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

from collections import defaultdict

def parent(node):
    """ Given a node in a tree, returns its parent node.

    node : str
        Node whose parent I'm interested in.

        Parent node of the input node or None if the input Node is already a
        root node.

    In truth, returning '' for root nodes would be acceptable. However,
    None values force us to think really hard about our assumptions at every
    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 : set
        Set of all subtrees that can be found in a given taxonomy.

        A cost for every possible node in the taxonomy.

    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
        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])
                # 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.

    leaves : set
        A set of selected subtrees from the overall category tree.

        A set containing the original subtrees plus all possible subtrees
        contained in these leaves.

    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)):
    return full_set

def h_loss(labels1, labels2, node_cost):
    """ Calculates the Hierarchical loss for the given two sets.

    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.

    If you want a loss between 0 and 1, the `nodes_to_cost` function implements
    such a function.

        Loss between the two given sets.

    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) - \
    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_2=set(['01.01', '02'])
    print(h_loss(node_1, node_2, weights))

The guerrilla guide to looking good online

I recently confirmed that high quality audio makes you sound smarter, which is exactly what I always wanted: a way to look smart without having to actually work for it. This discovery led me to an internet rabbit hole on how to look and sound good online, with this guide being the end result. If you are trapped inside online-meeting purgatory like me, hopefully this guide will give you a small edge in your next important meeting.

I divided this guide in three sections:

  • Video: while video is not as important as audio, it's the easiest one to improve. You may not know exactly how to better equalize your voice, but identifying which part of your face needs light is easy.
  • Audio: you can live with bad video (or no video at all) but bad audio is a different issue.
  • Delivery: once you are clearly seen and heard, let's talk about how to improve your message.

Before we start, a couple words of meta-advice: by caring about how you look and sound you are already ahead of everyone else who just turns their computer on and shows up. And since you can't improve when you don't know what is there to improve, your first step is to go get some feedback. If you can't find someone willing to have a meeting with you then you should at least have a test meeting alone. For instructions click here: Zoom, Teams, Jitsi.


As this guide points out you should ensure there is no strong light coming from behind you. Ideally you want a three-point lighting setup but a good compromise is a general strong light source (such as an open window or room light) plus diffuse light behind your monitor (either point a lamp towards the wall behind your monitor or a full-screen, white document opened on your screen). The next step up are ring lights, but we have other, more pressing issues to worry about first.

Your background comes next. Most meeting software nowadays includes a "blurry background" filter that you can use to hide what's going on behind you. These filters don't work as good as I wish they did, but they have nonetheless been a blessing for those of us sitting in shared living rooms. Still: consider re-orienting your camera (or your desk!) to keep a clear, distraction-free background.

Which brings us to the final point: the camera itself. Whichever camera you have around is likely to be fine. A more expensive camera will give better results, but they might not be worth the cost. Pro tip: if you have a DSLR camera laying around, it may also double as an amazing webcam too. Check your manual.

Pay attention to the camera angle. Keep your camera at eye level either by repositioning your webcam (if it's an external one) or by getting yourself a laptop stand (which you can also build out of cardboard). Say no to cameras looking at you from below!


Audio is tricky: it is more important than video during conferences but it's harder to tune adequately. Let's get the obvious out of the way: ideally you want a quiet room for your meeting, but there's only so much you can do with the rooms your apartment already has. So let's not dwell on that.

Unlike video, where you can get far with what you have, in audio you really, really want to have a better microphone. You don't have to go pro (in fact, an expensive mic can easily backfire by being too sensitive) but you should at least get a decent, dedicated one. If you have no idea of audio then I would recommend a USB mic - I have had bad experiences with microphones picking up line noise and USB should help with that. And since using your speakers is guaranteed to cause echo sooner or later, save yourself the trouble and get some headphones too.

If you want to tweak your voice even more you can try a software equalizer. There are plenty of guides around courtesy of the internet, but getting into details goes beyond this guide.


Once you have optimized your environment as much as possible, it is time to talk about delivery. That's a topic by itself, so I'll limit myself to two tips:

  1. Dress appropriately and keep a neat background. Bookshelves are particularly nice. If you show a messy room your audience will assume you are also a messy person with messy ideas, and no one wants that.
  2. You don't have to go and get a voice coach, but it might be worth your time to watch a couple videos on the topic. I have personally learned a lot from the Broadcast Voice Handbook but you might be better served by more casual online courses. Youtubers have created an explosion of content on that area, so it should be easy to find.