Machine Learning – Bias and Variance of a Decision Tree for Classification Explained

bias-variance tradeoffcartmachine learning

There are lot of discussion about bagging and boosting in the context of decision trees and how Random Forest and other methods helps to tackle bias and variance. But how exactly can I measure bias and variance in a decision tree ? If I am performing a k-fold cross validation, similar to a least squares regression can I calculate Bias and Variance ? I see many packages allows the user to input a parameter for folds in a Random Forest algorithm. Is the output of cross validation (which will be a confusion matrix) be used to create a bias and variance metric of some sort ? Can you please explain the bias and variance in the context of a binary classification using decision tree and how cross validation helps ?

Best Answer

There is nothing special about estimating bias and variance in ensemble methods (whether bagging or boosting). It is just like estimating them for any other supervised learner.

To estimate bias you start by assuming a fixed theoretical limit of accuracy, aka Bayesian Risk. Let's say this limit corresponds to 100% accuracy. Then you calculate the training error. The difference between the accuracy on training data and the the best achievable accuracy is an estimate of the bias. For example, if you get an 80% accuracy, then you have a bias problem.

Afterwards you calculate the accuracy on a test set that you kept aside (i.e. you didn't train on). The difference between training error/accuracy and the test error/accuracy is an estimate of variance.

More accurate estimates of the variance can be computed using k-fold cross validation.

[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

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

Random Forest – Why Bagged Trees Have Higher Bias than Single Decision Trees

I will accept the answer on 1) from Kunlun, but just to close this case, I will here give the conclusions on the two questions that I reached in my thesis (which were both accepted by my Supervisor):

1) More data produces better models, and since we only use part of the whole training data to train the model (bootstrap), higher bias occurs in each tree (Copy from the answer by Kunlun)

2) In the Random Forests algorithm, we limit the number of variables to split on in each split - i.e. we limit the number of variables to explain our data with. Again, higher bias occurs in each tree.

Conclusion: Both situations are a matter of limiting our ability to explain the population: First we limit the number of observations, then we limit the number of variables to split on in each split. Both limitations leads to higher bias in each tree, but often the variance reduction in the model overshines the bias increase in each tree, and thus Bagging and Random Forests tend to produce a better model than just a single decision tree.