# 3.1. Linear Regression¶ Open the notebook in SageMaker Studio Lab

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

As 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 develop a model for predicting house prices, we need to get
our hands on data consisting of sales, including the sales 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
(containing the data corresponding to one sale) is called an *example*
(or *data point*, *instance*, *sample*). The thing we are trying to
predict (price) is called a *label* (or *target*). The variables (age
and area) upon which the predictions are based are called *features* (or
*covariates*).

## 3.1.1. Basics¶

*Linear regression* may be both the simplest and most popular among the
standard tools for tackling regression problems. Dating back to the dawn
of the 19th century [Gauss, 1809, Legendre, 1805], linear
regression flows from a few simple assumptions. First, we assume that
the relationship between features \(\mathbf{x}\) and target
\(y\) is approximately linear, i.e., that the conditional mean
\(E[Y|X=\mathbf{x}]\) can be expressed as a weighted sum of the
features \(\mathbf{x}\). This setup allows that the target value may
still deviate from its expected value on account of observation noise.
Next, we can impose the assumption that any such noise is well-behaved,
following a Gaussian distribution. Typically, we will use \(n\) to
denote the number of examples in our dataset. We use superscripts to
enumerate samples and targets, and subscripts to index coordinates. More
concretely, \(\mathbf{x}^{(i)}\) denotes the \(i\)-th sample and
\(x_j^{(i)}\) denotes its \(j\)-th coordinate.

### 3.1.1.1. Model¶

At the heart of every solution is a model that describes how features can be transformed into an estimate of the target. The assumption of linearity means that the expected value of the target (price) can be expressed as a weighted sum of the features (area and age):

Here \(w_{\mathrm{area}}\) and \(w_{\mathrm{age}}\) are called
*weights*, and \(b\) is called a *bias* (or *offset* or
*intercept*). The weights determine the influence of each feature on our
prediction. The bias determines the value of the estimate when all
features are zero. Even though we will never see any newly-built homes
with precisely zero area, we still need the bias because it allows us to
express all linear functions of our features (versus restricting us to
lines that pass through the origin). 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 \(b\) that, on average, make our
model’s predictions fit the true prices observed in the data as closely
as possible.

In disciplines where it is common to focus on datasets with just a few features, explicitly expressing models long-form, as in (3.1.1), is common. In machine learning, we usually work with high-dimensional datasets, where it’s more convenient to employ compact linear algebra notation. When our inputs consist of \(d\) features, we can assign each an index (between \(1\) and \(d\)) and express our prediction \(\hat{y}\) (in general the “hat” symbol denotes an estimate) 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 via the dot product between \(\mathbf{w}\) and \(\mathbf{x}\):

In (3.1.3), the vector \(\mathbf{x}\) corresponds to
the features of a single 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 (Section 2.1.4) 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¶

Naturally, fitting our model to the data requires that we agree on some
measure of *fitness* (or, equivalently, of *unfitness*). *Loss
functions* quantify the distance between the *real* and *predicted*
values 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. For regression problems, the most common loss function is 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 proves to be notationally convenient, since it cancels out when we take the derivative of the loss. Because the training dataset is given to us, and thus out of our control, the empirical error is only a function of the model parameters. Below, we visualize the fit of a linear regression model in a problem with one-dimensional inputs (Fig. 3.1.1).

Note that large differences between estimates \(\hat{y}^{(i)}\) and targets \(y^{(i)}\) lead to even larger contributions to the loss, due to the quadratic form of the loss (this can be a double-edge sword. While it encourages the model to avoid large errors it can also lead to excessive sensitivity to anomalous data). 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¶

Unlike most of the models that we will cover, linear regression presents us with a surprisingly easy optimization problem. In particular, we can find the optimal parameters (as assessed on the training data) analytically by applying a simple formula as follows. First, 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 \(\|\mathbf{y} - \mathbf{X}\mathbf{w}\|^2\). So long as the design matrix \(\mathbf{X}\) has full rank (no feature is linearly dependent on the others), then there will be 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:

Solving for \(\mathbf{w}\) provides us with the optimal solution for the optimization problem. Note that this solution

will only be unique when the matrix \(\mathbf X^\top \mathbf X\) is invertible, i.e., when the columns of the design matrix are linearly independent [Golub & VanLoan, 1996].

