## Hierarchical loss for multi-label classification

Here's one of those problems that sounds complicated but, when you take a deep dive into it, turns out to be just as complicated as it sounds.

Suppose you build a classifier that takes a book and returns its classification according to the Dewey Decimal System. This classifier would take a book such as "The return of Sherlock Holmes" and classify it as, say, "Fiction".

Of course, life is rarely this easy. This book in particular is more often than not classified as 823.8, "Literature > English > Fiction > Victorian period 1837-1900". The stories, however, were written between 1903 and 1904, meaning that some librarians would rather file it under 823.912, "Literature > English > Fiction > Modern Period > 20th Century > 1901-1945".

Other books are more complicated. Tina Fey's autobiography Bossypants can be classified under any of the following categories:

- Arts and Recreation > Amusements and Recreation > Public Entertainments, TV, Movies > Biography And History > Biography
- Arts and Recreation > Amusements and Recreation > Stage presentations > Biography And History > Biography
- Literature > American And Canadian > Authors, American and American Miscellany > 21st Century

This is known as a *hierarchical multi-label classification problem*:

- It is
*hierarchical*because the expected classification is part of a hierarchy. We could argue whether Sherlock Holmes should be classified as "Victorian" or "Modern", but we would all agree that either case is not as bad as classifying it under "Natural Science and Mathematics > Chemistry". - It is
*multi-label*because there is more than one possible valid class. Tina Fey is*both*a Public entertainer and an American. There is no need to choose just one. - It is
*classification*because we need to choose the right bin for this book. - It is a
*problem*because I had to solve it this week and it wasn't easy.

There seems to be exactly one paper on this topic, Incremental algorithms for hierarchical classification, and is not as easy to read as one would like (and not just because it refers to Section 4 when in reality should be Section 5). Luckily, this survey on multi-label learning presents a simpler version.

I ended up writing a test implementation to ensure I had understood the solution correctly, and decided that it would be a shame to just throw it away. So here it is. This version separates levels in a tree with '.' characters and is optimized for clarity.

*Edit June 17: this algorithm doesn't work too well in practice. I'll write
about its shortcomings soon, but until then you should think twice about using
it as it is.*

*Edit June 26: Part II of this article is now
up*

```
#!/usr/bin/python
from collections import defaultdict
def parent(node):
""" Given a node in a tree, returns its parent node.
Parameters
----------
node : str
Node whose parent I'm interested in.
Returns
-------
str
Parent node of the input node or None if the input Node is already a
root node.
Notes
-----
In truth, returning '' for root nodes would be acceptable. However,
None values force us to think really hard about our assumptions at every
moment.
"""
parent_str = '.'.join(node.split('.')[:-1])
if parent_str == '':
parent_str = None
return parent_str
def nodes_to_cost(taxonomy):
""" Calculates the costs associated with errors for a specific node in a
taxonomy.
Parameters
----------
taxonomy : set
Set of all subtrees that can be found in a given taxonomy.
Returns
-------
dict
A cost for every possible node in the taxonomy.
References
----------
Implements the weight function from
Cesa-bianchi, N., Zaniboni, L., and Collins, M. "Incremental algorithms for
hierarchical classification". In Journal of Machine Learning Research,
pages 31–54. MIT Press, 2004.
"""
assert taxonomy == all_subtrees(taxonomy), \
"There are missing subnodes in the input taxonomy"
# Set of nodes at every depth
depth_to_nodes = defaultdict(set)
# How many children does a node have
num_children = defaultdict(int)
for node in taxonomy:
depth = len(node.split('.'))-1
depth_to_nodes[depth].add(node)
parent_node = parent(node)
if parent_node is not None:
num_children[parent_node] += 1
cost = dict()
for curr_depth in range(1+max(depth_to_nodes.keys())):
for node in depth_to_nodes[curr_depth]:
if curr_depth == 0:
# Base case: parent node
cost[node] = 1.0/len(depth_to_nodes[curr_depth])
else:
# General case: node guaranteed to have a parent
parent_node = parent(node)
cost[node] = cost[parent_node]/num_children[parent_node]
return cost
def all_subtrees(leaves):
""" Given a set of leafs, ensures that all possible subtrees are
included in the set too.
Parameters
----------
leaves : set
A set of selected subtrees from the overall category tree.
Returns
-------
set
A set containing the original subtrees plus all possible subtrees
contained in these leaves.
Notes
-----
Example: if leaves = {"01.02", "01.04.05"}, then the returned value is the
set {"01", "01.02", "01.04", "01.04.05"}.
"""
full_set = set()
for leave in leaves:
parts = leave.split('.')
for i in range(len(parts)):
full_set.add('.'.join(parts[:i+1]))
return full_set
def h_loss(labels1, labels2, node_cost):
""" Calculates the Hierarchical loss for the given two sets.
Parameters
----------
labels1 : set
First set of labels
labels2 : set
Second set of labels
node_cost : dict
A map between tree nodes and the weight associated with them.
Notes
-----
If you want a loss between 0 and 1, the `nodes_to_cost` function implements
such a function.
Returns
-------
float
Loss between the two given sets.
References
----------
The nicer reference of the algorithm is to be found in
Sorower, Mohammad S. "A literature survey on algorithms for multi-label
learning." Oregon State University, Corvallis (2010).
"""
# We calculate the entire set of subtrees, just in case.
all_labels1 = all_subtrees(labels1)
all_labels2 = all_subtrees(labels2)
# Symmetric difference between sets
sym_diff = all_labels1.union(all_labels2) - \
all_labels1.intersection(all_labels2)
loss = 0
for node in sym_diff:
parent_node = parent(node)
if parent_node not in sym_diff:
loss += node_cost[node]
return loss
if __name__ == '__main__':
# Simple usage example
taxonomy = set(["01", "01.01", "01.02", "01.03", "01.04", "01.05",
"02", "02.01", "02.02", "02.03", "02.03.01"])
weights = nodes_to_cost(taxonomy)
node_1=set(['01'])
node_2=set(['01.01', '02'])
print(h_loss(node_1, node_2, weights))
```