You’re constructing a Keras mannequin. For those who haven’t been doing deep studying for thus lengthy, getting the output activations and value operate proper may contain some memorization (or lookup). You may be making an attempt to recall the final pointers like so:

So with my cats and canine, I’m doing 2-class classification, so I’ve to make use of sigmoid activation within the output layer, proper, after which, it’s binary crossentropy for the associated fee operate…
Or: I’m doing classification on ImageNet, that’s multi-class, in order that was softmax for activation, after which, value ought to be categorical crossentropy…

It’s nice to memorize stuff like this, however realizing a bit in regards to the causes behind usually makes issues simpler. So we ask: Why is it that these output activations and value features go collectively? And, do they all the time need to?

In a nutshell

Put merely, we select activations that make the community predict what we would like it to foretell.
The fee operate is then decided by the mannequin.

It is because neural networks are usually optimized utilizing most probability, and relying on the distribution we assume for the output models, most probability yields completely different optimization targets. All of those targets then reduce the cross entropy (pragmatically: mismatch) between the true distribution and the anticipated distribution.

Let’s begin with the best, the linear case.

Regression

For the botanists amongst us, right here’s an excellent easy community meant to foretell sepal width from sepal size:

mannequin <- keras_model_sequential() %>%
  layer_dense(models = 32) %>%
  layer_dense(models = 1)

mannequin %>% compile(
  optimizer = "adam", 
  loss = "mean_squared_error"
)

mannequin %>% match(
  x = iris$Sepal.Size %>% as.matrix(),
  y = iris$Sepal.Width %>% as.matrix(),
  epochs = 50
)

Our mannequin’s assumption right here is that sepal width is generally distributed, given sepal size. Most frequently, we’re making an attempt to foretell the imply of a conditional Gaussian distribution:

[p(y|mathbf{x} = N(y; mathbf{w}^tmathbf{h} + b)]

In that case, the associated fee operate that minimizes cross entropy (equivalently: optimizes most probability) is imply squared error.
And that’s precisely what we’re utilizing as a price operate above.

Alternatively, we’d want to predict the median of that conditional distribution. In that case, we’d change the associated fee operate to make use of imply absolute error:

mannequin %>% compile(
  optimizer = "adam", 
  loss = "mean_absolute_error"
)

Now let’s transfer on past linearity.

Binary classification

We’re enthusiastic fowl watchers and wish an utility to inform us when there’s a fowl in our backyard – not when the neighbors landed their airplane, although. We’ll thus prepare a community to tell apart between two courses: birds and airplanes.

# Utilizing the CIFAR-10 dataset that conveniently comes with Keras.
cifar10 <- dataset_cifar10()

x_train <- cifar10$prepare$x / 255
y_train <- cifar10$prepare$y

is_bird <- cifar10$prepare$y == 2
x_bird <- x_train[is_bird, , ,]
y_bird <- rep(0, 5000)

is_plane <- cifar10$prepare$y == 0
x_plane <- x_train[is_plane, , ,]
y_plane <- rep(1, 5000)

x <- abind::abind(x_bird, x_plane, alongside = 1)
y <- c(y_bird, y_plane)

mannequin <- keras_model_sequential() %>%
  layer_conv_2d(
    filter = 8,
    kernel_size = c(3, 3),
    padding = "identical",
    input_shape = c(32, 32, 3),
    activation = "relu"
  ) %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_conv_2d(
    filter = 8,
    kernel_size = c(3, 3),
    padding = "identical",
    activation = "relu"
  ) %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
layer_flatten() %>%
  layer_dense(models = 32, activation = "relu") %>%
  layer_dense(models = 1, activation = "sigmoid")

mannequin %>% compile(
  optimizer = "adam", 
  loss = "binary_crossentropy", 
  metrics = "accuracy"
)

mannequin %>% match(
  x = x,
  y = y,
  epochs = 50
)

Though we usually speak about “binary classification,” the best way the end result is normally modeled is as a Bernoulli random variable, conditioned on the enter information. So:

[P(y = 1|mathbf{x}) = p, 0leq pleq1]

A Bernoulli random variable takes on values between (0) and (1). In order that’s what our community ought to produce.
One concept may be to simply clip all values of (mathbf{w}^tmathbf{h} + b) outdoors that interval. But when we do that, the gradient in these areas might be (0): The community can’t be taught.

