How do different cross-validation methods compare in terms of model variance and bias? 

My question is partly motivated by this thread: Optimal number of folds in $K$-fold cross-validation: is leave-one-out CV always the best choice?. The answer there suggests that models learned with leave-one-out cross-validation have higher variance than those learned with regular $K$-fold cross-validation, making leave-one-out CV a worse choice.

However, my intuition tells me that in leave-one-out CV one should see relatively lower variance between models than in the $K$-fold CV, since we are only shifting one data point across folds and therefore the training sets between folds overlap substantially.

Or going in the other direction, if $K$ is low in the $K$-fold CV, the training sets would be quite different across folds, and the resulting models are more likely to be different (hence higher variance).

If the above argument is right, why would models learned with leave-one-out CV have higher variance?


  why would models learned with leave-one-out CV have higher variance?


[TL:DR] A summary of recent posts and debates (July 2018)

This topic has been widely discussed both on this site, and in the scientific literature, with conflicting views, intuitions and conclusions. Back in 2013 when this question was first asked, the dominant view was that LOOCV leads to larger variance of the expected generalization error of a training algorithm producing models out of samples of size $n(K−1)/K$. 

This view, however, appears to be an incorrect generalization of a special case and I would argue that the correct answer is: "it depends..."

Paraphrasing Yves Grandvalet the author of a 2004 paper on the topic I would summarize the intuitive argument as follows:  


If cross-validation were averaging independent estimates: then leave-one-out CV one should see relatively lower variance between models since we are only shifting one data point across folds and therefore the training sets between folds overlap substantially.
This is not true when training sets are highly correlated: Correlation may increase with K and this increase is responsible for the overall increase of variance in the second scenario. Intuitively, in that situation, leave-one-out CV may be blind to instabilities that exist, but may not be triggered by changing a single point in the training data, which makes it highly variable to the realization of the training set.


Experimental simulations from myself and others on this site, as well as those of researchers in the papers linked below will show you that there is no universal truth on the topic. Most experiments have monotonically decreasing or constant variance with $K$, but some special cases show increasing variance with $K$. 

The rest of this answer proposes a simulation on a toy example and an informal literature review.  

[Update] You can find here an alternative simulation for an unstable model in the presence of outliers. 

Simulations from a toy example showing decreasing / constant variance

Consider the following toy example where we are fitting a degree 4 polynomial to a noisy sine curve. We expect this model to fare poorly for small datasets due to overfitting, as shown by the learning curve. 



Note that we plot 1 - MSE here to reproduce the illustration from ESLII page 243   

 Methodology

You can find the code for this simulation here. The approach was the following:


Generate 10,000 points from the distribution $sin(x) + \epsilon$ where the true variance of $\epsilon$ is known
Iterate $i$ times (e.g. 100 or 200 times). At each iteration, change the dataset by resampling $N$ points from the original distribution
For each data set $i$:


Perform K-fold cross validation for one value of $K$
Store the average Mean Square Error (MSE) across the K-folds

Once the loop over $i$ is complete, calculate the mean and standard deviation of the MSE across the $i$ datasets for the same value of $K$
Repeat the above steps for all $K$ in range $\{ 5,...,N\}$ all the way to Leave One Out CV (LOOCV)


Impact of $K$ on the Bias and Variance of the MSE across $i$ datasets.

Left Hand Side: Kfolds for 200 data points,  Right Hand Side: Kfolds for 40 data points



Standard Deviation of MSE (across data sets i) vs Kfolds 



From this simulation, it seems that:


For small number $N = 40$ of datapoints, increasing $K$ until $K=10$ or so significantly improves both the bias and the variance. For larger $K$ there is no effect on either bias or variance.
The intuition is that for too small effective training size, the polynomial model is very unstable, especially for $K \leq 5$ 
For larger $N = 200$ - increasing $K$ has no particular impact on both the bias and variance. 




An informal literature review

The following three papers investigate the bias and variance of cross validation


Ron Kohavi (1995)
Yves Grandvalet and Yoshua Bengio (2004)
Zhang and Yang (2015) 


Kohavi 1995

This paper is often refered to as the source for the argument that LOOC has higher variance. In section 1: 


  “For example, leave-oneout is almost unbiased, but it has high variance, leading to unreliable estimates (Efron 1983)" 


This statement is source of much confusion, because it seems to be from Efron in 1983, not Kohavi. Both Kohavi's theoretical argumentations and experimental results go against this statement:

Corollary 2 ( Variance in CV)


  Given a dataset and an inducer. If the inducer is stable under the perturbations caused by deleting the test instances for the folds in k-fold CV for various values of $k$, then the variance of the estimate will be the same


Experiment
In his experiment, Kohavi compares two algorithms: a C4.5 decision tree and a Naive Bayes classifier across multiple datasets from the UC Irvine repository. His results are below: LHS is accuracy vs folds (i.e. bias) and RHS is standard deviation vs folds



In fact, only the decision tree on three data sets clearly has higher variance for increasing K. Other results show decreasing or constant variance. 

Finally, although the conclusion could be worded more strongly, there is no argument for LOO having higher variance, quite the opposite. From section 6. Summary 


  "k-fold cross validation with moderate k values (10-20) reduces the variance... As k-decreases (2-5) and the samples get smaller, there is variance due to instability of the training sets themselves. 


Zhang and Yang

The authors take a strong view on this topic and clearly state in Section 7.1


  In fact, in least squares linear regression, Burman (1989) shows that among the k-fold CVs, in estimating the prediction error, LOO (i.e., n-fold CV) has the smallest asymptotic bias and variance.  ... 
  
  ... Then a theoretical calculation (Lu, 2007) shows that LOO has the smallest bias and variance at the same time among all delete-n CVs with all possible n_v deletions considered


Experimental results
Similarly, Zhang's experiments point in the direction of decreasing variance with K, as shown below for the True model and the wrong model for Figure 3 and Figure 5. 





The only experiment for which variance increases with $K$ is for the Lasso and SCAD models. This is explained as follows on page 31: 


  However, if model selection is involved, the performance of LOO worsens in variability as the model selection uncertainty gets higher due to large model space, small penalty coefficients and/or the use of data-driven penalty coefficients