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 almost all exciting aspects of deep learning.

### 3.1.1.4. Minibatch Stochastic Gradient Descent¶

Fortunately, even in cases where we cannot solve the models analytically, we can still often 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, even if the update steps might be very powerful [Liu & Nocedal, 1989]. Even worse, if there is a lot of redundancy in the training data, the benefit of a full update is even lower.

The other extreme is to consider only a single example at a time and to
take update steps based on one observation at a time. The resulting
algorithm, *stochastic gradient descent* (SGD) can be an effective
strategy [Bottou, 2010], even for large datasets. Unfortunately,
SGD has drawbacks, both computational and statistical. One problem
arises from the fact that processors are a lot faster multiplying and
adding numbers than they are at moving data from main memory to
processor cache. It is up to an order of magnitude more efficient to
perform a matrix-vector multiplication than a corresponding number of
vector-vector operations. This means that it can take a lot longer to
process one sample at a time compared to a full batch. A second problem
is that some of the layers, such as batch normalization (to be described
in Section 8.5), only work well when we have access to
more than one observation at a time.

The solution to both problems is to pick an intermediate strategy:
rather than taking a full batch or only a single sample at a time, we
take a *minibatch* of observations [Li et al., 2014b]. The
specific choice of the size of the said minibatch depends on many
factors, such as the amount of memory, the number of accelerators, the
choice of layers, and the total dataset size. Despite all of that, a
number between 32 and 256, preferably a multiple of a large power of
\(2\), is a good start. This leads us to *minibatch stochastic
gradient descent*.

In its most basic form, in each iteration \(t\), we first randomly
sample a minibatch \(\mathcal{B}_t\) consisting of a fixed number
\(|\mathcal{B}|\) of training examples. We then compute the
derivative (gradient) of the average loss on the minibatch with respect
to the model parameters. Finally, we multiply the gradient by a
predetermined small positive value \(\eta\), called the *learning
rate*, and subtract the resulting term from the current parameter
values. We can express the update as follows:

In summary, minibatch SGD proceeds as follows: (i) initialize the values of the model parameters, typically at random; (ii) iteratively sample random minibatches from the data, updating the parameters in the direction of the negative gradient. For quadratic losses and affine transformations, this has a closed-form expansion:

Since we pick a minibatch \(\mathcal{B}\) we need to normalize by
its size \(|\mathcal{B}|\). Frequently minibatch size and learning
rate are user-defined. Such tunable parameters that are not updated in
the training loop are called *hyperparameters*. They can be tuned
automatically by a number of techniques, such as Bayesian optimization
[Frazier, 2018]. In the end, the quality of the solution is
typically assessed on a separate *validation dataset* (or *validation
set*).

After training for some predetermined number of iterations (or until some other stopping criterion is 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, or even deterministic. Although the algorithm converges slowly towards the minimizers it typically cannot achieve it exactly in a finite number of steps. Moreover, the minibatches \(\mathcal{B}\) used to update the parameters are chosen at random. This breaks determinism.

Linear regression happens to be a learning problem with a global minimum
(whenever \(\mathbf{X}\) is full rank, or equivalently, whenever
\(\mathbf{X}^\top \mathbf{X}\) is invertible). However, the lost
surfaces for deep networks contain many saddle points and minima.
Fortunately, we typically don’t care about finding an exact set of
parameters but merely any set of parameters that leads to accurate
predictions (and thus low loss). In practice, deep learning
practitioners seldom struggle to find parameters that minimize the loss
*on training sets*
[Frankle & Carbin, 2018, Izmailov et al., 2018]. The
more formidable task is to find parameters that lead to accurate
predictions on previously unseen data, a challenge called
*generalization*. We return to these topics throughout the book.

