# 3.1. Linear Regression¶

*Regression* refers to a set of methods for modeling the relationship
between one or more independent variables and a dependent variable. In
the natural sciences and social sciences, the purpose of regression is
most often to *characterize* the relationship between the inputs and
outputs. Machine learning, on the other hand, is most often concerned
with *prediction*.

Regression problems pop up whenever we want to predict a numerical value. Common examples include predicting prices (of homes, stocks, etc.), predicting length of stay (for patients in the hospital), demand forecasting (for retail sales), among countless others. Not every prediction problem is a classic regression problem. In subsequent sections, we will introduce classification problems, where the goal is to predict membership among a set of categories.

## 3.1.1. Basic Elements of Linear Regression¶

*Linear regression* may be both the simplest and most popular among the
standard tools to regression. Dating back to the dawn of the 19th
century, linear regression flows from a few simple assumptions. First,
we assume that the relationship between the independent variables
\(\mathbf{x}\) and the dependent variable \(y\) is linear, i.e.,
that \(y\) can be expressed as a weighted sum of the elements in
\(\mathbf{x}\), given some noise on the observations. Second, we
assume that any noise is well-behaved (following a Gaussian
distribution).

To motivate the approach, let us start with a running example. Suppose
that we wish to estimate the prices of houses (in dollars) based on
their area (in square feet) and age (in years). To actually develop a
model for predicting house prices, we would need to get our hands on a
dataset consisting of sales for which we know the sale price, area, and
age for each home. In the terminology of machine learning, the dataset
is called a *training dataset* or *training set*, and each row (here the
data corresponding to one sale) is called an *example* (or *data point*,
*data instance*, *sample*). The thing we are trying to predict (price)
is called a *label* (or *target*). The independent variables (age and
area) upon which the predictions are based are called *features* (or
*covariates*).

Typically, we will use \(n\) to denote the number of examples in our dataset. We index the data examples by \(i\), denoting each input as \(\mathbf{x}^{(i)} = [x_1^{(i)}, x_2^{(i)}]^\top\) and the corresponding label as \(y^{(i)}\).

### 3.1.1.1. Linear Model¶

The linearity assumption just says that the target (price) can be expressed as a weighted sum of the features (area and age):

In (3.1.1), \(w_{\mathrm{area}}\) and
\(w_{\mathrm{age}}\) are called *weights*, and \(b\) is called a
*bias* (also called an *offset* or *intercept*). The weights determine
the influence of each feature on our prediction and the bias just says
what value the predicted price should take when all of the features take
value 0. Even if we will never see any homes with zero area, or that are
precisely zero years old, we still need the bias or else we will limit
the expressivity of our model. Strictly speaking,
(3.1.1) is an *affine transformation* of input
features, which is characterized by a *linear transformation* of
features via weighted sum, combined with a *translation* via the added
bias.

Given a dataset, our goal is to choose the weights \(\mathbf{w}\)
and the bias \(b\) such that on average, the predictions made
according to our model best fit the true prices observed in the data.
Models whose output prediction is determined by the affine
transformation of input features are *linear models*, where the affine
transformation is specified by the chosen weights and bias.

In disciplines where it is common to focus on datasets with just a few features, explicitly expressing models long-form like this is common. In machine learning, we usually work with high-dimensional datasets, so it is more convenient to employ linear algebra notation. When our inputs consist of \(d\) features, we express our prediction \(\hat{y}\) (in general the “hat” symbol denotes estimates) as

Collecting all features into a vector \(\mathbf{x} \in \mathbb{R}^d\) and all weights into a vector \(\mathbf{w} \in \mathbb{R}^d\), we can express our model compactly using a dot product:

In (3.1.3), the vector \(\mathbf{x}\) corresponds to
features of a single data example. We will often find it convenient to
refer to features of our entire dataset of \(n\) examples via the
*design matrix* \(\mathbf{X} \in \mathbb{R}^{n \times d}\). Here,
\(\mathbf{X}\) contains one row for every example and one column for
every feature.

For a collection of features \(\mathbf{X}\), the predictions \(\hat{\mathbf{y}} \in \mathbb{R}^n\) can be expressed via the matrix-vector product:

where broadcasting (see Section 2.1.3) is applied during the summation. Given features of a training dataset \(\mathbf{X}\) and corresponding (known) labels \(\mathbf{y}\), the goal of linear regression is to find the weight vector \(\mathbf{w}\) and the bias term \(b\) that given features of a new data example sampled from the same distribution as \(\mathbf{X}\), the new example’s label will (in expectation) be predicted with the lowest error.

