# 3.6. Generalization¶ Open the notebook in SageMaker Studio Lab

Consider two college students diligently preparing for their final exam.
Commonly, this preparation will consist of practicing and testing their
abilities by taking exams administered in previous years. Nonetheless,
doing well on past exams is no guarantee that they will excel when it
matters. For instance, imagine a student, Elephantine Ellie, whose
preparation consisted entirely of memorizing the answers to previous
years’ exam questions. Even if Ellie were endowed with an elephantine
memory, and thus could perfectly recall the answer to any *previously
seen* question, she might nevertheless freeze when faced with a new
(*previously unseen*) question. By comparison, imagine another student,
Inductive Irene, with comparably poor memorization skills, but a knack
for picking up patterns. Note that if the exam truly consisted of
recycled questions from a previous year, Ellie would handily outperform
Irene. Even if Irene’s inferred patterns yielded 90% accurate
predictions, they could never compete with Ellie’s 100% recall. However,
even if the exam consisted entirely of fresh questions, Irene might
maintain her 90% average.

As machine learning scientists, our goal is to discover *patterns*. But
how can we be sure that we have truly discovered a *general* pattern and
not simply memorized our data? Most of the time, our predictions are
only useful if our model discovers such a pattern. We do not want to
predict yesterday’s stock prices, but tomorrow’s. We do not need to
recognize already diagnosed diseases for previously seen patients, but
rather previously undiagnosed ailments in previously unseen patients.
This problem—how to discover patterns that *generalize*—is the
fundamental problem of machine learning, and arguably of all of
statistics. We might cast this problem as just one slice of a far
grander question that engulfs all of science: when are we ever justified
in making the leap from particular observations to more general
statements (Popper, 2005)?

In real life, we must fit out models using a finite collection of data.
The typical scales of that data vary wildly across domains. For many
important medical problem, we can only access a few thousand data
points. When studying rare diseases, we might be lucky to access
hundreds. By contrast, the largest public datasets consisting of labeled
photographs (e.g., ImageNet (Deng *et al.*, 2009)), contain
millions of images. And some unlabeled image collections such as the
Flickr YFC100M dataset can be even larger, containing over 100 million
images (Thomee *et al.*, 2016). However, even at this extreme
scale, the number of available data points remains infinitesimally small
compared to the space of all possible images at 1 megapixel resolution.
Whenever we work with finite samples, we must keep in mind the risk that
we might fit our training data, only to discover that we failed to
discover a generalizable pattern.

The phenomenon of fitting closer to our training data than to the
underlying distribution is called *overfitting*, and techniques for
combatting overfitting are often called *regularization* methods. While
there is no substitute for a proper introduction to statistical learning
theory (see Boucheron *et al.* (2005), Vapnik (1998)), we will give you
just enough intuition to get going. We will revisit generalization in
many chapters throughout the book, exploring both what is known about
the principles underlying generalization in various models, and also
heuristic techniques that have been found (empirically) to yield
improved generalization on tasks of practical interest.

## 3.6.1. Training Error and Generalization Error¶

In the standard supervised learning setting, we assume that the training
data and the test data are drawn *independently* from *identical*
distributions. This is commonly called the *IID assumption*. While this
assumption is strong, it is worth noting that absent any such assumption
we would be dead in the water. Why should we believe that training data
sampled from distribution \(P(X,Y)\) should tell us how to make
predictions on test data generated by a *different distribution*
\(Q(X,Y)\)? Making such leaps turns out to require strong
assumptions about how \(P\) and \(Q\) are related. Later on we
will discuss some assumptions that allow for shifts in distribution but
first we need to understand the IID case, where
\(P(\cdot) = Q(\cdot)\).

To begin with, we need to differentiate between the *training error*
\(R_\text{emp}\), which is a *statistic* calculated on the training
dataset, and the *generalization error* \(R\), which is an
*expectation* taken with respect to the underlying distribution. You can
think of the generalization error as what you would see if you applied
your model to an infinite stream of additional data examples drawn from
the same underlying data distribution. Formally the training error is
expressed as a *sum* (with the same notation in
Section 3.1):

while the generalization error is expressed as an integral:

Problematically, we can never calculate the generalization error
\(R\) exactly. Nobody ever tells us the precise form of the density
function \(p(\mathbf{x}, y)\). Moreover, we cannot sample an
infinite stream of data points. Thus, in practice, we must *estimate*
the generalization error by applying our model to an independent test
set constituted of a random selection of examples \(\mathbf{X}'\)
and labels \(\mathbf{y}'\) that were withheld from our training set.
This consists of applying the same formula as for calculating the
empirical training error but to a test set
\(\mathbf{X}', \mathbf{y}'\).

