A Few Useful Things to Know about Machine Learning

A Few Useful Things to Know about Machine Learning

Pedro Domingos
Department of Computer Science and Engineering
University of Washington
Seattle, WA 98195-2350, U.S.A.
pedrod@cs.washington.edu

ABSTRACT correct output yt for future examples xt (e.g., whether the
Machine learning algorithms can figure out how to perform spam filter correctly classifies previously unseen emails as
important tasks by generalizing from examples. This is of- spam or not spam).
ten feasible and cost-effective where manual programming
is not. As more data becomes available, more ambitious 2. LEARNING = REPRESENTATION +
problems can be tackled. As a result, machine learning is
widely used in computer science and other fields. However,
EVALUATION + OPTIMIZATION
Suppose you have an application that you think machine
developing successful machine learning applications requires
learning might be good for. The first problem facing you
a substantial amount of “black art” that is hard to find in
is the bewildering variety of learning algorithms available.
textbooks. This article summarizes twelve key lessons that
Which one to use? There are literally thousands available,
machine learning researchers and practitioners have learned.
and hundreds more are published each year. The key to not
These include pitfalls to avoid, important issues to focus on,
getting lost in this huge space is to realize that it consists
and answers to common questions.
of combinations of just three components. The components
are:
1. INTRODUCTION
Machine learning systems automatically learn programs from Representation. A classifier must be represented in some
data. This is often a very attractive alternative to manually formal language that the computer can handle. Con-
constructing them, and in the last decade the use of machine versely, choosing a representation for a learner is tan-
learning has spread rapidly throughout computer science tamount to choosing the set of classifiers that it can
and beyond. Machine learning is used in Web search, spam possibly learn. This set is called the hypothesis space
filters, recommender systems, ad placement, credit scoring, of the learner. If a classifier is not in the hypothesis
fraud detection, stock trading, drug design, and many other space, it cannot be learned. A related question, which
applications. A recent report from the McKinsey Global In- we will address in a later section, is how to represent
stitute asserts that machine learning (a.k.a. data mining or the input, i.e., what features to use.
predictive analytics) will be the driver of the next big wave of
innovation [15]. Several fine textbooks are available to inter- Evaluation. An evaluation function (also called objective
ested practitioners and researchers (e.g, [16, 24]). However, function or scoring function) is needed to distinguish
much of the “folk knowledge” that is needed to successfully good classifiers from bad ones. The evaluation function
develop machine learning applications is not readily avail- used internally by the algorithm may differ from the
able in them. As a result, many machine learning projects external one that we want the classifier to optimize, for
take much longer than necessary or wind up producing less- ease of optimization (see below) and due to the issues
than-ideal results. Yet much of this folk knowledge is fairly discussed in the next section.
easy to communicate. This is the purpose of this article. Optimization. Finally, we need a method to search among
the classifiers in the language for the highest-scoring
Many different types of machine learning exist, but for il- one. The choice of optimization technique is key to the
lustration purposes I will focus on the most mature and efficiency of the learner, and also helps determine the
widely used one: classification. Nevertheless, the issues I classifier produced if the evaluation function has more
will discuss apply across all of machine learning. A classi- than one optimum. It is common for new learners to
fier is a system that inputs (typically) a vector of discrete start out using off-the-shelf optimizers, which are later
and/or continuous feature values and outputs a single dis- replaced by custom-designed ones.
crete value, the class. For example, a spam filter classifies
email messages into “spam” or “not spam,” and its input may Table 1 shows common examples of each of these three com-
be a Boolean vector x = (x1 , . . . , xj , . . . , xd ), where xj = 1 ponents. For example, k-nearest neighbor classifies a test
if the jth word in the dictionary appears in the email and example by finding the k most similar training examples
xj = 0 otherwise. A learner inputs a training set of exam- and predicting the majority class among them. Hyperplane-
ples (xi , yi ), where xi = (xi,1 , . . . , xi,d ) is an observed input based methods form a linear combination of the features per
and yi is the corresponding output, and outputs a classifier. class and predict the class with the highest-valued combina-
The test of the learner is whether this classifier produces the tion. Decision trees test one feature at each internal node,

