Overfitting: Model complexity

The previous unit introduced the following model, which miscategorized a lot of trees in the test set:

Figure 16. The misbehaving complex model from the previous unit.

The preceding model contains a lot of complex shapes. Would a simpler model handle new data better? Suppose you replace the complex model with a ridiculously simple model--a straight line.

Figure 17. A much simpler model.

The simple model generalizes better than the complex model on new data. That is, the simple model made better predictions on the test set than the complex model.

Simplicity has been beating complexity for a long time. In fact, the preference for simplicity dates back to ancient Greece. Centuries later, a fourteenth-century friar named William of Occam formalized the preference for simplicity in a philosophy known as Occam's razor. This philosophy remains an essential underlying principle of many sciences, including machine learning.

Note: Complex models typically outperform simple models on the training set. However, simple models typically outperform complex models on the test set (which is more important).

Exercises: Check your understanding

You are developing a physics equation. Which of the following formulas conform more closely to Occam's Razor?

A formula with three variables.

Three variables is more Occam-friendly than twelve variables.

A formula with twelve variables.

Twelve variables seems overly complicated, doesn't it? The two most famous physics formulas of all time (F=ma and E=mc²) each involve only three variables.

You're on a brand-new machine learning project, about to select your first features. How many features should you pick?

Pick 1–3 features that seem to have strong predictive power.

It's best for your data collection pipeline to start with only one or two features. This will help you confirm that the ML model works as intended. Also, when you build a baseline from a couple of features, you'll feel like you're making progress!

Pick 4–6 features that seem to have strong predictive power.

You might eventually use this many features, but it's still better to start with fewer. Fewer features usually means fewer unnecessary complications.

Pick as many features as you can, so you can start observing which features have the strongest predictive power.

Start smaller. Every new feature adds a new dimension to your training dataset. When the dimensionality increases, the volume of the space increases so fast that the available training data become sparse. The sparser your data, the harder it is for a model to learn the relationship between the features that actually matter and the label. This phenomenon is called "the curse of dimensionality."

Regularization

Machine learning models must simultaneously meet two conflicting goals:

• Fit data well.
• Fit data as simply as possible.

One approach to keeping a model simple is to penalize complex models; that is, to force the model to become simpler during training. Penalizing complex models is one form of regularization.

A regularization analogy: Suppose every student in a lecture hall had a little buzzer that emitted a sound that annoyed the professor. Students would press the buzzer whenever the professor's lecture became too complicated. Annoyed, the professor would be forced to simplify the lecture. The professor would complain, "When I simplify, I'm not being precise enough." The students would counter with, "The only goal is to explain it simply enough that I understand it." Gradually, the buzzers would train the professor to give an appropriately simple lecture, even if the simpler lecture isn't as sufficiently precise.

Loss and complexity

So far, this course has suggested that the only goal when training was to minimize loss; that is:

$$\text{minimize(loss)}$$

As you've seen, models focused solely on minimizing loss tend to overfit. A better training optimization algorithm minimizes some combination of loss and complexity:

$$\text{minimize(loss + complexity)}$$

Unfortunately, loss and complexity are typically inversely related. As complexity increases, loss decreases. As complexity decreases, loss increases. You should find a reasonable middle ground where the model makes good predictions on both the training data and real-world data. That is, your model should find a reasonable compromise between loss and complexity.

What is complexity?

You've already seen a few different ways of quantifying loss. How would you quantify complexity? Start your exploration through the following exercise:

Exercise: Check your intuition

So far, we've been pretty vague about what complexity actually is. Which of the following ideas do you think would be reasonable complexity metrics?

Complexity is a function of the model's weights.

Yes, this is one way to measure some models' complexity. This metric is called L₁ regularization.

Complexity is a function of the square of the model's weights.

Yes, you can measure some models' complexity this way. This metric is called L₂ regularization.

Complexity is a function of the biases of all the features in the model.

Bias doesn't measure complexity.

Key terms:

• L₁ regularization
• L₂ regularization
• Regularization