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You’re constructing a Keras mannequin. If you happen to haven’t been doing deep studying for therefore lengthy, getting the output activations and value perform proper would possibly contain some memorization (or lookup). You is perhaps attempting to recall the overall tips 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 price perform…*

Or: *I’m doing classification on ImageNet, that’s multi-class, in order that was softmax for activation, after which, value must be categorical crossentropy…*

It’s tremendous to memorize stuff like this, however figuring out a bit concerning 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 must?

## In a nutshell

Put merely, we select activations that make the community predict what we wish it to foretell.

The price perform is then decided by the mannequin.

It’s because neural networks are usually optimized utilizing *most chance*, and relying on the distribution we assume for the output items, most chance yields totally different optimization goals. All of those goals then decrease the cross entropy (pragmatically: mismatch) between the true distribution and the expected 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:

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

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

In that case, the price perform that minimizes cross entropy (equivalently: optimizes most chance) is *imply squared error*.

And that’s precisely what we’re utilizing as a price perform above.

Alternatively, we’d want to predict the median of that conditional distribution. In that case, we’d change the price perform 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 chook watchers and need an software to inform us when there’s a chook in our backyard – not when the neighbors landed their airplane, although. We’ll thus practice a community to tell apart between two lessons: birds and airplanes.

```
# Utilizing the CIFAR-10 dataset that conveniently comes with Keras.
cifar10 <- dataset_cifar10()
x_train <- cifar10$practice$x / 255
y_train <- cifar10$practice$y
is_bird <- cifar10$practice$y == 2
x_bird <- x_train[is_bird, , ,]
y_bird <- rep(0, 5000)
is_plane <- cifar10$practice$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 = "similar",
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 = "similar",
activation = "relu"
) %>%
layer_max_pooling_2d(pool_size = c(2, 2)) %>%
layer_flatten() %>%
layer_dense(items = 32, activation = "relu") %>%
layer_dense(items = 1, activation = "sigmoid")
mannequin %>% compile(
optimizer = "adam",
loss = "binary_crossentropy",
metrics = "accuracy"
)
mannequin %>% match(
x = x,
y = y,
epochs = 50
)
```

Though we usually discuss “binary classification,” the best way the result is normally modeled is as a *Bernoulli random variable*, conditioned on the enter knowledge. 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 thought is perhaps to simply clip all values of (mathbf{w}^tmathbf{h} + b) exterior that interval. But when we do that, the gradient in these areas might be (0): The community can not be taught.

A greater means is to squish the whole incoming interval into the vary (0,1), utilizing the logistic *sigmoid* perform

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

As you possibly can see, the sigmoid perform saturates when its enter will get very giant, or very small. Is that this problematic?

It relies upon. Ultimately, what we care about is that if the price perform saturates. Have been we to decide on imply squared error right here, as within the regression process above, that’s certainly what may occur.

Nonetheless, if we comply with the overall precept of most chance/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 perform is `binary_crossentropy`

. For a single merchandise, the loss might be

- (- log(p)) when the bottom reality is 1
- (- log(1-p)) when the bottom reality is 0

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

What occurs after we distinguish between greater than two lessons?

## Multi-class classification

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

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

```
cifar10 <- dataset_cifar10()
x_train <- cifar10$practice$x / 255
y_train <- cifar10$practice$y
mannequin <- keras_model_sequential() %>%
layer_conv_2d(
filter = 8,
kernel_size = c(3, 3),
padding = "similar",
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 = "similar",
activation = "relu"
) %>%
layer_max_pooling_2d(pool_size = c(2, 2)) %>%
layer_flatten() %>%
layer_dense(items = 32, activation = "relu") %>%
layer_dense(items = 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 *softmax* mixed with *categorical crossentropy*. Why?

Once more, we wish a sound 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 a single-draw multinomial distribution (popularly often called “Multinoulli,” largely 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 can occur when *variations* between outputs change into very huge.

Additionally like with the sigmoid, a (log) in the price perform 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 perform that does this for us is named `categorical_crossentropy`

. We use sparse_categorical_crossentropy within the code which is similar as `categorical_crossentropy`

however doesn’t want conversion of integer labels to one-hot vectors.

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

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 usually are not simply there to provide sure desired distributions; they will additionally change relationships between values.

## Conclusion

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

Nonetheless, figuring out that background, it’s also possible to 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 essentially the most helpful, as we don’t wish to exaggerate variations between candidates. So right here, we’d use *sigmoid* on all output items as a substitute, to find out a likelihood of presence *per object*.

Goodfellow, Ian, Yoshua Bengio, and Aaron Courville. 2016. *Deep Studying*. MIT Press.

Murphy, Kevin. 2012. *Machine Studying: A Probabilistic Perspective*. MIT Press.

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