Table 1: The three components of learning algorithms.
Representation Evaluation Optimization
Instances Accuracy/Error rate Combinatorial optimization
K-nearest neighbor Precision and recall Greedy search
Support vector machines Squared error Beam search
Hyperplanes Likelihood Branch-and-bound
Naive Bayes Posterior probability Continuous optimization
Logistic regression Information gain Unconstrained
Decision trees K-L divergence Gradient descent
Sets of rules Cost/Utility Conjugate gradient
Propositional rules Margin Quasi-Newton methods
Logic programs Constrained
Neural networks Linear programming
Graphical models Quadratic programming
Bayesian networks
Conditional random fields

with one branch for each feature value, and have class predic- Algorithm 1 LearnDT(TrainSet)
tions at the leaves. Algorithm 1 shows a bare-bones decision if all examples in TrainSet have the same class y∗ then
tree learner for Boolean domains, using information gain and return MakeLeaf(y∗ )
greedy search [20]. InfoGain(xj ,y) is the mutual information if no feature xj has InfoGain(xj ,y) > 0 then
between feature xj and the class y. MakeNode(x,c0 ,c1 ) re- y∗ ← Most frequent class in TrainSet
turns a node that tests feature x and has c0 as the child for return MakeLeaf(y∗ )
x = 0 and c1 as the child for x = 1. x∗ ← argmaxxj InfoGain(xj ,y)
TS0 ← Examples in TrainSet with x∗ = 0
Of course, not all combinations of one component from each TS1 ← Examples in TrainSet with x∗ = 1
column of Table 1 make equal sense. For example, dis- return MakeNode(x∗ , LearnDT(TS0 ), LearnDT(TS1 ))
crete representations naturally go with combinatorial op-
timization, and continuous ones with continuous optimiza-
tion. Nevertheless, many learners have both discrete and Contamination of your classifier by test data can occur in
continuous components, and in fact the day may not be insidious ways, e.g., if you use test data to tune parameters
far when every single possible combination has appeared in and do a lot of tuning. (Machine learning algorithms have
some learner! lots of knobs, and success often comes from twiddling them
a lot, so this is a real concern.) Of course, holding out
Most textbooks are organized by representation, and it’s data reduces the amount available for training. This can
easy to overlook the fact that the other components are be mitigated by doing cross-validation: randomly dividing
equally important. There is no simple recipe for choosing your training data into (say) ten subsets, holding out each
each component, but the next sections touch on some of the one while training on the rest, testing each learned classifier
key issues. And, as we will see below, some choices in a on the examples it did not see, and averaging the results to
machine learning project may be even more important than see how well the particular parameter setting does.
the choice of learner.
In the early days of machine learning, the need to keep train-
3. IT’S GENERALIZATION THAT COUNTS ing and test data separate was not widely appreciated. This
The fundamental goal of machine learning is to generalize was partly because, if the learner has a very limited repre-
beyond the examples in the training set. This is because, sentation (e.g., hyperplanes), the difference between train-
no matter how much data we have, it is very unlikely that ing and test error may not be large. But with very flexible
we will see those exact examples again at test time. (No- classifiers (e.g., decision trees), or even with linear classifiers
tice that, if there are 100,000 words in the dictionary, the with a lot of features, strict separation is mandatory.
spam filter described above has 2100,000 possible different in-
puts.) Doing well on the training set is easy (just memorize Notice that generalization being the goal has an interesting
the examples). The most common mistake among machine consequence for machine learning. Unlike in most other op-
learning beginners is to test on the training data and have timization problems, we don’t have access to the function
the illusion of success. If the chosen classifier is then tested we want to optimize! We have to use training error as a sur-
on new data, it is often no better than random guessing. So, rogate for test error, and this is fraught with danger. How
if you hire someone to build a classifier, be sure to keep some to deal with it is addressed in some of the next sections. On
of the data to yourself and test the classifier they give you the positive side, since the objective function is only a proxy
on it. Conversely, if you’ve been hired to build a classifier, for the true goal, we may not need to fully optimize it; in
set some of the data aside from the beginning, and only use fact, a local optimum returned by simple greedy search may
it to test your chosen classifier at the very end, followed by be better than the global optimum.
learning your final classifier on the whole data.