A greater manner is to squish the whole incoming interval into the vary (0,1), utilizing the logistic sigmoid operate

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

As you possibly can see, the sigmoid operate saturates when its enter will get very massive, or very small. Is that this problematic?
It relies upon. In the long run, what we care about is that if the associated fee operate saturates. Had been we to decide on imply squared error right here, as within the regression job above, that’s certainly what may occur.

Nonetheless, if we comply with the final precept of most probability/cross entropy, the loss might be

[- log P (y|mathbf{x})]

the place the (log) undoes the (exp) within the sigmoid.

In Keras, the corresponding loss operate is binary_crossentropy. For a single merchandise, the loss might be

• (- log(p)) when the bottom fact is 1
• (- log(1-p)) when the bottom fact is 0

Right here, you possibly can see that when for a person instance, the community predicts the unsuitable class and is extremely assured about it, this instance will contributely very strongly to the loss.

What occurs once we distinguish between greater than two courses?

Multi-class classification

CIFAR-10 has 10 courses; so now we wish to resolve which of 10 object courses is current within the picture.

Right here first is the code: Not many variations to the above, however notice the modifications in activation and value operate.

cifar10 <- dataset_cifar10()

x_train <- cifar10$prepare$x / 255
y_train <- cifar10$prepare$y

mannequin <- keras_model_sequential() %>%
  layer_conv_2d(
    filter = 8,
    kernel_size = c(3, 3),
    padding = "identical",
    input_shape = c(32, 32, 3),
    activation = "relu"
  ) %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_conv_2d(
    filter = 8,
    kernel_size = c(3, 3),
    padding = "identical",
    activation = "relu"
  ) %>%
  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
  layer_flatten() %>%
  layer_dense(models = 32, activation = "relu") %>%
  layer_dense(models = 10, activation = "softmax")

mannequin %>% compile(
  optimizer = "adam",
  loss = "sparse_categorical_crossentropy",
  metrics = "accuracy"
)

mannequin %>% match(
  x = x_train,
  y = y_train,
  epochs = 50
)

So now we’ve got softmax mixed with categorical crossentropy. Why?

Once more, we would like a legitimate likelihood distribution: Chances for all disjunct occasions ought to sum to 1.

CIFAR-10 has one object per picture; so occasions are disjunct. Then we’ve got a single-draw multinomial distribution (popularly often known as “Multinoulli,” principally attributable to Murphy’s Machine studying(Murphy 2012)) that may be modeled by the softmax activation:

[softmax(mathbf{z})_i = frac{e^{z_i}}{sum_j{e^{z_j}}}]

Simply because the sigmoid, the softmax can saturate. On this case, that may occur when variations between outputs grow to be very large.
Additionally like with the sigmoid, a (log) in the associated fee operate undoes the (exp) that’s chargeable for saturation:

[log softmax(mathbf{z})_i = z_i – logsum_j{e^{z_j}}]

Right here (z_i) is the category we’re estimating the likelihood of – we see that its contribution to the loss is linear and thus, can by no means saturate.

In Keras, the loss operate that does this for us is known as categorical_crossentropy. We use sparse_categorical_crossentropy within the code which is identical as categorical_crossentropy however doesn’t want conversion of integer labels to one-hot vectors.

Let’s take a better take a look at what softmax does. Assume these are the uncooked outputs of our 10 output models:

Now that is what the normalized likelihood distribution appears like after taking the softmax:

Do you see the place the winner takes all within the title comes from? This is a vital level to remember: Activation features are usually not simply there to provide sure desired distributions; they will additionally change relationships between values.

Conclusion

We began this publish alluding to frequent heuristics, corresponding to “for multi-class classification, we use softmax activation, mixed with categorical crossentropy because the loss operate.” Hopefully, we’ve succeeded in exhibiting why these heuristics make sense.

Nonetheless, realizing that background, you can even infer when these guidelines don’t apply. For instance, say you wish to detect a number of objects in a picture. In that case, the winner-takes-all technique just isn’t probably the most helpful, as we don’t wish to exaggerate variations between candidates. So right here, we’d use sigmoid on all output models as an alternative, to find out a likelihood of presence per object.