# Softmax vs. Sigmoid functions

In Machine Learning, you deal with `softmax`

and `sigimoid`

functions often.
I wanted to provide some intuition when you should use one, over the other.

Suppose you have predictions as the output from neural net.

These are the predictions for cat, dog, cow, and zebra. They can be positive or negative (no `ReLU`

at the end).

### Softmax

If we plan to find exactly one value we should use `softmax`

function.
The character of this function is “there can be only one”.

Note the out values are in the `B`

column. Then for each `B`

value $x$ we do create $e^x$ in column `C`

.

What the `exp`

function do it will do:

- it will make the predictions positive
- what ever was max, it will stand out as max

The `softmax`

funciton:

Can be literally expressed as take the exponent value and divide it by the sum of all other exponents (~`34`

in the image). This will make one important feature of `softmax`

, that the sum off all softmax values will add to 1.

Just by peaking the `max`

value after the softmax we get out prediction.

### Sigmoid

Things are different for the `sigmoid`

function. This function can provide us with the top $n$ results based on the threshold.

If the threshold is e.g. `3`

from the image you can find two results greater than that number. We use the following formula to evaluate the `sigmoid`

function.

Exactly, the feature of `sigmoid`

is to emphasize multiple values, based on the threshold, and we use it for the multi-label classification problems.

### And in PyTorch…

In PyTorch you would use the `torch.nn.Softmax(dim=None)`

layer compute `softmax`

to an n-dimensional input tensor rescaling them so that the elements of the n-dimensional output tensor lie in the range [0,1] and sum to 1.

```
import torch.nn as nn
m = nn.Softmax(dim=0)
inp = torch.randn(2, 3)*2-1
print(inp)
out = m(inp)
print(out)
print(torch.sum(out))
# tensor([[-1.2928, -2.9990, -1.8886],
# [ 0.1079, -3.6320, -1.6835]])
# tensor([[0.1977, 0.6532, 0.4489],
# [0.8023, 0.3468, 0.5511]])
# tensor(3.)
```

Note you need to specify the dimension for `softmax`

, which is `dim=0`

in the previous example (dimension of columns). This is why the total sum will add to 3. since we have three columns.

But we can also functional version of `softmax`

. The previous example can be rewritten as:

```
import torch.nn.functional as F
inp = torch.randn(2, 3)*2-1
print(inp)
out = F.softmax(inp, dim=0)
print(out)
print(torch.sum(out))
# tensor([[ 0.9096, -2.5876, -2.2403],
# [-0.8566, 0.2757, -1.9268]])
# tensor([[0.8540, 0.0540, 0.4223],
# [0.1460, 0.9460, 0.5777]])
# tensor(3.)
```

There is also a special 2d `softmax`

that works on 4D tensors only, but you can always rewrite it using the regular `F.softmax`

.

```
m = nn.Softmax2d()
inp = torch.randn(1, 3, 2, 2)
out = m(inp)
out2 = F.softmax(inp, dim=1)
print(torch.equal(out, out2)) #True
```

For the `sigmoid`

function the things are quite clear, based on logits we get probabilities.

```
inp = torch.randn(1,5)
print(inp)
print(F.sigmoid(inp))
# tensor([[-0.4010, 0.0468, -0.4071, 0.6252, 1.0899]])
# tensor([[0.4011, 0.5117, 0.3996, 0.6514, 0.7484]])
```

### When to use one over the other?

We should use softmax if we do classification with one result, or single label classification (SLC). We should use sigmoid if we have multi-label classification case (MLC).

### Case of SLC:

Use log softmax followed by negative log likelihood loss (nll_loss). Here is the implementation of nll_loss:

```
def nll_loss(p, target):
return -p[range(target.shape[0]), target].mean()
```

There is one function called cross entropy loss in PyTorch that replaces both softmax and nll_loss.

```
lp = F.log_softmax(x, dim=-1)
loss = F.nll_loss(lp, target)
```

Which is equivalent to :

```
loss = F.cross_entropy(x, target)
```

Do not calculate log of softmax directly instead use log-sum-exp trick:

```
def log_softmax(x):
return x - x.exp().sum(-1).log().unsqueeze(-1)
```

### Case of MLC:

We use sigmoid and binary cross entropy functions in PyTorch that do broadcasting.

```
def sigmoid(x): return 1/(1 + (-x).exp())
def binary_cross_entropy(p, y): return -(p.log()*y + (1-y)*(1-p).log()).mean()
```

Sigmoid converts anything from (-inf, inf) into probability [0,1]. `binary_cross_entropy`

will take the log of this probability later.

We can forget about sigmoid if we use `F.binary_cross_entropy_with_logits`

function. This function takes logits directly.

```
F.sigmoid + F.binary_cross_entropy = F.binary_cross_entropy_with_logits
```

`F.sigmoid`

will take logits and you may be careful in here in general case
`logit(sigmoid(x))`

is not stable:

```
%matplotlib inline
import torch
torch.Tensor.ndim = property(lambda x: len(x.size()))
import matplotlib.pyplot as plt
x=torch.arange(-20, 20, 1e-4)
def sigmo(x):
return 1/(1+torch.exp(-x))
def logit(x):
return torch.log((x/(1-x)))
plt.plot(x, logit(x))
plt.xlabel("logit")
plt.show()
plt.close()
plt.plot(x, sigmo(x))
plt.xlabel("sigmoid")
plt.show()
plt.close()
plt.plot(x, logit(sigmo(x)))
plt.xlabel("logit(sigmoid(x))")
plt.show()
plt.close()
y = logit(sigmo(x))
plt.plot(x, ((y-x+1e-5)/(x+1e-3)))
plt.xlabel("(logit(sigmo(x)-x)/x")
plt.show()
plt.close()
```

Still PyTorch implementation of `F.binary_cross_entropy_with_logits`

should be numerically stable.

### An example in SLC

```
batch_size, n_classes = 10, 5
x = torch.randn(batch_size, n_classes)
print("x:",x)
target = torch.randint(n_classes, size=(batch_size,), dtype=torch.long)
print("target:",target)
def log_softmax(x):
return x - x.exp().sum(-1).log().unsqueeze(-1)
def nll_loss(p, target):
return -p[range(target.shape[0]), target].mean()
pred = log_softmax(x)
print ("pred:", pred)
ohe = torch.zeros(batch_size, n_classes)
ohe[range(ohe.shape[0]), target]=1
print("ohe:",ohe)
pe = pred[range(target.shape[0]), target]
print("pe:",pe)
mean = pred[range(target.shape[0]), target].mean()
print("mean:",mean)
negmean = -mean
print("negmean:", negmean)
loss = nll_loss(pred, target)
print("loss:",loss)
```