4. DATA ALONE IS NOT ENOUGH Low High
Generalization being the goal has another major consequence: Variance Variance
data alone is not enough, no matter how much of it you have.
Consider learning a Boolean function of (say) 100 variables
from a million examples. There are 2100 − 106 examples
whose classes you don’t know. How do you figure out what
those classes are? In the absence of further information, High
there is just no way to do this that beats flipping a coin. This Bias
observation was first made (in somewhat different form) by
the philosopher David Hume over 200 years ago, but even
today many mistakes in machine learning stem from failing
to appreciate it. Every learner must embody some knowl-
edge or assumptions beyond the data it’s given in order to
generalize beyond it. This was formalized by Wolpert in
his famous “no free lunch” theorems, according to which no Low
learner can beat random guessing over all possible functions Bias
to be learned [25].

This seems like rather depressing news. How then can we
ever hope to learn anything? Luckily, the functions we want Figure 1: Bias and variance in dart-throwing.
to learn in the real world are not drawn uniformly from
the set of all mathematically possible functions! In fact,
very general assumptions—like smoothness, similar exam- quirks in the data. This problem is called overfitting, and is
ples having similar classes, limited dependences, or limited the bugbear of machine learning. When your learner outputs
complexity—are often enough to do very well, and this is a a classifier that is 100% accurate on the training data but
large part of why machine learning has been so successful. only 50% accurate on test data, when in fact it could have
Like deduction, induction (what learners do) is a knowledge output one that is 75% accurate on both, it has overfit.
lever: it turns a small amount of input knowledge into a
large amount of output knowledge. Induction is a vastly Everyone in machine learning knows about overfitting, but
more powerful lever than deduction, requiring much less in- it comes in many forms that are not immediately obvious.
put knowledge to produce useful results, but it still needs One way to understand overfitting is by decomposing gener-
more than zero input knowledge to work. And, as with any alization error into bias and variance [9]. Bias is a learner’s
lever, the more we put in, the more we can get out. tendency to consistently learn the same wrong thing. Vari-
ance is the tendency to learn random things irrespective of
A corollary of this is that one of the key criteria for choos- the real signal. Figure 1 illustrates this by an analogy with
ing a representation is which kinds of knowledge are easily throwing darts at a board. A linear learner has high bias,
expressed in it. For example, if we have a lot of knowledge because when the frontier between two classes is not a hyper-
about what makes examples similar in our domain, instance- plane the learner is unable to induce it. Decision trees don’t
based methods may be a good choice. If we have knowl- have this problem because they can represent any Boolean
edge about probabilistic dependencies, graphical models are function, but on the other hand they can suffer from high
a good fit. And if we have knowledge about what kinds of variance: decision trees learned on different training sets
preconditions are required by each class, “IF . . . THEN . . .” generated by the same phenomenon are often very different,
rules may be the the best option. The most useful learners when in fact they should be the same. Similar reasoning
in this regard are those that don’t just have assumptions applies to the choice of optimization method: beam search
hard-wired into them, but allow us to state them explicitly, has lower bias than greedy search, but higher variance, be-
vary them widely, and incorporate them automatically into cause it tries more hypotheses. Thus, contrary to intuition,
the learning (e.g., using first-order logic [21] or grammars a more powerful learner is not necessarily better than a less
[6]). powerful one.

In retrospect, the need for knowledge in learning should not Figure 2 illustrates this.1 Even though the true classifier
be surprising. Machine learning is not magic; it can’t get is a set of rules, with up to 1000 examples naive Bayes is
something from nothing. What it does is get more from more accurate than a rule learner. This happens despite
less. Programming, like all engineering, is a lot of work: naive Bayes’s false assumption that the frontier is linear!
we have to build everything from scratch. Learning is more Situations like this are common in machine learning: strong
like farming, which lets nature do most of the work. Farm- false assumptions can be better than weak true ones, because
ers combine seeds with nutrients to grow crops. Learners a learner with the latter needs more data to avoid overfitting.
combine knowledge with data to grow programs.

1
5. OVERFITTING HAS MANY FACES Training examples consist of 64 Boolean features and a
What if the knowledge and data we have are not sufficient Boolean class computed from them according to a set of “IF
. . . THEN . . .” rules. The curves are the average of 100 runs
to completely determine the correct classifier? Then we run with different randomly generated sets of rules. Error bars
the risk of just hallucinating a classifier (or parts of it) that are two standard deviations. See Domingos and Pazzani [10]
is not grounded in reality, and is simply encoding random for details.

