Overfitting

[3]: 45 Underfitting occurs when a mathematical model cannot adequately capture the underlying structure of the data.

A function class that is too large, in a suitable sense, relative to the dataset size is likely to overfit.

To lessen the chance or amount of overfitting, several techniques are available (e.g., model comparison, cross-validation, regularization, early stopping, pruning, Bayesian priors, or dropout).

Burnham & Anderson, in their much-cited text on model selection, argue that to avoid overfitting, we should adhere to the "Principle of Parsimony".

Is the monkey who typed Hamlet actually a good writer?In regression analysis, overfitting occurs frequently.

The goal is that the algorithm will also perform well on predicting the output when fed "validation data" that was not encountered during its training.

For an example where there are too many adjustable parameters, consider a dataset where training data for y can be adequately predicted by a linear function of two independent variables.

Everything else being equal, the more difficult a criterion is to predict (i.e., the higher its uncertainty), the more noise exists in past information that needs to be ignored.

Other negative consequences include: The optimal function usually needs verification on bigger or completely new datasets.

This matrix can be represented topologically as a complex network where direct and indirect influences between variables are visualized.

Underfitting is the inverse of overfitting, meaning that the statistical model or machine learning algorithm is too simplistic to accurately capture the patterns in the data.

In this case, bias in the parameter estimators is often substantial, and the sampling variance is underestimated, both factors resulting in poor confidence interval coverage.

Underfitted models tend to miss important treatment effects in experimental settings.There are multiple ways to deal with underfitting: Benign overfitting describes the phenomenon of a statistical model that seems to generalize well to unseen data, even when it has been fit perfectly on noisy training data (i.e., obtains perfect predictive accuracy on the training set).

The phenomenon is of particular interest in deep neural networks, but is studied from a theoretical perspective in the context of much simpler models, such as linear regression.

In other words, the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size.

Figure 1.  The green line represents an overfitted model and the black line represents a regularized model. While the green line best follows the training data, it is too dependent on that data and is likely to have a higher error rate on new unseen data, illustrated by black-outlined dots, compared to the black line.
Figure 2.  Noisy (roughly linear) data is fitted to a linear function and a polynomial function. Although the polynomial function is a perfect fit, the linear function can be expected to generalize better: If the two functions were used to extrapolate beyond the fitted data, the linear function should make better predictions.
Figure 3.  The blue dashed line represents an underfitted model. A straight line can never fit a parabola. This model is too simple.
Figure 4. Overfitting/overtraining in supervised learning (e.g., a neural network ). Training error is shown in blue, and validation error in red, both as a function of the number of training cycles. If the validation error increases (positive slope) while the training error steadily decreases (negative slope), then a situation of overfitting may have occurred. The best predictive and fitted model would be where the validation error has its global minimum.
Figure 5.  The red line represents an underfitted model of the data points represented in blue. We would expect to see a parabola shaped line to represent the curvature of the data points.
Figure 6.  The blue line represents a fitted model of the data points represented in green.