Even if we believe that the best model for predicting \(y\) given \(\mathbf{x}\) is linear, we would not expect to find a real-world dataset of \(n\) examples where \(y^{(i)}\) exactly equals \(\mathbf{w}^\top \mathbf{x}^{(i)}+b\) for all \(1 \leq i \leq n\). For example, whatever instruments we use to observe the features \(\mathbf{X}\) and labels \(\mathbf{y}\) might suffer small amount of measurement error. Thus, even when we are confident that the underlying relationship is linear, we will incorporate a noise term to account for such errors.

Before we can go about searching for the best *parameters* (or *model
parameters*) \(\mathbf{w}\) and \(b\), we will need two more
things: (i) a quality measure for some given model; and (ii) a procedure
for updating the model to improve its quality.

### 3.1.1.2. Loss Function¶

Before we start thinking about how to *fit* data with our model, we need
to determine a measure of *fitness*. The *loss function* quantifies the
distance between the *real* and *predicted* value of the target. The
loss will usually be a non-negative number where smaller values are
better and perfect predictions incur a loss of 0. The most popular loss
function in regression problems is the squared error. When our
prediction for an example \(i\) is \(\hat{y}^{(i)}\) and the
corresponding true label is \(y^{(i)}\), the squared error is given
by:

The constant \(\frac{1}{2}\) makes no real difference but will prove notationally convenient, canceling out when we take the derivative of the loss. Since the training dataset is given to us, and thus out of our control, the empirical error is only a function of the model parameters. To make things more concrete, consider the example below where we plot a regression problem for a one-dimensional case as shown in Fig. 3.1.1.

Note that large differences between estimates \(\hat{y}^{(i)}\) and observations \(y^{(i)}\) lead to even larger contributions to the loss, due to the quadratic dependence. To measure the quality of a model on the entire dataset of \(n\) examples, we simply average (or equivalently, sum) the losses on the training set.

When training the model, we want to find parameters (\(\mathbf{w}^*, b^*\)) that minimize the total loss across all training examples:

### 3.1.1.3. Analytic Solution¶

Linear regression happens to be an unusually simple optimization problem. Unlike most other models that we will encounter in this book, linear regression can be solved analytically by applying a simple formula. To start, we can subsume the bias \(b\) into the parameter \(\mathbf{w}\) by appending a column to the design matrix consisting of all ones. Then our prediction problem is to minimize \(\|\mathbf{y} - \mathbf{X}\mathbf{w}\|^2\). There is just one critical point on the loss surface and it corresponds to the minimum of the loss over the entire domain. Taking the derivative of the loss with respect to \(\mathbf{w}\) and setting it equal to zero yields the analytic (closed-form) solution:

While simple problems like linear regression may admit analytic solutions, you should not get used to such good fortune. Although analytic solutions allow for nice mathematical analysis, the requirement of an analytic solution is so restrictive that it would exclude all of deep learning.

### 3.1.1.4. Minibatch Stochastic Gradient Descent¶

Even in cases where we cannot solve the models analytically, it turns out that we can still train models effectively in practice. Moreover, for many tasks, those difficult-to-optimize models turn out to be so much better that figuring out how to train them ends up being well worth the trouble.

The key technique for optimizing nearly any deep learning model, and
which we will call upon throughout this book, consists of iteratively
reducing the error by updating the parameters in the direction that
incrementally lowers the loss function. This algorithm is called
*gradient descent*.

The most naive application of gradient descent consists of taking the
derivative of the loss function, which is an average of the losses
computed on every single example in the dataset. In practice, this can
be extremely slow: we must pass over the entire dataset before making a
single update. Thus, we will often settle for sampling a random
minibatch of examples every time we need to compute the update, a
variant called *minibatch stochastic gradient descent*.

In each iteration, we first randomly sample a minibatch \(\mathcal{B}\) consisting of a fixed number of training examples. We then compute the derivative (gradient) of the average loss on the minibatch with regard to the model parameters. Finally, we multiply the gradient by a predetermined positive value \(\eta\) and subtract the resulting term from the current parameter values.