Cross-validation can help to combat overfitting, for example
by using it to choose the best size of decision tree to learn. 80
Bayes
But it’s no panacea, since if we use it to make too many C4.5

Test-Set Accuracy (%)
parameter choices it can itself start to overfit [17]. 75

Besides cross-validation, there are many methods to combat 70
overfitting. The most popular one is adding a regulariza-
tion term to the evaluation function. This can, for exam- 65
ple, penalize classifiers with more structure, thereby favoring
smaller ones with less room to overfit. Another option is to 60
perform a statistical significance test like chi-square before
adding new structure, to decide whether the distribution of 55
the class really is different with and without this structure.
These techniques are particularly useful when data is very 50
scarce. Nevertheless, you should be skeptical of claims that 10 100 1000 10000
a particular technique “solves” the overfitting problem. It’s Number of Examples
easy to avoid overfitting (variance) by falling into the op-
posite error of underfitting (bias). Simultaneously avoiding Figure 2: Naive Bayes can outperform a state-of-
both requires learning a perfect classifier, and short of know- the-art rule learner (C4.5rules) even when the true
ing it in advance there is no single technique that will always classifier is a set of rules.
do best (no free lunch).

A common misconception about overfitting is that it is caused trillion examples, the latter covers only a fraction of about
by noise, like training examples labeled with the wrong class. 10−18 of the input space. This is what makes machine learn-
This can indeed aggravate overfitting, by making the learner ing both necessary and hard.
draw a capricious frontier to keep those examples on what
it thinks is the right side. But severe overfitting can occur More seriously, the similarity-based reasoning that machine
even in the absence of noise. For instance, suppose we learn a learning algorithms depend on (explicitly or implicitly) breaks
Boolean classifier that is just the disjunction of the examples down in high dimensions. Consider a nearest neighbor clas-
labeled “true” in the training set. (In other words, the classi- sifier with Hamming distance as the similarity measure, and
fier is a Boolean formula in disjunctive normal form, where suppose the class is just x1 ∧ x2 . If there are no other fea-
each term is the conjunction of the feature values of one tures, this is an easy problem. But if there are 98 irrele-
specific training example). This classifier gets all the train- vant features x3 , . . . , x100 , the noise from them completely
ing examples right and every positive test example wrong, swamps the signal in x1 and x2 , and nearest neighbor effec-
regardless of whether the training data is noisy or not. tively makes random predictions.

The problem of multiple testing [13] is closely related to over- Even more disturbing is that nearest neighbor still has a
fitting. Standard statistical tests assume that only one hy- problem even if all 100 features are relevant! This is because
pothesis is being tested, but modern learners can easily test in high dimensions all examples look alike. Suppose, for
millions before they are done. As a result what looks signif- instance, that examples are laid out on a regular grid, and
icant may in fact not be. For example, a mutual fund that consider a test example xt . If the grid is d-dimensional, xt ’s
beats the market ten years in a row looks very impressive, 2d nearest examples are all at the same distance from it.
until you realize that, if there are 1000 funds and each has So as the dimensionality increases, more and more examples
a 50% chance of beating the market on any given year, it’s become nearest neighbors of xt , until the choice of nearest
quite likely that one will succeed all ten times just by luck. neighbor (and therefore of class) is effectively random.
This problem can be combatted by correcting the signifi-
cance tests to take the number of hypotheses into account, This is only one instance of a more general problem with
but this can lead to underfitting. A better approach is to high dimensions: our intuitions, which come from a three-
control the fraction of falsely accepted non-null hypotheses, dimensional world, often do not apply in high-dimensional
known as the false discovery rate [3]. ones. In high dimensions, most of the mass of a multivari-
ate Gaussian distribution is not near the mean, but in an
increasingly distant “shell” around it; and most of the vol-
6. INTUITION FAILS IN HIGH ume of a high-dimensional orange is in the skin, not the pulp.
DIMENSIONS If a constant number of examples is distributed uniformly in
After overfitting, the biggest problem in machine learning a high-dimensional hypercube, beyond some dimensionality
is the curse of dimensionality. This expression was coined most examples are closer to a face of the hypercube than
by Bellman in 1961 to refer to the fact that many algo- to their nearest neighbor. And if we approximate a hyper-
rithms that work fine in low dimensions become intractable sphere by inscribing it in a hypercube, in high dimensions
when the input is high-dimensional. But in machine learn- almost all the volume of the hypercube is outside the hyper-
ing it refers to much more. Generalizing correctly becomes sphere. This is bad news for machine learning, where shapes
exponentially harder as the dimensionality (number of fea- of one type are often approximated by shapes of another.
tures) of the examples grows, because a fixed-size training
set covers a dwindling fraction of the input space. Even with Building a classifier in two or three dimensions is easy; we
a moderate dimension of 100 and a huge training set of a

