5.1. Multilayer Perceptrons
Open the notebook in Colab
Open the notebook in Colab
Open the notebook in Colab
Open the notebook in SageMaker Studio Lab

In Section 4, we introduced softmax regression (Section 4.1), implementing the algorithm from scratch (Section 4.4) and using high-level APIs (Section 4.5). This allowed us to train classifiers capable of recognizing 10 categories of clothing from low-resolution images. Along the way, we learned how to wrangle data, coerce our outputs into a valid probability distribution, apply an appropriate loss function, and minimize it with respect to our model’s parameters. Now that we have mastered these mechanics in the context of simple linear models, we can launch our exploration of deep neural networks, the comparatively rich class of models with which this book is primarily concerned.

5.1.1. Hidden Layers

We described affine transformations in Section 3.1.1.1 as linear transformations with added bias. To begin, recall the model architecture corresponding to our softmax regression example, illustrated in Fig. 4.1.1. This model maps inputs directly to outputs via a single affine transformation, followed by a softmax operation. If our labels truly were related to the input data by a simple affine transformation, then this approach would be sufficient. However, linearity (in affine transformations) is a strong assumption.

5.1.1.1. Limitations of Linear Models

For example, linearity implies the weaker assumption of monotonicity, i.e., that any increase in our feature must either always cause an increase in our model’s output (if the corresponding weight is positive), or always cause a decrease in our model’s output (if the corresponding weight is negative). Sometimes that makes sense. For example, if we were trying to predict whether an individual will repay a loan, we might reasonably assume that all other things being equal, an applicant with a higher income would always be more likely to repay than one with a lower income. While monotonic, this relationship likely is not linearly associated with the probability of repayment. An increase in income from \$0 to \$50,000 likely corresponds to a bigger increase in likelihood of repayment than an increase from \$1 million to \$1.05 million. One way to handle this might be to post-process our outcome such that linearity becomes more plausible, by using the logistic map (and thus the logarithm of the probability of outcome).

Note that we can easily come up with examples that violate monotonicity. Say for example that we want to predict health as a function of body temperature. For individuals with a body temperature above 37°C (98.6°F), higher temperatures indicate greater risk. However, for individuals with body temperatures below 37°C, lower temperatures indicate greater risk! Again, we might resolve the problem with some clever preprocessing, such as using the distance from 37°C as a feature.

But what about classifying images of cats and dogs? Should increasing the intensity of the pixel at location (13, 17) always increase (or always decrease) the likelihood that the image depicts a dog? Reliance on a linear model corresponds to the implicit assumption that the only requirement for differentiating cats vs. dogs is to assess the brightness of individual pixels. This approach is doomed to fail in a world where inverting an image preserves the category.

And yet despite the apparent absurdity of linearity here, as compared with our previous examples, it is less obvious that we could address the problem with a simple preprocessing fix. That is, because the significance of any pixel depends in complex ways on its context (the values of the surrounding pixels). While there might exist a representation of our data that would take into account the relevant interactions among our features, on top of which a linear model would be suitable, we simply do not know how to calculate it by hand. With deep neural networks, we used observational data to jointly learn both a representation via hidden layers and a linear predictor that acts upon that representation.

This problem of nonlinearity has been studied for at least a century (Fisher, 1928). For instance, decision trees in their most basic form use a sequence of binary decisions to decide upon class membership (Quinlan, 2014). Likewise, kernel methods have been used for many decades to model nonlinear dependencies (Aronszajn, 1950). This has found its way, e.g., into nonparametric spline models (Wahba, 1990) and kernel methods (Schölkopf and Smola, 2002). It is also something that the brain solves quite naturally. After all, neurons feed into other neurons which, in turn, feed into other neurons again (y Cajal and Azoulay, 1894). Consequently we have a sequence of relatively simple transformations.

5.1.1.2. Incorporating Hidden Layers

We can overcome the limitations of linear models by incorporating one or more hidden layers. The easiest way to do this is to stack many fully connected layers on top of each other. Each layer feeds into the layer above it, until we generate outputs. We can think of the first \(L-1\) layers as our representation and the final layer as our linear predictor. This architecture is commonly called a multilayer perceptron, often abbreviated as MLP (Fig. 5.1.1).

../_images/mlp.svg

Fig. 5.1.1 An MLP with a hidden layer of 5 hidden units.

