiclass classification (i.e. Softmax) - is that if your softmax is too large (e.g. if you are using a one-hot word embedding of a dictionary size of 10K or more) - it can be inefficient to train it. What you can do instead is take a small part of your training-set and use it to train only a small part of your sigmoids. This is the main idea behind Negative Sampling.

$\endgroup$
  • $\begingroup$ The functions are not equivalent because the softmax network is constrained to produce a probability distribution over the classes as the outputs: the vector is non-negative and sums to 1. The $C$ sigmoid units are non-negative, but they can sum to any number between 0 and $C$; it is not a valid probability distribution. This distinction is crucial to characterizing how the two functions differ. $\endgroup$ – Reinstate Monica Sep 23 at 20:59
  • $\begingroup$ What is your definition of equivalent? Mine is: you can use either for multiclass classification without any problem. Also - any multiclass classification that uses softmax can be transformed to a one-vs-all binary classifications that use sigmoids. Why should I care about the distributions of the outputs summing to 1? $\endgroup$ – David Refaeli Sep 24 at 12:06
  • $\begingroup$ Your argument about multi-label classification shows why sigmoid and softmax are not equivalent. When using softmax, increasing the probability of one class decreases the total probability of all other classes (because of sum-to-1). Using sigmoid, increasing the probability of one class does not change the total probability of the other classes. This observation is the reason that sigmoid is plausible for multi-label classification: a single example can belong to $0, 1, 2, \dots , C$ classes. Sum-to-1 is also the reason that softmax is not suitable for multi-label classification. $\endgroup$ – Reinstate Monica Sep 24 at 13:00
  • $\begingroup$ I lost you. For all practical purposes that I know of, multiple sigmoids = 1 softmax. I even added the case of negative sampling, where multiple sigmoids actually have an advantage over a softmax. $\endgroup$ – David Refaeli Sep 29 at 15:15

protected by Reinstate Monica Sep 23 at 21:48

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.