Submitted by
Assigned_Reviewer_3
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper outlines theoretical methods for gauging
the relevance of a particular variable when using forests of randomized
trees (using pary splits and discrete variables). Given any (of the
commonly used) impurity measures, several theorems are developed
describing the MDI (importance) of a variable in an asymptotic case. The
asymptotic condition is weakened and less rigorous (but still useful)
claims are made about randomized trees that resemble those used in
"typical" Random Forests. Finally, an experiment verifies some of the
theorems and helps give some more intuition.
In terms of quality
and clarity, the paper is very well written and is relatively easy to
follow. In my opinion, some of the theorems could use a little more
intuition and motivation.
I am not an expert in the field, but in
terms of originality and significance, this seems to be the first paper
attempting to analyze the importance of variables in the context of
randomized trees  which is useful (however it would help to give more
motivation for this point in the paper).
Since the theorems apply
to asymptotic cases with fairly specific tree structures, it would greatly
help to demonstrate (at least empirically) with realworld experiments how
much the results deviate from the theory in the case of (smaller) finite
trees, different number of splits per node, different variable subset
sizes K, and different impurity measures. The 7segment experiment is nice
but doesn't help quantify the deviations.
I would have also liked
to see a bit more motivation and description of prior work to help put
this work in context. Q2: Please summarize your review in
12 sentences
A well written paper outlining several theoretical
measures (albeit applicable in very specific conditions). Would help to
have more motivation and some analysis of the deviation of the results
from those using ideal conditions. Submitted by
Assigned_Reviewer_5
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper attempts to analyze theoretically a popular
measure of variable importance, computed by random forests (RF) or
extratrees. The gap between the practical success of RF, in particular
for feature selection, and its theoretical understanding is so huge that
the author's efforts should be praised.
The main results of this
paper show that, in a particular RF model (discrete attributes, totally
randomized and fully developed), we can obtain an exact estimate of the RF
importance in terms of mutual information. This is a very nice result,
which allows to understand what is estimated by the RF algorithm. The
extension to slightly more realistic models (pruned RF, or RF where at
each node, the best among K candidate splits is chosen) shows that the
situation is more complicated in these cases, but give useful hints at
what are the mechanisms which play a role. The experiments illustrate
nicely the main findings of the paper.
With a little more effort,
one assumes that results (bounds) for a finite mixture of trees could be
obtained too.
There is still a gap between the theory of RF and
their practical use, but this paper offers a nice improvement over the
stateoftheart which is almost void for RF feature importance. It would
certainly be of interest for people interested in understanding the good
practical behavior of RF. Q2: Please summarize your
review in 12 sentences
A very nice theoretical analysis of the variable
importance computed by some simplified models of RF. Only caveat is that
extending the theory to more realistic models of RF is difficult, but this
paper is among the first to say something about it. Submitted
by Assigned_Reviewer_6
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
Variable importance measures are often used to
highlight key predictor variables in treebased ensemble models. However,
there has generally been a lack of a theoretical understanding of these
measures. In this paper, the authors study the theoretical properties of
the Mean Decrease Impurity (MDI) importance measures (such as the Gini
importance). Through most of the paper they use the Shanon entropy as the
impurity measure but show that the theorems and results are applicable to
any impurity measure. They begin with an asymptotic analysis of totally
randomized fullydeveloped tree ensembles learned using an infinitely
large ensemble. They show that under these conditions the variable
importance for all variables provide a threelevel decomposition of the
information contained in the set of input variables about the output
variable. First, they show that the information decomposes as a sum of
importance scores of all input variables. Second, the importance of each
variable decomposes as a sum over all degrees of interactions of that
variable with other variables. The third level of decomposition for each
degree is over all possible combinatorial interactions of that degree.
Next, the authors show that that the MDI importance of a variable is equal
to zero if and only if the variable is irrelevant and that the MDI
importance of a relevant variable is invariant with respect to the removal
or the addition of irrelevant variables. The authors then analyze the
properties of the variable importance measures in depthpruned ensembles
of randomized trees. They show that as long as the pruning depth is
greater than the total number of true relevant variables, these relevant
variables still obtain a strictly positive importance score (albeit
different from their values in fullydeveloped trees) and the irrelevant
variables continue to obtain an importance of zero. Finally, for the case
of ensembles consisting of nontotally randomized trees such as random
forests, where top scoring predictors are selected at splits from a random
subset of predictors, they show strong masking effects by the strongest
variables resulting in over or underestimation of importance scores. They
finally show the relevance of their results through a simple yet
insightful example of digit recognition.
The paper is very
accessible and written in a very understandable manner. At the same time,
the theorems and results discussed are very interesting and provide some
nice theoretical insights into variable importance measures from idealized
asymptotic models to more realistic models. The masking effects by the
strongest predictors and the inability of the variable importance to
appropriately account for interactions and dependencies between variables
for randomforest like models are very relevant to practical analyses and
may be amplified even more in cases involving correlated or partially
correlated variables. I think the paper lays down some nice initial
results and a general framework to analyze more realistic models in the
future and potentially point to better strategies for learning trees or
computing more accurate importance scores. The paper will definitely be of
interest to the general machine learning
community. Q2: Please summarize your review in 12
sentences
The paper provides an insightful theoretical analysis
of variable importance measures in treebased ensemble models ranging from
idealized asymptotic models to realistic randomforest like models. It
provides a key framework to study these types of measures and potentially
improve them.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
Dear reviewers,
We would like to thank you for
your positive reviews. We agree with the assessment of our submission.
Best regards,
The authors.
