1. We fit a full classification tree model $T_k$ of given depth $k$ to data using the CART algorithm, and prune the tree by finding $E(k, \alpha) = min_{T\subset Tk} Err(T) + \alpha |T|$. Here, $Err(T)$ is the training error of a tree $T$ on the training data, $|T|$ is the number of leaves in $T$ and $\alpha$ is a given parameter. Which of the following is sometimes false?

$\color{blue}{(a) E(k+ 1,0) \geq E (k,0)}$

(b) $E(k+ 1,0) \leq E(k,0)$

(c) $E(k+ \alpha+1) \geq E(k,\alpha)$

(d) All three statements above are always true

$\color{blue}{Explanation:}$ By supplying the depths $k$ and $k + 1$, the Cart algorithm will return us with a tree of depth $k, T,$ and a tree of depth $k + 1, \widetilde {T}$. It is important to note that $T$ and $\widetilde {T}$ share the same first $k - 1 $ levels.

Now the pruning starts, any possible pruning of $T$ can be achieved by pruning $\widetilde {T}$, but not vise versa. Therefore $E(k, \alpha) \geq E(k + 1, \alpha)$ for any $\alpha \in R_+.$$ See Lecture 7.

So this is taken from an exam I just did. I'd like to know if there are any instances same as in the image where the CART algorithm could use a negative alpha and thus encourage a larger tree? Or does the algorithm state that alpha must be a non negative integer at all times?


3 Answers 3


Making an additional split will always decrease $Err$ (until pure leaves are reached), so any negative $\alpha$ will have the same minimizing $T$ as $\alpha=0$. So we might as well take $\alpha\geq0$. (But there is no requirement that $\alpha$ be an integer.)

Erm, unless you implement things so that even a pure node can be split, in which case a negative $\alpha$ will make the tree split even these, until every leaf consists of a single sample (and duplicates thereof) or the maximum depth is reached.


Or does the algorithm state that alpha must be a non negative integer at all times?

As far as I know that is actually the case:

In "The Elements of Statistical Learning" (1) on page 308 the authors write:

The tuning parameter $\alpha \geq 0$ governs the tradeoff between tree size and its goodness of fit to the data.

Also, the scikit-learn documentation says:

Minimal cost-complexity pruning is an algorithm used to prune a tree to avoid over-fitting, described in Chapter 3 of [BRE]. This algorithm is parameterized by $\alpha \geq 0$ known as the complexity parameter.

Both sources refer to "Classification and Regression Trees" (2) as the original source.

(1) "The Elements of Statistical Learning"; Hastie et al; 2nd Ed; 2008

(2) "Classification and Regression Trees"; Breiman, et al; 1984


𝛼 is a parameter used for regularization which is controlling the tree from being overfit , an inherent issue with CART algorithms. A negative 𝛼 would lead to a larger tree which may accentuate the overfitting issue


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