%matplotlib inline
import math
import numpy as np
from d2l import torch as d2l2.6 Generalization
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, Extraordinary Ellie, whose preparation consisted entirely of memorizing the answers to previous years’ exam questions. Even if Ellie were endowed with an extraordinary 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?
In real life, we must fit our models using a finite collection of data. The typical scales of that data vary wildly across domains. For many important medical problems, 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 a 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 it is no substitute for a proper introduction to statistical learning theory (Section 3.6 gives a first, rigorous taste; see also Vapnik (1998); Boucheron et al. (2005)), 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.
2.6.1 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, 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 \(R_\textrm{emp}\), 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 an average over the finite training sample (with the same notation as Section 2.1):
\[R_\textrm{emp}[\mathbf{X}, \mathbf{y}, f] = \frac{1}{n} \sum_{i=1}^n l(\mathbf{x}^{(i)}, y^{(i)}, f(\mathbf{x}^{(i)})), \tag{2.6.1}\]
while the generalization error (also called the risk) is expressed as an integral:
\[R[P, f] = E_{(\mathbf{x}, y) \sim P} [l(\mathbf{x}, y, f(\mathbf{x}))] = \int \int l(\mathbf{x}, y, f(\mathbf{x})) p(\mathbf{x}, y) \;d\mathbf{x} dy. \tag{2.6.2}\]
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 that was used for calculating the empirical training error but to a test set \(\mathbf{X}', \mathbf{y}'\).
Crucially, when we evaluate our model on the test set, we are working with a fixed model (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).
2.6.1.1 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 (the difference \(R - R_\textrm{emp}\) between the generalization error and the training error) 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 low training error is evidence that it has captured a real pattern, provided the sample is large relative to the class’s capacity (Section 26.6). 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 (Popper 2005). 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. For squared-error regression, with expectation taken over repeated training sets drawn by the same process, the classical bias-variance decomposition makes the trade-off precise: a model too simple to capture the signal makes a systematic error (high bias, i.e., underfitting), while a model flexible enough to chase the noise in a particular training set varies wildly from one dataset to the next (high variance, i.e., overfitting). Their sum, plus an irreducible noise floor \(\sigma^2\), is the expected test error, which traces the U-shaped curve of Figure 2.6.1; we derive the decomposition formally in Section 26.5. 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 (Schölkopf 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. 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 with confidence 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.
This classical picture, in which more capacity (the richness of the model class) means more overfitting, is, however, incomplete for the heavily overparametrized models at the heart of modern deep learning. Once a model is large enough to interpolate its training data (drive training error to zero), pushing capacity even higher often makes test error fall again rather than rise: the double descent phenomenon (Belkin et al. 2019; Nakkiran et al. 2021). We take up this modern story, and why the classical complexity intuition breaks down, in Section 4.5; for a quantitative treatment that reproduces the double-descent curve from scratch, see Section 26.6.
2.6.2 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 only a small 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 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.
2.6.2.1 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 \(d\)
\[\hat{y}= \sum_{i=0}^d x^i w_i\]
for estimating 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 compare the relationship between polynomial degree (model complexity) and both underfitting and overfitting in Figure 2.6.1.
To see this concretely, let us generate data from a known cubic and fit polynomials of growing degree to a small training set.
%matplotlib inline
import math
import numpy as np
from d2l import tensorflow as d2l%matplotlib inline
import math
import numpy as np
from d2l import jax as d2l%matplotlib inline
import math
import numpy as np
from d2l import mxnet as d2lWe draw inputs \(x\) uniformly from \([-1, 1]\), build a design matrix whose \(i\)-th column is \(x^i\), and generate labels from a degree-3 target \(y = 5 + 1.2 x - 3.4 x^2 + 5.6 x^3\) plus a little Gaussian noise. We deliberately keep the training set small so that high-degree models have room to overfit.
np.random.seed(0)
max_degree = 20 # highest polynomial degree we will fit
n_train, n_test = 20, 100 # few training points, so high degrees overfit
true_w = np.zeros(max_degree)
true_w[:4] = np.array([5, 1.2, -3.4, 5.6])
x = np.random.uniform(-1, 1, size=n_train + n_test)
poly = np.power(x.reshape(-1, 1), np.arange(max_degree)) # column i holds x**i
labels = poly @ true_w + np.random.normal(scale=0.1, size=n_train + n_test)Fitting the first \(d+1\) columns by least squares gives the best degree-\(d\) polynomial; we record its loss on both the training and the held-out test split.
def fit_degree(d):
cols = slice(0, d + 1)
w, *_ = np.linalg.lstsq(poly[:n_train, cols], labels[:n_train], rcond=None)
err = poly[:, cols] @ w - labels
return (err[:n_train] ** 2).mean(), (err[n_train:] ** 2).mean()A degree-1 polynomial is too rigid to capture a cubic, so it errs on both splits (underfitting); degree 3 matches the true model, with low error on both; and a degree-19 polynomial has enough freedom to interpolate the 20 training points almost exactly, driving training error toward zero while test error explodes (overfitting).