This MLP has 4 inputs, 3 outputs, and its hidden layer contains 5 hidden units. Since the input layer does not involve any calculations, producing outputs with this network requires implementing the computations for both the hidden and output layers; thus, the number of layers in this MLP is 2. Note that both layers are fully connected. Every input influences every neuron in the hidden layer, and each of these in turn influences every neuron in the output layer. Alas, we are not quite done yet.

5.1.1.3. From Linear to Nonlinear

As before, we denote by the matrix \(\mathbf{X} \in \mathbb{R}^{n \times d}\) a minibatch of \(n\) examples where each example has \(d\) inputs (features). For a one-hidden-layer MLP whose hidden layer has \(h\) hidden units, we denote by \(\mathbf{H} \in \mathbb{R}^{n \times h}\) the outputs of the hidden layer, which are hidden representations. Since the hidden and output layers are both fully connected, we have hidden-layer weights \(\mathbf{W}^{(1)} \in \mathbb{R}^{d \times h}\) and biases \(\mathbf{b}^{(1)} \in \mathbb{R}^{1 \times h}\) and output-layer weights \(\mathbf{W}^{(2)} \in \mathbb{R}^{h \times q}\) and biases \(\mathbf{b}^{(2)} \in \mathbb{R}^{1 \times q}\). This allows us to calculate the outputs \(\mathbf{O} \in \mathbb{R}^{n \times q}\) of the one-hidden-layer MLP as follows:

(5.1.1)\[\begin{split}\begin{aligned} \mathbf{H} & = \mathbf{X} \mathbf{W}^{(1)} + \mathbf{b}^{(1)}, \\ \mathbf{O} & = \mathbf{H}\mathbf{W}^{(2)} + \mathbf{b}^{(2)}. \end{aligned}\end{split}\]

Note that after adding the hidden layer, our model now requires us to track and update additional sets of parameters. So what have we gained in exchange? You might be surprised to find out that—in the model defined above—we gain nothing for our troubles! The reason is plain. The hidden units above are given by an affine function of the inputs, and the outputs (pre-softmax) are just an affine function of the hidden units. An affine function of an affine function is itself an affine function. Moreover, our linear model was already capable of representing any affine function.

To see this formally we can just collapse out the hidden layer in the above definition, yielding an equivalent single-layer model with parameters \(\mathbf{W} = \mathbf{W}^{(1)}\mathbf{W}^{(2)}\) and \(\mathbf{b} = \mathbf{b}^{(1)} \mathbf{W}^{(2)} + \mathbf{b}^{(2)}\):

(5.1.2)\[\mathbf{O} = (\mathbf{X} \mathbf{W}^{(1)} + \mathbf{b}^{(1)})\mathbf{W}^{(2)} + \mathbf{b}^{(2)} = \mathbf{X} \mathbf{W}^{(1)}\mathbf{W}^{(2)} + \mathbf{b}^{(1)} \mathbf{W}^{(2)} + \mathbf{b}^{(2)} = \mathbf{X} \mathbf{W} + \mathbf{b}.\]

In order to realize the potential of multilayer architectures, we need one more key ingredient: a nonlinear activation function \(\sigma\) to be applied to each hidden unit following the affine transformation. For instance, a popular choice is the ReLU (Rectified Linear Unit) activation function (Nair and Hinton, 2010) \(\sigma(x) = \mathrm{max}(0, x)\) operating on its arguments element-wise. The outputs of activation functions \(\sigma(\cdot)\) are called activations. In general, with activation functions in place, it is no longer possible to collapse our MLP into a linear model:

(5.1.3)\[\begin{split}\begin{aligned} \mathbf{H} & = \sigma(\mathbf{X} \mathbf{W}^{(1)} + \mathbf{b}^{(1)}), \\ \mathbf{O} & = \mathbf{H}\mathbf{W}^{(2)} + \mathbf{b}^{(2)}.\\ \end{aligned}\end{split}\]

Since each row in \(\mathbf{X}\) corresponds to an example in the minibatch, with some abuse of notation, we define the nonlinearity \(\sigma\) to apply to its inputs in a row-wise fashion, i.e., one example at a time. Note that we used the same notation for softmax when we denoted a row-wise operation in Section 4.1.1.3. Quite frequently the activation functions we use apply not merely row-wise but element-wise. That means that after computing the linear portion of the layer, we can calculate each activation without looking at the values taken by the other hidden units.