We can express the update mathematically as follows (\(\partial\) denotes the partial derivative):

To summarize, steps of the algorithm are the following: (i) we initialize the values of the model parameters, typically at random; (ii) we iteratively sample random minibatches from the data, updating the parameters in the direction of the negative gradient. For quadratic losses and affine transformations, we can write this out explicitly as follows:

Note that \(\mathbf{w}\) and \(\mathbf{x}\) are vectors in
(3.1.10). Here, the more elegant vector
notation makes the math much more readable than expressing things in
terms of coefficients, say \(w_1, w_2, \ldots, w_d\). The set
cardinality \(|\mathcal{B}|\) represents the number of examples in
each minibatch (the *batch size*) and \(\eta\) denotes the *learning
rate*. We emphasize that the values of the batch size and learning rate
are manually pre-specified and not typically learned through model
training. These parameters that are tunable but not updated in the
training loop are called *hyperparameters*. *Hyperparameter tuning* is
the process by which hyperparameters are chosen, and typically requires
that we adjust them based on the results of the training loop as
assessed on a separate *validation dataset* (or *validation set*).

After training for some predetermined number of iterations (or until some other stopping criteria are met), we record the estimated model parameters, denoted \(\hat{\mathbf{w}}, \hat{b}\). Note that even if our function is truly linear and noiseless, these parameters will not be the exact minimizers of the loss because, although the algorithm converges slowly towards the minimizers it cannot achieve it exactly in a finite number of steps.

Linear regression happens to be a learning problem where there is only
one minimum over the entire domain. However, for more complicated
models, like deep networks, the loss surfaces contain many minima.
Fortunately, for reasons that are not yet fully understood, deep
learning practitioners seldom struggle to find parameters that minimize
the loss *on training sets*. The more formidable task is to find
parameters that will achieve low loss on data that we have not seen
before, a challenge called *generalization*. We return to these topics
throughout the book.

### 3.1.1.5. Making Predictions with the Learned Model¶

Given the learned linear regression model
\(\hat{\mathbf{w}}^\top \mathbf{x} + \hat{b}\), we can now estimate
the price of a new house (not contained in the training data) given its
area \(x_1\) and age \(x_2\). Estimating targets given features
is commonly called *prediction* or *inference*.

We will try to stick with *prediction* because calling this step
*inference*, despite emerging as standard jargon in deep learning, is
somewhat of a misnomer. In statistics, *inference* more often denotes
estimating parameters based on a dataset. This misuse of terminology is
a common source of confusion when deep learning practitioners talk to
statisticians.

## 3.1.2. Vectorization for Speed¶

When training our models, we typically want to process whole minibatches of examples simultaneously. Doing this efficiently requires that we vectorize the calculations and leverage fast linear algebra libraries rather than writing costly for-loops in Python.

```
%matplotlib inline
import math
import time
from mxnet import np
from d2l import mxnet as d2l
```

```
%matplotlib inline
import math
import time
import numpy as np
import torch
from d2l import torch as d2l
```

```
%matplotlib inline
import math
import time
import numpy as np
import tensorflow as tf
from d2l import tensorflow as d2l
```

To illustrate why this matters so much, we can consider two methods for
adding vectors. To start we instantiate two 10000-dimensional vectors
containing all ones. In one method we will loop over the vectors with a
Python for-loop. In the other method we will rely on a single call to
`+`

.

```
n = 10000
a = np.ones(n)
b = np.ones(n)
```

```
n = 10000
a = torch.ones(n)
b = torch.ones(n)
```

```
n = 10000
a = tf.ones(n)
b = tf.ones(n)
```

Since we will benchmark the running time frequently in this book, let us define a timer.

```
class Timer: #@save
"""Record multiple running times."""
def __init__(self):
self.times = []
self.start()
def start(self):
"""Start the timer."""
self.tik = time.time()
def stop(self):
"""Stop the timer and record the time in a list."""
self.times.append(time.time() - self.tik)
return self.times[-1]
def avg(self):
"""Return the average time."""
return sum(self.times) / len(self.times)
def sum(self):
"""Return the sum of time."""
return sum(self.times)
def cumsum(self):
"""Return the accumulated time."""
return np.array(self.times).cumsum().tolist()
```

Now we can benchmark the workloads. First, we add them, one coordinate at a time, using a for-loop.

```
c = np.zeros(n)
timer = Timer()
for i in range(n):
c[i] = a[i] + b[i]
f'{timer.stop():.5f} sec'
```

```
'4.45868 sec'
```

```
c = torch.zeros(n)
timer = Timer()
for i in range(n):
c[i] = a[i] + b[i]
f'{timer.stop():.5f} sec'
```

```
'0.09248 sec'
```

```
c = tf.Variable(tf.zeros(n))
timer = Timer()
for i in range(n):
c[i].assign(a[i] + b[i])
f'{timer.stop():.5f} sec'
```

```
'4.70278 sec'
```

Alternatively, we rely on the reloaded `+`

operator to compute the
elementwise sum.

```
timer.start()
d = a + b
f'{timer.stop():.5f} sec'
```

```
'0.00037 sec'
```

```
timer.start()
d = a + b
f'{timer.stop():.5f} sec'
```

```
'0.00028 sec'
```

```
timer.start()
d = a + b
f'{timer.stop():.5f} sec'
```

```
'0.00030 sec'
```

You probably noticed that the second method is dramatically faster than the first. Vectorizing code often yields order-of-magnitude speedups. Moreover, we push more of the mathematics to the library and need not write as many calculations ourselves, reducing the potential for errors.

