When selecting $K$ for K-Fold CV, we should consider a bias-variance trade-off. K-Fold CV provides $K$ estimates of the model error. The mean of the errors indicates the model's bias. A lower mean value indicates a higher accuracy of the model. The model variance is calculated based on the standard deviation of the errors. A lower variance indicates a low variation in the model performance. Usually, it is difficult to reduce both model bias and variance.

K-Fold Cross-Validation Bias-Variance Tradeoff

K-Fold CV is usually the method of choice for CV, and functions as a sort of generalization of LOOCV, because:
1. It is much less computationally expensive than LOOCV.
2. It minimizes the **Bias-Variance Tradeoff**.

K-Fold Cross-Validation vs. Leave-One-Out Cross-Validation

$$5 \leq K \leq 10$$ is commonly used in Macine Learning. A too-small $K$ may lead to high bias, and a too-large $K$ may result in high variance. 

$$5 \leq K \leq 10$$ in K-Fold Cross-Validation

When evaluating numerous hyperparameter configurations, the risk of multiple testing increases, leading to situations where validation performance appears favorable simply by chance rather than reflecting true generalization capability. However, when applied to a sufficiently large dataset with a standard range of hyperparameters, $$K$$-fold cross-validation tends to be reasonably resilient against this issue, providing a more robust estimate of true error than a single validation split.

Multiple Testing Resilience in K-Fold Cross-Validation

Throughout the training and model selection process, continuously monitoring both training and validation errors is crucial for diagnosing learning behavior. If a model exhibits an extremely low training error for a specific hyperparameter configuration while simultaneously yielding a considerably higher error during $$K$$-fold cross-validation, it serves as a strong indicator of overfitting. This divergence demonstrates that the model is memorizing the training data rather than generalizing well to unseen validation folds.

Identifying Overfitting via Cross-Validation

To perform $$K$$-fold cross-validation on a dataset with $$n$$ observations, the training set is separated into $$K$$ sets by randomly sampling $$n/K$$ observations from the dataset into each fold. The model of interest is then trained on $$K-1$$ folds and validated on the left-out fold; that is, once the model is trained, we use it to make predictions on the group that was left out of training. This process repeats $$K$$ times. To evaluate the model's overall performance, we calculate the average validation error across all $$K$$ tests. $$K$$-fold cross-validation provides a robust estimate of the empirical testing error of a model and is particularly useful for selecting model designs and adjusting hyperparameters.

University of Michigan - Ann Arbor

Claude

1. Validation set
2. Leave-One-Out Cross-Validation
3. k-fold Cross-Validation

Cross-Validation resampling methods

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Dive into Deep Learning

If we have $n$ data points in our training set, we train the model on $n-1$ data points. Then, we make a prediction on the observation that we left out. We repeat this process repeats $n$ times, and we will average the performance of our model for each of the predictions that we make. This average will give us model performance.

Leave-One-Out Cross-Validation

K-Fold Cross-Validation

The advantage of k-fold CV is that it often gives more accurate estimates of the test error rate than does LOOCV. This has to do with a bias-variance trade-off.
The validation set approach can lead to overestimates of the test error rate, since in this approach the training set used to fit the statistical learning method contains only half the observations of the entire data set ( it means the model does not have enough information about the data set.)
LOOCV will give approximately unbiased estimates of the test error since each training set contains n − 1 observation, which is almost as many as the number of observations in the full data set.
K-fold CV for, say, k = 5 or k = 10 will lead to an intermediate level of
bias, since each training set contains (k − 1)n/k observations—fewer than
in the LOOCV approach, but substantially more than in the validation set
approach.

Note:
LOOCV has a higher variance than does K-fold CV with k < n.

Bias-Variance Trade Off Comparison

Stratified folds each contain a proportion of classes that matches the overal dataset. Now, all cclasses will be fairly represented in the test set.

Stratified Cross-Validation

Which of the following is true of cross-validation?

A validation dataset (or validation set) is a separate partition of data used to assess the quality of a machine learning model's solution. After training a model iteratively on the training data, its predictive performance is evaluated on this distinct validation dataset to measure how well it has learned the underlying patterns. This evaluation is critical for automatically tuning hyperparameters without relying on the training set, ensuring the model generalizes appropriately to unseen data.

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