Which statistical learning method usually results in increased training MSE with flexibility?

The concept of increased training mean squared error (MSE) with flexibility relates closely to the behavior of statistical models as they increase in complexity. High flexibility methods, such as decision trees or neural networks with many nodes, can capture intricate patterns and relationships within a dataset. However, their ability to fit the training data can also lead to overfitting, where the model learns not just the underlying trends but also the noise present in the training data.

As a result, while these high flexibility methods may perform exceptionally well on the training set by minimizing training error, they can lead to higher training MSE due to overfitting. Essentially, this increased flexibility allows the model to be more adaptable to the training coefficients, but it may compromise the model's ability to generalize well, thus potentially increasing MSE even within the training data.

In contrast, regularization techniques, model averaging, and low complexity methods generally tend to reduce the impact of overfitting and promote more generalizable models by controlling for excessive flexibility. This characteristic distinguishes high flexibility methods from the rest and underscores why they would result in increased training MSE despite their adaptive capabilities.