Pedro Domingos wrote this wonderful introductory overview of machine learning which teaches twelve short lessons about classifiers:

- LEARNING = REPRESENTATION + EVALUATION + OPTIMIZATION
- IT’S GENERALIZATION THAT COUNTS
- DATA ALONE IS NOT ENOUGH
- OVERFITTING HAS MANY FACES
- INTUITION FAILS IN HIGH DIMENSIONS
- THEORETICAL GUARANTEES ARE NOT WHAT THEY SEEM
- FEATURE ENGINEERING IS THE KEY
- MORE DATA BEATS A CLEVERER ALGORITHM
- LEARN MANY MODELS, NOT JUST ONE
- SIMPLICITY DOES NOT IMPLY ACCURACY
- REPRESENTABLE DOES NOT IMPLY LEARNABLE
- CORRELATION DOES NOT IMPLY CAUSATION

The lessons are augmented with excellent examples, so I highly recommend reading the simply written, short article.

I put together some notes and quotes, on and from these lessons which I will share below.

LEARNING = REPRESENTATION + EVALUATION + OPTIMIZATION – Creating a classifiers involves three key ideas: representation, an objective function, and optimization. For example, for $k$-means classification, the domain is divided up by a Voronoi diagram and each region has a different label or possibly a mixture of labels. This is the representation. The objective function does not involve the labels—only the input data. The objective function is the sum of the squared distances from input data to the “center” of each region to which the datum belongs. The optimization technique is greedy alternation between optimizing the center of each region and the assignment of data to regions.

IT’S GENERALIZATION THAT COUNTS – “The fundamental goal of machine learning is to generalize beyond the examples in the training set.” Merely learning the training set is both trivial and insufficient. Often doing well on the training set is a bad indicator of performance outside the training set. Generalization is the only way to label inputs that are not in the training set. Unfortunately, Wolpert’s “no free lunch” theorems demonstrate that no generalization method can always beat random guessing for learning problems without structure (i.e. Martin-Löf random classification problems do not admit generalization.)

OVERFITTING HAS MANY FACES – If we do not have enough data or time to determine the correct representation or the correct number of parameters, then we introduce either bias errors when we use the wrong representation or overfitting errors when we have too many parameters (i.e. Bias-Variance Trade-off). The fact that trying too many hypotheses has the same dangers as using too many parameters implies “contrary to intuition, a more powerful learner is not necessarily better than a less powerful one.” and “strong false assumptions can be better than weak true ones, because a learner with the latter needs more data to avoid overﬁtting.” Cross validation, statistical significance tests, minimizing the false discovery rate, and regularization terms prevent most overfitting. Overfitting can even occur when all the data is correctly labelled.

INTUITION FAILS IN HIGH DIMENSIONS – Most algorithms have more trouble in higher dimensional spaces and our intuition fails in higher dimensional spaces. Nearest Neighbor for simple classification problems tends to fail in more than 100 dimensions. Our intuition built on experience with 3 dimensional spaces fails to understand facts like a hypersphere with radius 0.95 has less than 1% of the volume of a hypersphere of radius 1 in a 200 dimensional space. So for a Gaussian in such a space, the majority of the probability is not near the mean, instead it lies in a “shell” about the mean. The volume of a hypersphere is much less than the smallest hypercube that contains it. These counter-intuitive facts reek havoc for several machine learning algorithms. It is even hard to fix or adjust the algorithm because we often can’t visualize the problem in high dimensional spaces. So dimensionality reduction via manifold learning, PCA, LDA, ICA, and feature selection becomes very important.

THEORETICAL GUARANTEES ARE NOT WHAT THEY SEEM – Domingos repeats the well known argument that if bad classifiers are wrong with probability greater than $\epsilon$ and the number of hypothetical classifiers is $H$, then $n = {1\over\epsilon}(\ln(H) + \ln({1\over\delta}))$ randomly chosen test examples are sufficient to eliminate all of the bad classifiers with probability greater than $1- \delta$. This works great as long at $H$ is not large, but often $H$ is large. The number of possible classifiers for $k$ binary features is $2^{2^k}$, so the bound above is sometimes not useful. The number of hypotheses can also be effectively infinite. Similarly, if the number of bad classifiers is large compared to the number of good classifiers, then $\delta$ needs to be very small. Other performance bounds are true only in the limit (e.g. infinite data) or contain unknown constants, so they merely give the user hope rather than a guarantee that the resulting classifier is correct.

FEATURE ENGINEERING IS THE KEY – Machine learning is often an iterative process of generating features, choosing algorithms, and analyzing the results. Creating useful features is often difficult and can require domain knowledge or luck, so “there is ultimately no replacement for the smarts you put into feature engineering.”

LEARN MANY MODELS, NOT JUST ONE – Use ensemble learning including bagging, boosting, bootstrap, stacking, Bayesian optimization, mixture of experts, and plain old averaging/voting. These ensembles of other well-known methods work and win contests like the Netflix prize. Bayesian model averaging (BMA) is theoretically optimal, but it tend to heavily weight one of the learners, so it should not be considered ensemble learning.

MORE DATA BEATS A CLEVERER ALGORITHM – If you are only going to read one section, read this section. The title itself is self-explanatory, but Domingos adds several interesting insights in this section like,

“…even though in principle more data means that more complex classiﬁers can be learned, in practice simpler classiﬁers wind up being used, because complex ones take too long to learn.”

“As a rule, it pays to try the simplest learners ﬁrst (e.g., naive Bayes before logistic regression, k-nearest neighbor before support vector machines). More sophisticated learners are seductive, but they are usually harder to use, because they have more knobs you need to turn to get good results, and because their internals are more opaque.”,

“Part of the reason using cleverer algorithms has a smaller payoﬀ than you might expect is that, to a ﬁrst approximation, they all do the same. This is surprising when you consider representations as different as, say, sets of rules and neural networks. But in fact propositional rules are readily encoded as neural networks, and similar relationships hold between other representations. All learners essentially work by grouping nearby examples into the same class; the key difference is in the meaning of “nearby.” “,

“In the end, the biggest bottleneck is not data or CPU cycles, but human cycles. In research papers, learners are typically compared on measures of accuracy and computational cost. But human eﬀort saved and insight gained, although harder to measure, are often more important. This favors learners that produce human-understandable output (e.g., rule sets). And the organizations that make the most of machine 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 experts and application domain ones.”,

and

“Learners can be divided into two major types: those whose representation has a ﬁxed size, like linear classiﬁers, and those whose representation can grow with the data, like decision trees.”

He says the variable size learners can, in theory, learn any class, but they often fall into local optima, require too much time, or fall to the curse of dimensionality, so these methods require a lot of experimentation and thinking on the part of the analyst.

Lastly, Generative models are more fun.