d
can find a reasonable frontier between examples of different total number of functions is 22 . And even for hypothesis
classes just by visual inspection. (It’s even been said that if spaces that are “merely” exponential, the bound is still very
people could see in high dimensions machine learning would loose, because the union bound is very pessimistic. For ex-
not be necessary.) But in high dimensions it’s hard to under- ample, if there are 100 Boolean features and the hypothesis
stand what is happening. This in turn makes it difficult to space is decision trees with up to 10 levels, to guarantee
design a good classifier. Naively, one might think that gath- δ = ǫ = 1% in the bound above we need half a million ex-
ering more features never hurts, since at worst they provide amples. But in practice a small fraction of this suffices for
no new information about the class. But in fact their bene- accurate learning.
fits may be outweighed by the curse of dimensionality.
Further, we have to be careful about what a bound like this
Fortunately, there is an effect that partly counteracts the means. For instance, it does not say that, if your learner
curse, which might be called the “blessing of non-uniformity.” returned a hypothesis consistent with a particular training
In most applications examples are not spread uniformly thr- set, then this hypothesis probably generalizes well. What
oughout the instance space, but are concentrated on or near it says is that, given a large enough training set, with high
a lower-dimensional manifold. For example, k-nearest neigh- probability your learner will either return a hypothesis that
bor works quite well for handwritten digit recognition even generalizes well or be unable to find a consistent hypothesis.
though images of digits have one dimension per pixel, be- The bound also says nothing about how to select a good hy-
cause the space of digit images is much smaller than the pothesis space. It only tells us that, if the hypothesis space
space of all possible images. Learners can implicitly take contains the true classifier, then the probability that the
advantage of this lower effective dimension, or algorithms learner outputs a bad classifier decreases with training set
for explicitly reducing the dimensionality can be used (e.g., size. If we shrink the hypothesis space, the bound improves,
[22]). but the chances that it contains the true classifier shrink
also. (There are bounds for the case where the true classi-
fier is not in the hypothesis space, but similar considerations
7. THEORETICAL GUARANTEES ARE apply to them.)
NOT WHAT THEY SEEM
Another common type of theoretical guarantee is asymp-
Machine learning papers are full of theoretical guarantees.
totic: given infinite data, the learner is guaranteed to output
The most common type is a bound on the number of ex-
the correct classifier. This is reassuring, but it would be rash
amples needed to ensure good generalization. What should
to choose one learner over another because of its asymptotic
you make of these guarantees? First of all, it’s remarkable
guarantees. In practice, we are seldom in the asymptotic
that they are even possible. Induction is traditionally con-
regime (also known as “asymptopia”). And, because of the
trasted with deduction: in deduction you can guarantee that
bias-variance tradeoff we discussed above, if learner A is bet-
the conclusions are correct; in induction all bets are off. Or
ter than learner B given infinite data, B is often better than
such was the conventional wisdom for many centuries. One
A given finite data.
of the major developments of recent decades has been the
realization that in fact we can have guarantees on the re-
The main role of theoretical guarantees in machine learning
sults of induction, particularly if we’re willing to settle for
is not as a criterion for practical decisions, but as a source of
probabilistic guarantees.
understanding and driving force for algorithm design. In this
capacity, they are quite useful; indeed, the close interplay
The basic argument is remarkably simple [5]. Let’s say a
of theory and practice is one of the main reasons machine
classifier is bad if its true error rate is greater than ǫ. Then
learning has made so much progress over the years. But
the probability that a bad classifier is consistent with n ran-
caveat emptor: learning is a complex phenomenon, and just
dom, independent training examples is less than (1 − ǫ)n .
because a learner has a theoretical justification and works in
Let b be the number of bad classifiers in the learner’s hy-
practice doesn’t mean the former is the reason for the latter.
pothesis space H. The probability that at least one of them
is consistent is less than b(1 − ǫ)n , by the union bound.
Assuming the learner always returns a consistent classifier, 8. FEATURE ENGINEERING IS THE KEY
the probability that this classifier is bad is then less than At the end of the day, some machine learning projects suc-
|H|(1 − ǫ)n , where we have used the fact that b ≤ |H|. So ceed and some fail. What makes the difference? Easily
if we want this probability to be less than δ, it suffices to the most important factor is the features used. If you have
make n > ln(δ/|H|)/ ln(1 − ǫ) ≥ 1 ln |H| + ln 1 .
` ´
ǫ δ
many independent features that each correlate well with the
class, learning is easy. On the other hand, if the class is
Unfortunately, guarantees of this type have to be taken with a very complex function of the features, you may not be
a large grain of salt. This is because the bounds obtained in able to learn it. Often, the raw data is not in a form that is
this way are usually extremely loose. The wonderful feature amenable to learning, but you can construct features from it
of the bound above is that the required number of examples that are. This is typically where most of the effort in a ma-
only grows logarithmically with |H| and 1/δ. Unfortunately, chine learning project goes. It is often also one of the most
most interesting hypothesis spaces are doubly exponential in interesting parts, where intuition, creativity and “black art”
the number of features d, which still leaves us needing a num- are as important as the technical stuff.
ber of examples exponential in d. For example, consider the
space of Boolean functions of d Boolean variables. If there First-timers are often surprised by how little time in a ma-
are e possible different examples, there are 2e possible dif- chine learning project is spent actually doing machine learn-
ferent functions, so since there are 2d possible examples, the ing. But it makes sense if you consider how time-consuming