### 3.1.1.5. Predictions¶

Given the model \(\hat{\mathbf{w}}^\top \mathbf{x} + \hat{b}\), we
can now make *predictions* for a new example, e.g., to predict the sales
price of a previously unseen house given its area \(x_1\) and age
\(x_2\). Deep learning practitioners have taken to calling the
prediction phase *inference* but this is a bit of a
misnomer—*inference* refers broadly to any conclusion reached on the
basis of evidence, including both the values of the parameters and the
likely label for an unseen instance. If anything, in the statistics
literature *inference* more often denotes parameter inference and this
overloading of terminology creates unnecessary confusion when deep
learning practitioners talk to statisticians. In the following we will
stick to *prediction* whenever possible.

## 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
import numpy as np
import torch
from d2l import torch as d2l
```

```
%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 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 10,000-dimensional vectors
containing all ones. In one method, we loop over the vectors with a
Python for-loop. In the other method, we rely on a single call to `+`

.

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

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

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

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

```
c = torch.zeros(n)
t = time.time()
for i in range(n):
c[i] = a[i] + b[i]
f'{time.time() - t:.5f} sec'
```

```
'0.11987 sec'
```

```
c = np.zeros(n)
t = time.time()
for i in range(n):
c[i] = a[i] + b[i]
f'{time.time() - t:.5f} sec'
```

```
'4.66569 sec'
```

```
c = tf.Variable(tf.zeros(n))
t = time.time()
for i in range(n):
c[i].assign(a[i] + b[i])
f'{time.time() - t:.5f} sec'
```

```
'9.52192 sec'
```

Alternatively, we rely on the reloaded `+`

operator to compute the
elementwise sum.

```
t = time.time()
d = a + b
f'{time.time() - t:.5f} sec'
```

```
'0.00017 sec'
```

```
t = time.time()
d = a + b
f'{time.time() - t:.5f} sec'
```

```
'0.00054 sec'
```

```
t = time.time()
d = a + b
f'{time.time() - t:.5f} sec'
```

```
'0.00033 sec'
```

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 and increasing portability of the code.

## 3.1.3. The Normal Distribution and Squared Loss¶

So far we’ve given a fairly functional motivation of the squared loss objective: the optimal parameters return the conditional expectation \(E[Y|X]\) whenever the underlying pattern is truly linear, and the loss assigns outsize penalties for outliers. We can also provide a more formal motivation for the squared loss objective by making probabilistic assumptions about the distribution of noise.

Linear regression was invented at the turn of the 19th century. While it
has long been debated whether Gauss or Legendre first thought up the
idea, it was Gauss who also discovered the normal distribution (also
called the *Gaussian*). It turns out that the normal distribution and
linear regression with squared loss share a deeper connection than
common parentage.

To begin, recall that a normal distribution with mean \(\mu\) and variance \(\sigma^2\) (standard deviation \(\sigma\)) is given as

