Overfitting manifests itself when you have a model that does not generalize well. Say that you achieve a classification accuracy rate on your training data of 95 percent, but when you test its accuracy on another set of data, the accuracy falls to 50 percent. This would be considered a high variance. If we had a case of 60 percent accuracy on the train data and 59 percent accuracy on the test data, we now have a low variance but a high bias. This bias-variance trade-off is fundamental to machine learning and model complexity.

Let's nail down the definitions. A bias error is the difference between the value or class that we predict and the actual value or class in our training data. A variance error is the amount by which the predicted value or class in our training set differs from the predicted value or class versus the other datasets. Of course, our goal is to minimize the total error (bias + variance), but how does that relate to model complexity?

For the sake of argument, let's say that we are trying to predict a value and we build a simple linear model with our train data. As this is a simple model, we could expect a high bias, while on the other hand, it would have a low variance between the train and test data. Now, let's try including polynomial terms in the linear model or build decision trees. The models are more complex and should reduce the bias. However, as the bias decreases, the variance, at some point, begins to expand and generalizability is diminished. You can see this phenomena in the following illustration. Any machine learning effort should strive to achieve the optimal trade-off between the bias and variance, which is easier said than done.

We will look at methods to combat this problem and optimize the model complexity, including cross-validation (Chapter 2, Linear Regression - The Blocking and Tackling of Machine Learning. through Chapter 7, Neural Networks) and regularization (Chapter 4, Advanced Feature Selection in Linear Models).