it is to gather data, integrate it, clean it and pre-process it, N. Bayes
and how much trial and error can go into feature design.
Also, machine learning is not a one-shot process of build- kNN
ing a data set and running a learner, but rather an iterative SVM
process of running the learner, analyzing the results, modi-
fying the data and/or the learner, and repeating. Learning
is often the quickest part of this, but that’s because we’ve
already mastered it pretty well! Feature engineering is more
difficult because it’s domain-specific, while learners can be
largely general-purpose. However, there is no sharp frontier
between the two, and this is another reason the most useful
learners are those that facilitate incorporating knowledge. D. Tree
Of course, one of the holy grails of machine learning is to
automate more and more of the feature engineering process.
One way this is often done today is by automatically gen- Figure 3: Very different frontiers can yield similar
erating large numbers of candidate features and selecting class predictions. (+ and − are training examples of
the best by (say) their information gain with respect to the two classes.)
class. But bear in mind that features that look irrelevant
in isolation may be relevant in combination. For example, and neural networks. But in fact propositional rules are
if the class is an XOR of k input features, each of them by readily encoded as neural networks, and similar relation-
itself carries no information about the class. (If you want ships hold between other representations. All learners es-
to annoy machine learners, bring up XOR.) On the other sentially work by grouping nearby examples into the same
hand, running a learner with a very large number of fea- class; the key difference is in the meaning of “nearby.” With
tures to find out which ones are useful in combination may non-uniformly distributed data, learners can produce widely
be too time-consuming, or cause overfitting. So there is ul- different frontiers while still making the same predictions in
timately no replacement for the smarts you put into feature the regions that matter (those with a substantial number of
engineering. training examples, and therefore also where most test ex-
amples are likely to appear). This also helps explain why
9. MORE DATA BEATS A CLEVERER powerful learners can be unstable but still accurate. Fig-
ALGORITHM ure 3 illustrates this in 2-D; the effect is much stronger in
Suppose you’ve constructed the best set of features you high dimensions.
can, but the classifiers you’re getting are still not accurate
enough. What can you do now? There are two main choices: As a rule, it pays to try the simplest learners first (e.g., naive
design a better learning algorithm, or gather more data Bayes before logistic regression, k-nearest neighbor before
(more examples, and possibly more raw features, subject to support vector machines). More sophisticated learners are
the curse of dimensionality). Machine learning researchers seductive, but they are usually harder to use, because they
are mainly concerned with the former, but pragmatically have more knobs you need to turn to get good results, and
the quickest path to success is often to just get more data. because their internals are more opaque.
As a rule of thumb, a dumb algorithm with lots and lots of
data beats a clever one with modest amounts of it. (After Learners can be divided into two major types: those whose
all, machine learning is all about letting data do the heavy representation has a fixed size, like linear classifiers, and
lifting.) those whose representation can grow with the data, like deci-
sion trees. (The latter are sometimes called non-parametric
This does bring up another problem, however: scalability. learners, but this is somewhat unfortunate, since they usu-
In most of computer science, the two main limited resources ally wind up learning many more parameters than paramet-
are time and memory. In machine learning, there is a third ric ones.) Fixed-size learners can only take advantage of so
one: training data. Which one is the bottleneck has changed much data. (Notice how the accuracy of naive Bayes asymp-
from decade to decade. In the 1980’s it tended to be data. totes at around 70% in Figure 2.) Variable-size learners can
Today it is often time. Enormous mountains of data are in principle learn any function given sufficient data, but in
available, but there is not enough time to process it, so it practice they may not, because of limitations of the algo-
goes unused. This leads to a paradox: even though in prin- rithm (e.g., greedy search falls into local optima) or compu-
ciple more data means that more complex classifiers can be tational cost. Also, because of the curse of dimensionality,
learned, in practice simpler classifiers wind up being used, no existing amount of data may be enough. For these rea-
because complex ones take too long to learn. Part of the sons, clever algorithms—those that make the most of the
answer is to come up with fast ways to learn complex classi- data and computing resources available—often pay off in
fiers, and indeed there has been remarkable progress in this the end, provided you’re willing to put in the effort. There
direction (e.g., [11]). is no sharp frontier between designing learners and learn-
ing classifiers; rather, any given piece of knowledge could be
Part of the reason using cleverer algorithms has a smaller encoded in the learner or learned from data. So machine
payoff than you might expect is that, to a first approxima- learning projects often wind up having a significant compo-
tion, they all do the same. This is surprising when you nent of learner design, and practitioners need to have some
consider representations as different as, say, sets of rules expertise in it [12].

