Model Complexity and Accuracy

Imagine a model that gets every single training example right. It feels like a success, and it is tempting to call the job finished. In supervised learning that feeling is misleading. A model exists to make predictions about cases it has never seen, and a perfect score on the cases it already studied says little about that goal.

The belief behind the mistake is easy to understand. We normally build a model so that it predicts the training data accurately, and we hope that accuracy carries over to new data. Often that hope is justified when the training and test sets have enough in common. The problem is that the training score alone cannot tell us whether the hope is justified in a particular case.

To see how this plays out, consider a made-up scenario. A novice data scientist wants to predict whether a customer will buy a boat, using records of past boat buyers and customers known not to be interested. The goal is to send promotional emails only to people who are likely to purchase. The records the scientist has available cover 12 people whose answers are already known.

Work through the memorizer one request at a time. Its behavior has two cases: an age it has stored, and an age it has never seen. The trace below shows that the training score and the answer for a new customer come from completely different sources.

A training score measures how well a model reproduces answers it was shown. Generalization measures how well it predicts answers it was not shown. These two quantities are different, and they can differ enormously. The memorizer shows the gap at its most extreme: a perfect training score alongside no real ability to predict new customers.

The diagram shows where generalization is actually tested. Training examples are used to fit the model, but the model is judged on predictions for new data, which are compared with answers the model never used during fitting. A high score in the first stage tells you only that fitting worked.

After studying the table, the novice scientist proposed this rule: if the customer is older than 45 and has fewer than 3 children or is not divorced, then they want to buy a boat. Asked how well it works, the scientist said it is 100 percent accurate. On the table itself, that claim is true. But the scientist's real aim was never to describe these 12 people, whose answers are already known. The aim is to predict new customers, and a rule that is perfect on the known table may fail on people outside it.

There is also a way to fit the table perfectly without finding any real rule. Because no age appears twice in the data, one could list the ages of the boat buyers and declare that exactly those people want a boat. That rule is also perfectly accurate on the table, but it is memorization with a different label. Both the novice's rule and this lookup are warnings: many rules can explain the training data perfectly, and perfect explanation of the training data is not the same as a rule that works on new customers.

The before side of the comparison is the memorizer's structure: a lookup of stored pairs that scores perfectly on what it has seen. The after side describes a model whose rule can answer for cases outside the stored list. The source's point is that the training score cannot tell these two apart, so the score must be paired with a check on examples the model did not study.

Model complexity is the lever that connects these ideas. If we allow ourselves to build very complex models, we can always be as accurate as we like on the training set. A model that is too complex fits the training data too closely and may fail to generalize. This situation is called overfitting. The novice's rule is an example: its clause 'or is not divorced' depends on a single customer, so it is unlikely to hold up for new customers.

The opposite failure is underfitting. Here the model is too simple to capture the underlying patterns in the training data. It does not fit the training data well, and so it also does not generalize well. Overfitting and underfitting sit at opposite ends of a complexity range, and a model that generalizes well lies somewhere between them.

Read the two columns against each other. Both failure modes produce poor predictions on new data, but they arise for different reasons and look different on the training data. An underfit model is weak on the training data as well as on new data. An overfit model can look excellent on the training data and still fail on new data, which is why the training score is the misleading signal.

The table below sets the two failure modes side by side so that a model can be checked against each row. Use the complexity of the model together with its behavior on training data and on new data.

The practical habit this topic builds is to refuse to judge a model by its training score alone. Each check below follows directly from the definition of generalization.