To build more general MLPs, we can continue stacking such hidden layers, e.g., \(\mathbf{H}^{(1)} = \sigma_1(\mathbf{X} \mathbf{W}^{(1)} + \mathbf{b}^{(1)})\) and \(\mathbf{H}^{(2)} = \sigma_2(\mathbf{H}^{(1)} \mathbf{W}^{(2)} + \mathbf{b}^{(2)})\), one atop another, yielding ever more expressive models.

5.1.1.4. Universal Approximators

We know that the brain is capable of very sophisticated statistical analysis. As such, it is worth asking, just how powerful a deep network could be. This question has been answered multiple times, e.g., in Cybenko (1989) in the context of MLPs, and in Micchelli (1984) in the context of reproducing kernel Hilbert spaces in a way that could be seen as radial basis function (RBF) networks with a single hidden layer. These (and related results) suggest that even with a single-hidden-layer network, given enough nodes (possibly absurdly many), and the right set of weights, we can model any function. Actually learning that function is the hard part, though. You might think of your neural network as being a bit like the C programming language. The language, like any other modern language, is capable of expressing any computable program. But actually coming up with a program that meets your specifications is the hard part.

Moreover, just because a single-hidden-layer network can learn any function does not mean that you should try to solve all of your problems with single-hidden-layer networks. In fact, in this case kernel methods are way more effective, since they are capable of solving the problem exactly even in infinite dimensional spaces (Kimeldorf and Wahba, 1971, Schölkopf et al., 2001). In fact, we can approximate many functions much more compactly by using deeper (vs. wider) networks (Simonyan and Zisserman, 2014). We will touch upon more rigorous arguments in subsequent chapters.

5.1.2. Activation Functions

Activation functions decide whether a neuron should be activated or not by calculating the weighted sum and further adding bias with it. They are differentiable operators to transform input signals to outputs, while most of them add non-linearity. Because activation functions are fundamental to deep learning, let’s briefly survey some common activation functions.

%matplotlib inline
import torch
from d2l import torch as d2l
%matplotlib inline
from mxnet import autograd, np, npx
from d2l import mxnet as d2l

npx.set_np()
%matplotlib inline
import tensorflow as tf
from d2l import tensorflow as d2l

5.1.2.1. ReLU Function

The most popular choice, due to both simplicity of implementation and its good performance on a variety of predictive tasks, is the rectified linear unit (ReLU) (Nair and Hinton, 2010). ReLU provides a very simple nonlinear transformation. Given an element \(x\), the function is defined as the maximum of that element and \(0\):

(5.1.4)\[\operatorname{ReLU}(x) = \max(x, 0).\]

Informally, the ReLU function retains only positive elements and discards all negative elements by setting the corresponding activations to 0. To gain some intuition, we can plot the function. As you can see, the activation function is piecewise linear.

x = torch.arange(-8.0, 8.0, 0.1, requires_grad=True)
y = torch.relu(x)
d2l.plot(x.detach(), y.detach(), 'x', 'relu(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_15_0.svg
x = np.arange(-8.0, 8.0, 0.1)
x.attach_grad()
with autograd.record():
    y = npx.relu(x)
d2l.plot(x, y, 'x', 'relu(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_18_0.svg
x = tf.Variable(tf.range(-8.0, 8.0, 0.1), dtype=tf.float32)
y = tf.nn.relu(x)
d2l.plot(x.numpy(), y.numpy(), 'x', 'relu(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_21_0.svg

When the input is negative, the derivative of the ReLU function is 0, and when the input is positive, the derivative of the ReLU function is 1. Note that the ReLU function is not differentiable when the input takes value precisely equal to 0. In these cases, we default to the left-hand-side derivative and say that the derivative is 0 when the input is 0. We can get away with this because the input may never actually be zero (mathematicians would say that it’s nondifferentiable on a set of measure zero). There is an old adage that if subtle boundary conditions matter, we are probably doing (real) mathematics, not engineering. That conventional wisdom may apply here, or at least, the fact that we are not performing constrained optimization (Mangasarian, 1965, Rockafellar, 1970). We plot the derivative of the ReLU function plotted below.