In the end, the biggest bottleneck is not data or CPU cycles, those produced by (say) bagging or boosting: the latter are
but human cycles. In research papers, learners are typically fairly even, while the former are extremely skewed, to the
compared on measures of accuracy and computational cost. point where the single highest-weight classifier usually dom-
But human effort saved and insight gained, although harder inates, making BMA effectively equivalent to just selecting
to measure, are often more important. This favors learn- it [8]. A practical consequence of this is that, while model
ers that produce human-understandable output (e.g., rule ensembles are a key part of the machine learning toolkit,
sets). And the organizations that make the most of ma- BMA is seldom worth the trouble.
chine learning are those that have in place an infrastructure
that makes experimenting with many different learners, data
sources and learning problems easy and efficient, and where
there is a close collaboration between machine learning ex-
perts and application domain ones. 11. SIMPLICITY DOES NOT IMPLY
ACCURACY
10. LEARN MANY MODELS, NOT JUST Occam’s razor famously states that entities should not be
ONE multiplied beyond necessity. In machine learning, this is of-
In the early days of machine learning, everyone had their fa- ten taken to mean that, given two classifiers with the same
vorite learner, together with some a priori reasons to believe training error, the simpler of the two will likely have the
in its superiority. Most effort went into trying many varia- lowest test error. Purported proofs of this claim appear reg-
tions of it and selecting the best one. Then systematic em- ularly in the literature, but in fact there are many counter-
pirical comparisons showed that the best learner varies from examples to it, and the “no free lunch” theorems imply it
application to application, and systems containing many dif- cannot be true.
ferent learners started to appear. Effort now went into try-
ing many variations of many learners, and still selecting just We saw one counter-example in the previous section: model
the best one. But then researchers noticed that, if instead ensembles. The generalization error of a boosted ensem-
of selecting the best variation found, we combine many vari- ble continues to improve by adding classifiers even after the
ations, the results are better—often much better—and at training error has reached zero. Another counter-example is
little extra effort for the user. support vector machines, which can effectively have an infi-
nite number of parameters without overfitting. Conversely,
Creating such model ensembles is now standard [1]. In the the function sign(sin(ax)) can discriminate an arbitrarily
simplest technique, called bagging, we simply generate ran- large, arbitrarily labeled set of points on the x axis, even
dom variations of the training set by resampling, learn a though it has only one parameter [23]. Thus, contrary to in-
classifier on each, and combine the results by voting. This tuition, there is no necessary connection between the number
works because it greatly reduces variance while only slightly of parameters of a model and its tendency to overfit.
increasing bias. In boosting, training examples have weights,
and these are varied so that each new classifier focuses on A more sophisticated view instead equates complexity with
the examples the previous ones tended to get wrong. In the size of the hypothesis space, on the basis that smaller
stacking, the outputs of individual classifiers become the in- spaces allow hypotheses to be represented by shorter codes.
puts of a “higher-level” learner that figures out how best to Bounds like the one in the section on theoretical guarantees
combine them. above might then be viewed as implying that shorter hy-
potheses generalize better. This can be further refined by
Many other techniques exist, and the trend is toward larger assigning shorter codes to the hypothesis in the space that
and larger ensembles. In the Netflix prize, teams from all we have some a priori preference for. But viewing this as
over the world competed to build the best video recom- “proof” of a tradeoff between accuracy and simplicity is cir-
mender system (http://netflixprize.com). As the competi- cular reasoning: we made the hypotheses we prefer simpler
tion progressed, teams found that they obtained the best by design, and if they are accurate it’s because our prefer-
results by combining their learners with other teams’, and ences are accurate, not because the hypotheses are “simple”
merged into larger and larger teams. The winner and runner- in the representation we chose.
up were both stacked ensembles of over 100 learners, and
combining the two ensembles further improved the results. A further complication arises from the fact that few learners
Doubtless we will see even larger ones in the future. search their hypothesis space exhaustively. A learner with a
larger hypothesis space that tries fewer hypotheses from it
Model ensembles should not be confused with Bayesian model is less likely to overfit than one that tries more hypotheses
averaging (BMA). BMA is the theoretically optimal approach from a smaller space. As Pearl [18] points out, the size of
to learning [4]. In BMA, predictions on new examples are the hypothesis space is only a rough guide to what really
made by averaging the individual predictions of all classifiers matters for relating training and test error: the procedure
in the hypothesis space, weighted by how well the classifiers by which a hypothesis is chosen.
explain the training data and how much we believe in them
a priori. Despite their superficial similarities, ensembles and Domingos [7] surveys the main arguments and evidence on
BMA are very different. Ensembles change the hypothesis the issue of Occam’s razor in machine learning. The conclu-
space (e.g., from single decision trees to linear combinations sion is that simpler hypotheses should be preferred because
of them), and can take a wide variety of forms. BMA assigns simplicity is a virtue in its own right, not because of a hy-
weights to the hypotheses in the original space according to pothetical connection with accuracy. This is probably what
a fixed formula. BMA weights are extremely different from Occam meant in the first place.