## 3.1.3. The Normal Distribution and Squared Loss¶

While you can already get your hands dirty using only the information above, in the following we can more formally motivate the squared loss objective via assumptions about the distribution of noise.

Linear regression was invented by Gauss in 1795, who also discovered the
normal distribution (also called the *Gaussian*). It turns out that the
connection between the normal distribution and linear regression runs
deeper than common parentage. To refresh your memory, the probability
density of a normal distribution with mean \(\mu\) and variance
\(\sigma^2\) (standard deviation \(\sigma\)) is given as

Below we define a Python function to compute the normal distribution.

```
def normal(x, mu, sigma):
p = 1 / math.sqrt(2 * math.pi * sigma**2)
return p * np.exp(-0.5 / sigma**2 * (x - mu)**2)
```

We can now visualize the normal distributions.

```
# Use numpy again for visualization
x = np.arange(-7, 7, 0.01)
# Mean and standard deviation pairs
params = [(0, 1), (0, 2), (3, 1)]
d2l.plot(x, [normal(x, mu, sigma) for mu, sigma in params], xlabel='x',
ylabel='p(x)', figsize=(4.5, 2.5),
legend=[f'mean {mu}, std {sigma}' for mu, sigma in params])
```

```
# Use numpy again for visualization
x = np.arange(-7, 7, 0.01)
# Mean and standard deviation pairs
params = [(0, 1), (0, 2), (3, 1)]
d2l.plot(x, [normal(x, mu, sigma) for mu, sigma in params], xlabel='x',
ylabel='p(x)', figsize=(4.5, 2.5),
legend=[f'mean {mu}, std {sigma}' for mu, sigma in params])
```

```
# Use numpy again for visualization
x = np.arange(-7, 7, 0.01)
# Mean and standard deviation pairs
params = [(0, 1), (0, 2), (3, 1)]
d2l.plot(x, [normal(x, mu, sigma) for mu, sigma in params], xlabel='x',
ylabel='p(x)', figsize=(4.5, 2.5),
legend=[f'mean {mu}, std {sigma}' for mu, sigma in params])
```

As we can see, changing the mean corresponds to a shift along the \(x\)-axis, and increasing the variance spreads the distribution out, lowering its peak.

One way to motivate linear regression with the mean squared error loss function (or simply squared loss) is to formally assume that observations arise from noisy observations, where the noise is normally distributed as follows:

Thus, we can now write out the *likelihood* of seeing a particular
\(y\) for a given \(\mathbf{x}\) via

Now, according to the principle of maximum likelihood, the best values
of parameters \(\mathbf{w}\) and \(b\) are those that maximize
the *likelihood* of the entire dataset:

Estimators chosen according to the principle of maximum likelihood are
called *maximum likelihood estimators*. While, maximizing the product of
many exponential functions, might look difficult, we can simplify things
significantly, without changing the objective, by maximizing the log of
the likelihood instead. For historical reasons, optimizations are more
often expressed as minimization rather than maximization. So, without
changing anything we can minimize the *negative log-likelihood*
\(-\log P(\mathbf y \mid \mathbf X)\). Working out the mathematics
gives us:

Now we just need one more assumption that \(\sigma\) is some fixed constant. Thus we can ignore the first term because it does not depend on \(\mathbf{w}\) or \(b\). Now the second term is identical to the squared error loss introduced earlier, except for the multiplicative constant \(\frac{1}{\sigma^2}\). Fortunately, the solution does not depend on \(\sigma\). It follows that minimizing the mean squared error is equivalent to maximum likelihood estimation of a linear model under the assumption of additive Gaussian noise.

## 3.1.4. From Linear Regression to Deep Networks¶

So far we only talked about linear models. While neural networks cover a much richer family of models, we can begin thinking of the linear model as a neural network by expressing it in the language of neural networks. To begin, let us start by rewriting things in a “layer” notation.

### 3.1.4.1. Neural Network Diagram¶

Deep learning practitioners like to draw diagrams to visualize what is happening in their models. In Fig. 3.1.2, we depict our linear regression model as a neural network. Note that these diagrams highlight the connectivity pattern such as how each input is connected to the output, but not the values taken by the weights or biases.

