4.1. Multilayer Perceptrons¶ Open the notebook in SageMaker Studio Lab
In Section 3, we introduced softmax regression (Section 3.4), implementing the algorithm from scratch (Section 3.6) and using high-level APIs (Section 3.7), and training classifiers to recognize 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.
4.1.2. 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 us briefly survey some common activation functions.
%matplotlib inline
from mxnet import autograd, np, npx
from d2l import mxnet as d2l
npx.set_np()
%matplotlib inline
import torch
from d2l import torch as d2l
%matplotlib inline
import tensorflow as tf
from d2l import tensorflow as d2l
4.1.2.1. 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). ReLU provides a very simple nonlinear transformation. Given an element \(x\), the function is defined as the maximum of that element and \(0\):
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 = 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))
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))
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))
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. 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. We plot the derivative of the ReLU function plotted below.
y.backward()
d2l.plot(x, x.grad, 'x', 'grad of relu', figsize=(5, 2.5))
[19:46:50] src/base.cc:49: GPU context requested, but no GPUs found.
y.backward(torch.ones_like(x), retain_graph=True)
d2l.plot(x.detach(), x.grad, 'x', 'grad of relu', figsize=(5, 2.5))
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))
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 (). This variation adds a linear term to ReLU, so some information still gets through, even when the argument is negative:
4.1.2.2. 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):
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. 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. In later chapters 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.
with autograd.record():
y = npx.sigmoid(x)
d2l.plot(x, y, 'x', 'sigmoid(x)', figsize=(5, 2.5))
y = torch.sigmoid(x)
d2l.plot(x.detach(), y.detach(), 'x', 'sigmoid(x)', figsize=(5, 2.5))
y = tf.nn.sigmoid(x)
d2l.plot(x.numpy(), y.numpy(), 'x', 'sigmoid(x)', figsize=(5, 2.5))
The derivative of the sigmoid function is given by the following equation:
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.
y.backward()
d2l.plot(x, x.grad, 'x', 'grad of sigmoid', figsize=(5, 2.5))
# 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))
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))
4.1.2.3. 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:
We plot the tanh function below. Note that as the 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.
with autograd.record():
y = np.tanh(x)
d2l.plot(x, y, 'x', 'tanh(x)', figsize=(5, 2.5))
y = torch.tanh(x)
d2l.plot(x.detach(), y.detach(), 'x', 'tanh(x)', figsize=(5, 2.5))
y = tf.nn.tanh(x)
d2l.plot(x.numpy(), y.numpy(), 'x', 'tanh(x)', figsize=(5, 2.5))
The derivative of the tanh function is:
The derivative of tanh function 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 the input moves away from 0 in either direction, the derivative of the tanh function approaches 0.
y.backward()
d2l.plot(x, x.grad, 'x', 'grad of tanh', figsize=(5, 2.5))
# 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))
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))
In 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 thousands of lines of C and Fortran.
4.1.3. Summary¶
MLP adds one or multiple fully-connected hidden layers between the output and input layers and transforms the output of the hidden layer via an activation function.
Commonly-used activation functions include the ReLU function, the sigmoid function, and the tanh function.
4.1.4. Exercises¶
Compute the derivative of the pReLU activation function.
Show that an MLP using only ReLU (or pReLU) constructs a continuous piecewise linear function.
Show that \(\operatorname{tanh}(x) + 1 = 2 \operatorname{sigmoid}(2x)\).
Assume that we have a nonlinearity that applies to one minibatch at a time. What kinds of problems do you expect this to cause?