12. REPRESENTABLE DOES NOT IMPLY Many researchers believe that causality is only a convenient
LEARNABLE fiction. For example, there is no notion of causality in phys-
Essentially all representations used in variable-size learners ical laws. Whether or not causality really exists is a deep
have associated theorems of the form “Every function can philosophical question with no definitive answer in sight,
be represented, or approximated arbitrarily closely, using but the practical points for machine learners are two. First,
this representation.” Reassured by this, fans of the repre- whether or not we call them “causal,” we would like to pre-
sentation often proceed to ignore all others. However, just dict the effects of our actions, not just correlations between
because a function can be represented does not mean it can observable variables. Second, if you can obtain experimental
be learned. For example, standard decision tree learners data (for example by randomly assigning visitors to different
cannot learn trees with more leaves than there are training versions of a Web site), then by all means do so [14].
examples. In continuous spaces, representing even simple
functions using a fixed set of primitives often requires an 14. CONCLUSION
infinite number of components. Further, if the hypothesis Like any discipline, machine learning has a lot of “folk wis-
space has many local optima of the evaluation function, as dom” that can be hard to come by, but is crucial for suc-
is often the case, the learner may not find the true function cess. This article summarized some of the most salient items.
even if it is representable. Given finite data, time and mem- Of course, it’s only a complement to the more conventional
ory, standard learners can learn only a tiny subset of all pos- study of machine learning. Check out http://www.cs.washin-
sible functions, and these subsets are different for learners gton.edu/homes/pedrod/class for a complete online machine
with different representations. Therefore the key question is learning course that combines formal and informal aspects.
not “Can it be represented?”, to which the answer is often There’s also a treasure trove of machine learning lectures at
trivial, but “Can it be learned?” And it pays to try different http://www.videolectures.net. A good open source machine
learners (and possibly combine them). learning toolkit is Weka [24]. Happy learning!

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