Crucially, when we evaluate our classifier on the test set, we are
working with a *fixed* classifier (it does not depend on the sample of
the test set), and thus estimating its error is simply the problem of
mean estimation. However the same cannot be said for the training set.
Note that the model we wind up with depends explicitly on the selection
of the training set and thus the training error will in general be a
biased estimate of the true error on the underlying population. The
central question of generalization is then when should we expect our
training error to be close to the population error (and thus the
generalization error).

### 3.6.1.1. Model Complexity¶

In classical theory, when we have simple models and abundant data, the training and generalization errors tend to be close. However, when we work with more complex models and/or fewer examples, we expect the training error to go down but the generalization gap to grow. This should not be surprising. Imagine a model class so expressive that for any dataset of \(n\) examples, we can find a set of parameters that can perfectly fit arbitrary labels, even if randomly assigned. In this case, even if we fit our training data perfectly, how can we conclude anything about the generalization error? For all we know, our generalization error might be no better than random guessing.

In general, absent any restriction on our model class, we cannot
conclude based on fitting the training data alone that our model has
discovered any generalizable pattern (Vapnik *et al.*, 1994). On
the other hand, if our model class was not capable of fitting arbitrary
labels, then it must have discovered a pattern. Learning-theoretic ideas
about model complexity derived some inspiration from the ideas of Karl
Popper, an influential philosopher of science, who formalized the
criterion of falsifiability. According to Popper, a theory that can
explain any and all observations is not a scientific theory at all!
After all, what has it told us about the world if it has not ruled out
any possibility? In short, what we want is a hypothesis that *could not*
explain any observations we might conceivably make and yet nevertheless
happens to be compatible with those observations that we *in fact* make.

Now what precisely constitutes an appropriate notion of model complexity
is a complex matter. Often, models with more parameters are able to fit
a greater number of arbitrarily assigned labels. However, this is not
necessarily true. For instance, kernel methods operate in spaces with
infinite numbers of parameters, yet their complexity is controlled by
other means (Scholkopf and Smola, 2002). One notion of complexity
that often proves useful is the range of values that the parameters can
take. Here, a model whose parameters are permitted to take arbitrary
values would be more complex. We will revisit this idea in the next
section, when we introduce *weight decay*, your first practical
regularization technique. Notably, it can be difficult to compare
complexity among members of substantially different model classes (say,
decision trees vs. neural networks).

At this point, we must stress another important point that we will
revisit when introducing deep neural networks. When a model is capable
of fitting arbitrary labels, low training error does not necessarily
imply low generalization error. *However, it does not necessarily imply
high generalization error either!* All we can say confidently is that
low training error alone is not enough to certify low generalization
error. Deep neural networks turn out to be just such models: while they
generalize well in practice, they are too powerful to allow us to
conclude much on the basis of training error alone. In these cases we
must rely more heavily on our holdout data to certify generalization
after the fact. Error on the holdout data, i.e., validation set, is
called the *validation error*.

## 3.6.2. Underfitting or Overfitting?¶

When we compare the training and validation errors, we want to be
mindful of two common situations. First, we want to watch out for cases
when our training error and validation error are both substantial but
there is a little gap between them. If the model is unable to reduce the
training error, that could mean that our model is too simple (i.e.,
insufficiently expressive) to capture the pattern that we are trying to
model. Moreover, since the *generalization gap*
(\(R_\text{emp} - R\)) between our training and generalization
errors is small, we have reason to believe that we could get away with a
more complex model. This phenomenon is known as *underfitting*.

On the other hand, as we discussed above, we want to watch out for the
cases when our training error is significantly lower than our validation
error, indicating severe *overfitting*. Note that overfitting is not
always a bad thing. In deep learning especially, the best predictive
models often perform far better on training data than on holdout data.
Ultimately, we usually care about driving the generalization error
lower, and only care about the gap insofar as it becomes an obstacle to
that end. Note that if the training error is zero, then the
generalization gap is precisely equal to the generalization error and we
can make progress only by reducing the gap.

### 3.6.2.1. Polynomial Curve Fitting¶

To illustrate some classical intuition about overfitting and model complexity, consider the following: given training data consisting of a single feature \(x\) and a corresponding real-valued label \(y\), we try to find the polynomial of degree \(d\)

to estimate the label \(y\). This is just a linear regression problem where our features are given by the powers of \(x\), the model’s weights are given by \(w_i\), and the bias is given by \(w_0\) since \(x^0 = 1\) for all \(x\). Since this is just a linear regression problem, we can use the squared error as our loss function.