For the neural network shown in Fig. 3.1.2, the
inputs are \(x_1, \ldots, x_d\), so the *number of inputs* (or
*feature dimensionality*) in the input layer is \(d\). The output of
the network in Fig. 3.1.2 is \(o_1\), so the
*number of outputs* in the output layer is 1. Note that the input values
are all *given* and there is just a single *computed* neuron. Focusing
on where computation takes place, conventionally we do not consider the
input layer when counting layers. That is to say, the *number of layers*
for the neural network in Fig. 3.1.2 is 1. We can
think of linear regression models as neural networks consisting of just
a single artificial neuron, or as single-layer neural networks.

Since for linear regression, every input is connected to every output
(in this case there is only one output), we can regard this
transformation (the output layer in Fig. 3.1.2) as a
*fully-connected layer* or *dense layer*. We will talk a lot more about
networks composed of such layers in the next chapter.

### 3.1.4.2. Biology¶

Since linear regression (invented in 1795) predates computational
neuroscience, it might seem anachronistic to describe linear regression
as a neural network. To see why linear models were a natural place to
begin when the cyberneticists/neurophysiologists Warren McCulloch and
Walter Pitts began to develop models of artificial neurons, consider the
cartoonish picture of a biological neuron in Fig. 3.1.3,
consisting of *dendrites* (input terminals), the *nucleus* (CPU), the
*axon* (output wire), and the *axon terminals* (output terminals),
enabling connections to other neurons via *synapses*.

Information \(x_i\) arriving from other neurons (or environmental
sensors such as the retina) is received in the dendrites. In particular,
that information is weighted by *synaptic weights* \(w_i\)
determining the effect of the inputs (e.g., activation or inhibition via
the product \(x_i w_i\)). The weighted inputs arriving from multiple
sources are aggregated in the nucleus as a weighted sum
\(y = \sum_i x_i w_i + b\), and this information is then sent for
further processing in the axon \(y\), typically after some nonlinear
processing via \(\sigma(y)\). From there it either reaches its
destination (e.g., a muscle) or is fed into another neuron via its
dendrites.

Certainly, the high-level idea that many such units could be cobbled together with the right connectivity and right learning algorithm, to produce far more interesting and complex behavior than any one neuron alone could express owes to our study of real biological neural systems.

At the same time, most research in deep learning today draws little
direct inspiration in neuroscience. We invoke Stuart Russell and Peter
Norvig who, in their classic AI text book *Artificial Intelligence: A
Modern Approach* [Russell & Norvig, 2016], pointed out that
although airplanes might have been *inspired* by birds, ornithology has
not been the primary driver of aeronautics innovation for some
centuries. Likewise, inspiration in deep learning these days comes in
equal or greater measure from mathematics, statistics, and computer
science.

## 3.1.5. Summary¶

Key ingredients in a machine learning model are training data, a loss function, an optimization algorithm, and quite obviously, the model itself.

Vectorizing makes everything better (mostly math) and faster (mostly code).

Minimizing an objective function and performing maximum likelihood estimation can mean the same thing.

Linear regression models are neural networks, too.

## 3.1.6. Exercises¶

Assume that we have some data \(x_1, \ldots, x_n \in \mathbb{R}\). Our goal is to find a constant \(b\) such that \(\sum_i (x_i - b)^2\) is minimized.

Find a analytic solution for the optimal value of \(b\).

How does this problem and its solution relate to the normal distribution?

Derive the analytic solution to the optimization problem for linear regression with squared error. To keep things simple, you can omit the bias \(b\) from the problem (we can do this in principled fashion by adding one column to \(\mathbf X\) consisting of all ones).

Write out the optimization problem in matrix and vector notation (treat all the data as a single matrix, and all the target values as a single vector).

Compute the gradient of the loss with respect to \(w\).

Find the analytic solution by setting the gradient equal to zero and solving the matrix equation.

When might this be better than using stochastic gradient descent? When might this method break?

Assume that the noise model governing the additive noise \(\epsilon\) is the exponential distribution. That is, \(p(\epsilon) = \frac{1}{2} \exp(-|\epsilon|)\).

Write out the negative log-likelihood of the data under the model \(-\log P(\mathbf y \mid \mathbf X)\).

Can you find a closed form solution?

Suggest a stochastic gradient descent algorithm to solve this problem. What could possibly go wrong (hint: what happens near the stationary point as we keep on updating the parameters)? Can you fix this?