y.backward(torch.ones_like(x), retain_graph=True)
d2l.plot(x.detach(), x.grad, 'x', 'grad of relu', figsize=(5, 2.5))
../_images/output_mlp_76f463_27_0.svg
y.backward()
d2l.plot(x, x.grad, 'x', 'grad of relu', figsize=(5, 2.5))
[22:56:42] src/base.cc:49: GPU context requested, but no GPUs found.
../_images/output_mlp_76f463_30_1.svg
with tf.GradientTape() as t:
    y = tf.nn.relu(x)
d2l.plot(x.numpy(), t.gradient(y, x).numpy(), 'x', 'grad of relu',
         figsize=(5, 2.5))
../_images/output_mlp_76f463_33_0.svg

The reason for using ReLU is that its derivatives are particularly well behaved: either they vanish or they just let the argument through. This makes optimization better behaved and it mitigated the well-documented problem of vanishing gradients that plagued previous versions of neural networks (more on this later).

Note that there are many variants to the ReLU function, including the parameterized ReLU (pReLU) function (He et al., 2015). This variation adds a linear term to ReLU, so some information still gets through, even when the argument is negative:

(5.1.5)\[\operatorname{pReLU}(x) = \max(0, x) + \alpha \min(0, x).\]

5.1.2.2. Sigmoid Function

The sigmoid function transforms its inputs, for which values lie in the domain \(\mathbb{R}\), to outputs that lie on the interval (0, 1). For that reason, the sigmoid is often called a squashing function: it squashes any input in the range (-inf, inf) to some value in the range (0, 1):

(5.1.6)\[\operatorname{sigmoid}(x) = \frac{1}{1 + \exp(-x)}.\]

In the earliest neural networks, scientists were interested in modeling biological neurons which either fire or do not fire. Thus the pioneers of this field, going all the way back to McCulloch and Pitts, the inventors of the artificial neuron, focused on thresholding units (McCulloch and Pitts, 1943). A thresholding activation takes value 0 when its input is below some threshold and value 1 when the input exceeds the threshold.

When attention shifted to gradient based learning, the sigmoid function was a natural choice because it is a smooth, differentiable approximation to a thresholding unit. Sigmoids are still widely used as activation functions on the output units, when we want to interpret the outputs as probabilities for binary classification problems: you can think of the sigmoid as a special case of the softmax. However, the sigmoid has mostly been replaced by the simpler and more easily trainable ReLU for most use in hidden layers. Much of this has to do with the fact that the sigmoid poses challenges for optimization (LeCun et al., 1998) since its gradient vanishes for large positive and negative arguments. This can lead to plateaus that are difficult to escape from. Nonetheless sigmoids are important. In later chapters (e.g., Section 10.1) on recurrent neural networks, we will describe architectures that leverage sigmoid units to control the flow of information across time.

Below, we plot the sigmoid function. Note that when the input is close to 0, the sigmoid function approaches a linear transformation.

y = torch.sigmoid(x)
d2l.plot(x.detach(), y.detach(), 'x', 'sigmoid(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_39_0.svg
with autograd.record():
    y = npx.sigmoid(x)
d2l.plot(x, y, 'x', 'sigmoid(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_42_0.svg
y = tf.nn.sigmoid(x)
d2l.plot(x.numpy(), y.numpy(), 'x', 'sigmoid(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_45_0.svg

The derivative of the sigmoid function is given by the following equation:

(5.1.7)\[\frac{d}{dx} \operatorname{sigmoid}(x) = \frac{\exp(-x)}{(1 + \exp(-x))^2} = \operatorname{sigmoid}(x)\left(1-\operatorname{sigmoid}(x)\right).\]

The derivative of the sigmoid function is plotted below. Note that when the input is 0, the derivative of the sigmoid function reaches a maximum of 0.25. As the input diverges from 0 in either direction, the derivative approaches 0.