for name, d in [('underfitting (degree 1) ', 1),
('just right (degree 3) ', 3),
('overfitting (degree 19)', 19)]:
train_mse, test_mse = fit_degree(d)
print(f'{name}: train {train_mse:8.4f} test {test_mse:12.4f}')underfitting (degree 1) : train 1.6698 test 1.7318
just right (degree 3) : train 0.0136 test 0.0129
overfitting (degree 19): train 0.0000 test 49271508224071.6328Sweeping the degree from 1 to 19 traces out exactly the U-shaped test-error curve that Figure 2.6.1 sketches: error first falls as the model gains the capacity to represent the signal, then rises as the surplus capacity is spent fitting noise. Training error, by contrast, only ever decreases.
degrees = list(range(1, max_degree))
mse = np.array([fit_degree(d) for d in degrees])
d2l.plot(degrees, [mse[:, 0], mse[:, 1]], xlabel='polynomial degree',
ylabel='loss', legend=['train', 'test'], yscale='log')
So far we have named bias and variance; this experiment has everything we need to compute them. Because we know the noiseless target \(f(x) = 5 + 1.2 x - 3.4 x^2 + 5.6 x^3\), we can redraw the training noise many times, refit the degree-\(d\) polynomial on each draw, and ask two questions on the held-out inputs: how far is the average fit from the truth (squared bias), and how much does the fit fluctuate across draws (variance)?
f = poly @ true_w # noiseless target on all inputs
bias2, var = [], []
for d in range(1, 15):
preds = []
for _ in range(200): # 200 fresh draws of training noise
y_tr = f[:n_train] + np.random.normal(scale=0.1, size=n_train)
w, *_ = np.linalg.lstsq(poly[:n_train, :d+1], y_tr, rcond=None)
preds.append(poly[n_train:, :d+1] @ w)
preds = np.stack(preds)
bias2.append(((preds.mean(0) - f[n_train:]) ** 2).mean())
var.append(preds.var(0).mean())
d2l.plot(list(range(1, 15)), [bias2, var, np.array(bias2) + np.array(var)],
xlabel='polynomial degree', ylabel='error', yscale='log',
legend=['bias^2', 'variance', 'bias^2 + variance'])
The U-curve now visibly decomposes. Squared bias dominates for degrees below 3 and collapses to essentially zero the moment the model class contains the truth; variance is tiny at first but grows relentlessly with surplus capacity, exploding as the polynomial gains the freedom to chase each draw’s noise. Their sum (which, up to the irreducible noise floor \(\sigma^2 = 0.01\), is the expected test error) is lowest exactly where the two failure modes trade off, at degree 3. This is a numerical instance of the decomposition proved in Section 26.5.
2.6.2.2 Dataset Size
Beyond model complexity, 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 when the learning procedure and data distribution are held fixed. This is a tendency, not a monotonic law: unstable procedures and interpolation thresholds can produce temporary increases, including the sample-wise double descent discussed later. 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.
2.6.3 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 of 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, as documented when researchers rebuilt fresh test sets for long-standing benchmarks and watched accuracy drop (Recht et al. 2019). This effect is visible, e.g., for image classification and optical character recognition.
The common practice for addressing 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 business 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.
2.6.3.1 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. The procedure is illustrated in Figure 2.6.2.
How should we choose \(K\)? The choice trades off bias, variance, and compute. Each fold’s model is trained on only \((K-1)/K\) of the data. If the learning curve improves monotonically with sample size, its error is higher than that of the model finally trained on all the data, making the cross-validation estimate pessimistic. This conclusion is not guaranteed for unstable or non-monotone learning procedures. Taking \(K = n\) (leave-one-out cross-validation) all but eliminates this bias, but at a steep price: it requires \(n\) model fits, and the \(n\) training sets are nearly identical, so the fold errors are highly correlated and their average tends to have higher variance; in fact no general unbiased estimator of the cross-validation variance exists (Bengio and Grandvalet 2004). The standard compromise, \(K = 5\) or \(K = 10\), keeps the bias modest, averages over reasonably distinct training sets, and costs only \(5\)–\(10\) fits, which is why these values dominate practice (Kohavi 1995). (Exercises 4 and 5 ask you to reason through the cost and the bias.)
2.6.4 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; here, 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 the learning procedure and data distribution fixed, more data usually improves generalization, but the curve need not be monotone;
- 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.
2.6.5 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 for measuring how complex the class of functions is? Hint: consider 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.
- Re-run the polynomial-fitting experiment above with
n_trainset to 10, 40, and 100. At what degree does the test loss start to climb in each case? Relate your finding to the rule of thumb that more complex models require more data.