A higher-order polynomial function is more complex than a lower-order polynomial function, since the higher-order polynomial has more parameters and the model function’s selection range is wider. Fixing the training dataset, higher-order polynomial functions should always achieve lower (at worst, equal) training error relative to lower degree polynomials. In fact, whenever each data example has a distinct value of \(x\), a polynomial function with degree equal to the number of data examples can fit the training set perfectly. We visualize the relationship between polynomial degree (model complexity) and underfitting vs. overfitting in Fig. 3.6.1.

### 3.6.2.2. Dataset Size¶

As the above bound already indicates, another big consideration to bear in mind is dataset size. Fixing our model, the fewer samples we have in the training dataset, the more likely (and more severely) we are to encounter overfitting. As we increase the amount of training data, the generalization error typically decreases. Moreover, in general, more data never hurts. For a fixed task and data distribution, model complexity should not increase more rapidly than the amount of data. Given more data, we might attempt to fit a more complex model. Absent sufficient data, simpler models may be more difficult to beat. For many tasks, deep learning only outperforms linear models when many thousands of training examples are available. In part, the current success of deep learning owes considerably to the abundance of massive datasets arising from Internet companies, cheap storage, connected devices, and the broad digitization of the economy.

## 3.6.3. Model Selection¶

Typically, we select our final model, only after evaluating multiple
models that differ in various ways (different architectures, training
objectives, selected features, data preprocessing, learning rates,
etc.). Choosing among many models is aptly called *model selection*.

In principle, we should not touch our test set until after we have
chosen all our hyperparameters. Were we to use the test data in the
model selection process, there is a risk that we might overfit the test
data. Then we would be in serious trouble. If we overfit our training
data, there is always the evaluation on test data to keep us honest. But
if we overfit the test data, how would we ever know? See
Ong *et al.* (2005) for an example how this can lead to absurd
results even for models where the complexity can be tightly controlled.

Thus, we should never rely on the test data for model selection. And yet we cannot rely solely on the training data for model selection either because we cannot estimate the generalization error on the very data that we use to train the model.

In practical applications, the picture gets muddier. While ideally we would only touch the test data once, to assess the very best model or to compare a small number of models with each other, real-world test data is seldom discarded after just one use. We can seldom afford a new test set for each round of experiments. In fact, recycling benchmark data for decades can have a significant impact on the development of algorithms, e.g., for image classification and optical character recognition.

The common practice to address the problem of *training on the test set*
is to split our data three ways, incorporating a *validation set* in
addition to the training and test datasets. The result is a murky
practice where the boundaries between validation and test data are
worryingly ambiguous. Unless explicitly stated otherwise, in the
experiments in this book we are really working with what should rightly
be called training data and validation data, with no true test sets.
Therefore, the accuracy reported in each experiment of the book is
really the validation accuracy and not a true test set accuracy.

### 3.6.3.1. Cross-Validation¶

When training data is scarce, we might not even be able to afford to
hold out enough data to constitute a proper validation set. One popular
solution to this problem is to employ \(K\)*-fold
cross-validation*. Here, the original training data is split into
\(K\) non-overlapping subsets. Then model training and validation
are executed \(K\) times, each time training on \(K-1\) subsets
and validating on a different subset (the one not used for training in
that round). Finally, the training and validation errors are estimated
by averaging over the results from the \(K\) experiments.

## 3.6.4. Summary¶

This section explored some of the underpinnings of generalization in machine learning. Some of these ideas become complicated and counterintuitive when we get to deeper models, there, models are capable of overfitting data badly, and the relevant notions of complexity can be both implicit and counterintuitive (e.g., larger architectures with more parameters generalizing better). We leave you with a few rules of thumb:

Use validation sets (or \(K\)

*-fold cross-validation*) for model selection;More complex models often require more data;

Relevant notions of complexity include both the number of parameters and the range of values that they are allowed to take;

Keeping all else equal, more data almost always leads to better generalization;

This entire talk of generalization is all predicated on the IID assumption. If we relax this assumption, allowing for distributions to shift between the train and testing periods, then we cannot say anything about generalization absent a further (perhaps milder) assumption.

## 3.6.5. Exercises¶

When can you solve the problem of polynomial regression exactly?

Give at least five examples where dependent random variables make treating the problem as IID data inadvisable.

Can you ever expect to see zero training error? Under which circumstances would you see zero generalization error?

Why is \(K\)-fold cross-validation very expensive to compute?

Why is the \(K\)-fold cross-validation error estimate biased?

The VC dimension is defined as the maximum number of points that can be classified with arbitrary labels \(\{\pm 1\}\) by a function of a class of functions. Why might this not be a good idea to measure how complex the class of functions is? Hint: what about the magnitude of the functions?

Your manager gives you a difficult dataset on which your current algorithm does not perform so well. How would you justify to him that you need more data? Hint: you cannot increase the data but you can decrease it.