# Clear out previous gradients
x.grad.data.zero_()
y.backward(torch.ones_like(x),retain_graph=True)
d2l.plot(x.detach(), x.grad, 'x', 'grad of sigmoid', figsize=(5, 2.5))
../_images/output_mlp_76f463_51_0.svg
y.backward()
d2l.plot(x, x.grad, 'x', 'grad of sigmoid', figsize=(5, 2.5))
../_images/output_mlp_76f463_54_0.svg
with tf.GradientTape() as t:
    y = tf.nn.sigmoid(x)
d2l.plot(x.numpy(), t.gradient(y, x).numpy(), 'x', 'grad of sigmoid',
         figsize=(5, 2.5))
../_images/output_mlp_76f463_57_0.svg

5.1.2.3. Tanh Function

Like the sigmoid function, the tanh (hyperbolic tangent) function also squashes its inputs, transforming them into elements on the interval between -1 and 1:

(5.1.8)\[\operatorname{tanh}(x) = \frac{1 - \exp(-2x)}{1 + \exp(-2x)}.\]

We plot the tanh function below. Note that as input nears 0, the tanh function approaches a linear transformation. Although the shape of the function is similar to that of the sigmoid function, the tanh function exhibits point symmetry about the origin of the coordinate system (Kalman and Kwasny, 1992).

y = torch.tanh(x)
d2l.plot(x.detach(), y.detach(), 'x', 'tanh(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_63_0.svg
with autograd.record():
    y = np.tanh(x)
d2l.plot(x, y, 'x', 'tanh(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_66_0.svg
y = tf.nn.tanh(x)
d2l.plot(x.numpy(), y.numpy(), 'x', 'tanh(x)', figsize=(5, 2.5))
../_images/output_mlp_76f463_69_0.svg

The derivative of the tanh function is:

(5.1.9)\[\frac{d}{dx} \operatorname{tanh}(x) = 1 - \operatorname{tanh}^2(x).\]

It is plotted below. As the input nears 0, the derivative of the tanh function approaches a maximum of 1. And as we saw with the sigmoid function, as input moves away from 0 in either direction, the derivative of the tanh function approaches 0.

# Clear out previous gradients.
x.grad.data.zero_()
y.backward(torch.ones_like(x),retain_graph=True)
d2l.plot(x.detach(), x.grad, 'x', 'grad of tanh', figsize=(5, 2.5))
../_images/output_mlp_76f463_75_0.svg
y.backward()
d2l.plot(x, x.grad, 'x', 'grad of tanh', figsize=(5, 2.5))
../_images/output_mlp_76f463_78_0.svg
with tf.GradientTape() as t:
    y = tf.nn.tanh(x)
d2l.plot(x.numpy(), t.gradient(y, x).numpy(), 'x', 'grad of tanh',
         figsize=(5, 2.5))
../_images/output_mlp_76f463_81_0.svg

5.1.3. Summary

We now know how to incorporate nonlinearities to build expressive multilayer neural network architectures. As a side note, your knowledge already puts you in command of a similar toolkit to a practitioner circa 1990. In some ways, you have an advantage over anyone working in the 1990s, because you can leverage powerful open-source deep learning frameworks to build models rapidly, using only a few lines of code. Previously, training these networks required researchers to code up layers and derivatives explicitly in C, Fortran, or even Lisp (in the case of LeNet).

A secondary benefit is that ReLU is significantly more amenable to optimization than the sigmoid or the tanh function. One could argue that this was one of the key innovations that helped the resurgence of deep learning over the past decade. Note, though, that research in activation functions has not stopped. For instance, the Swish activation function \(\sigma(x) = x \operatorname{sigmoid}(\beta x)\) as proposed in (Ramachandran et al., 2017) can yield better accuracy in many cases.

5.1.4. Exercises

  1. Show that adding layers to a linear deep network, i.e., a network without nonlinearity \(\sigma\) can never increase the expressive power of the network. Give an example where it actively reduces it.

  2. Compute the derivative of the pReLU activation function.

  3. Compute the derivative of the Swish activation function \(x \operatorname{sigmoid}(\beta x)\).

  4. Show that an MLP using only ReLU (or pReLU) constructs a continuous piecewise linear function.

  5. Sigmoid and tanh are very similar.

    1. Show that \(\operatorname{tanh}(x) + 1 = 2 \operatorname{sigmoid}(2x)\).

    2. Prove that the function classes parametrized by both nonlinearities are identical. Hint: affine layers have bias terms, too.

  6. Assume that we have a nonlinearity that applies to one minibatch at a time, such as the batch normalization (Ioffe and Szegedy, 2015). What kinds of problems do you expect this to cause?

  7. Provide an example where the gradients vanish for the sigmoid activation function.