Below we define a 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 * (x - mu)**2 / sigma**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.asnumpy(), [normal(x, mu, sigma).asnumpy() 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])
```

Note that 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 squared loss is to assume that observations arise from noisy measurements, 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

As such, the likelihood factorizes. 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:

The equality follows since all pairs \((\mathbf{x}^{(i)}, y^{(i)})\)
were drawn independently of each other. 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 logarithm 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*, which we can
express as follows:

If we assume that \(\sigma\) is fixed, we can ignore the first term, because it does not depend on \(\mathbf{w}\) or \(b\). 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 \(\sigma\) either. 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. Linear Regression as a Neural Network¶

While linear models are not sufficiently rich to express the many complicated neural networks that we will introduce in this book, neural networks are rich enough to subsume linear models as neural networks in which every feature is represented by an input neuron, all of which are connected directly to the output.

Fig. 3.1.2 depicts linear regression as a neural network. The diagram highlights the connectivity pattern such as how each input is connected to the output, but not the specific values taken by the weights or biases.

The inputs are \(x_1, \ldots, x_d\). We refer to \(d\) as the
*number of inputs* or *feature dimensionality* in the input layer. The
output of the network is \(o_1\). Because we are just trying to
predict a single numerical value, we have only one output neuron. Note
that the input values are all *given*. There is just a single *computed*
neuron. In summary, we can think of linear regression as a single-layer
fully connected neural network. We will encounter networks with far more
layers in future chapters.

### 3.1.4.1. Biology¶

Because linear regression predates computational neuroscience, it might
seem anachronistic to describe linear regression in terms of neural
networks. Nonetheless, they were a natural place to start when the
cyberneticists and 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) 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
\(y = \sum_i x_i w_i + b\), possibly subject to some nonlinear
postprocessing via \(\sigma(y)\). This information is then sent via
the axon to the axon terminals, where it reaches its destination (e.g.,
an actuator such as a muscle) or it is fed into another neuron via its
dendrites.

Certainly, the high-level idea that many such units could be combined
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 inspiration from a much
wider source. We invoke Stuart Russell and Peter Norvig
[Russell & Norvig, 2016] who 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, linguistics, psychology, statistics, computer
science, and many other fields.

## 3.1.5. Summary¶

In this section, we introduced traditional linear regression, where the parameters of a linear function are chosen to minimize squared loss on the training set. We also motivated this choice of objective both via some practical considerations and through an interpretation of linear regression as maximimum likelihood estimation under an assumption of linearity and Gaussian noise. After discussing both computational considerations and connections to statistics, we showed how such linear models could be expressed as simple neural networks where the inputs are directly wired to the output(s). While we will soon move past linear models altogether, they are sufficient to introduce most of the components that all of our models require: parametric forms, differentiable objectives, optimization via minibatch stochastic gradient descent, and ultimately, evaluation on previously unseen data.

## 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 an analytic solution for the optimal value of \(b\).

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

What if we change the loss from \(\sum_i (x_i - b)^2\) to \(\sum_i |x_i-b|\)? Can you find the optimal solution for \(b\)?

Prove that the affine functions that can be expressed by \(\mathbf{x}^\top \mathbf{w} + b\) are equivalent to linear functions on \((\mathbf{x}, 1)\).

Assume that you want to find quadratic functions of \(\mathbf{x}\), i.e., \(f(\mathbf{x}) = b + \sum_i w_i x_i + \sum_{j \leq i} w_{ij} x_{i} x_{j}\). How would you formulate this in a deep network?

Recall that one of the conditions for the linear regression problem to be solvable was that the design matrix \(\mathbf{X}^\top \mathbf{X}\) has full rank.

What happens if this is not the case?

How could you fix it? What happens if you add a small amount of coordinate-wise independent Gaussian noise to all entries of \(\mathbf{X}\)?

What is the expected value of the design matrix \(\mathbf{X}^\top \mathbf{X}\) in this case?

What happens with stochastic gradient descent when \(\mathbf{X}^\top \mathbf{X}\) doesn’t have full rank?

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 minibatch 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?

Assume that we want to design a neural network with two layers by composing two linear layers. That is, the output of the first layer becomes the input of the second layer. Why would such a naive composition not work?

What happens if you want to use regression for realistic price estimation of houses or stock prices?

Show that the additive Gaussian noise assumption is not appropriate. Hint: can we have negative prices? What about fluctuations?

Why would regression to the logarithm of the price be much better, i.e., \(y = \log \text{price}\)?

What do you need to worry about when dealing with pennystock, i.e., stock with very low prices? Hint: can you trade at all possible prices? Why is this a bigger problem for cheap stock?

For more information review the celebrated Black-Scholes model for option pricing [Black & Scholes, 1973].

Suppose we want to use regression to estimate the

*number*of apples sold in a grocery store.What are the problems with a Gaussian additive noise model? Hint: you are selling apples, not oil.

The Poisson distribution captures distributions over counts. It is given by \(p(k|\lambda) = \lambda^k e^{-\lambda}/k!\). Here \(\lambda\) is the rate function and \(k\) is the number of events you see. Prove that \(\lambda\) is the expected value of counts \(k\).

Design a loss function associated with the Poisson distribution.

Design a loss function for estimating \(\log \lambda\) instead.