Maths, machine learning, brute force and elegance

Back when I was at the International Maths Olympiad Training Camp in Mumbai in 1999, the biggest insult one could hurl at a peer was to describe the latter’s solution to a problem as being a “brute force solution”. Brute force solutions, which were often ungainly, laboured and unintuitive were supposed to be the last resort, to be used only if one were thoroughly unable to implement an “elegant solution” to the problem.

Mathematicians love and value elegance. While they might be comfortable with esoteric formulae and the Greek alphabet, they are always on the lookout for solutions that are, at least to the trained eye, intuitive to perceive and understand. Among other things, it is the belief that it is much easier to get an intuitive understanding for an elegant solution.

When all the parts of the solution seem to fit so well into each other, with no loose ends, it is far easier to accept the solution as being correct (even if you don’t understand it fully). Brute force solutions, on the other hand, inevitably leave loose ends and appreciating them can be a fairly massive task, even to trained mathematicians.

In the conventional view, though, non-mathematicians don’t have much fondness for elegance. A solution is a solution, and a problem solved is a problem solved.

With the coming of big data and increased computational power, however, the tables are getting turned. In this case, the more mathematical people, who are more likely to appreciate “machine learning” algorithms recommend “leaving it to the system” – to unleash the brute force of computational power at the problem so that the “best model” can be found, and later implemented.

And in this case, it is the “half-blood mathematicians” like me, who are aware of complex algorithms but are unsure of letting the system take over stuff end-to-end, who bat for elegance – to look at data, massage it, analyse it and then find that one simple method or transformation that can throw immense light on the problem, effectively solving it!

The world moves in strange ways.

Understanding different kinds of art

There are some kinds of art that I intuitively understand – like an elegant mathematical proof, or a beautiful combination in a game of chess; a Sachin Tendulkar straight drive, or a long-distance beautifully threaded pass by Xabi Alonso. I can easily appreciate a well-done-up home when I see it. Some music makes me go delirious, and there have been times when I’ve actually started rolling on the floor in ecstasy after listening to certain songs.

But there is art that I simply don’t get. Poetry – for example – I’ve never got what is the big deal with that. To me it just looks like a bunch of sentences broken up in random ways, which is supposed to make it sound nice. In fact, I’ve argued earlier that poetry is a vestige of the pre-writing era.

It is the same with “literature”. Some people read books or articles because they are just “written beautifully”.  I absolutely fail to appreciate that phrase. As long as something is explained simply and intuitively, it is enough for me. In fact, when a writer tries to get too cute and makes a conscious effort to “write beautifully” it puts me off, for it makes the reading less intuitive. As a consequence, there’s hardly any fiction I’ve read in the last 5-6 years.

I was thinking of this last evening when I went to watch this dance show called “Prayog 4” here in Bangalore. I think it was good – the three performances looked extremely well choreographed and well-coordinated, and the dancers seemed to have put in considerable effort into the production. They were all supremely fit and were literally doing gymnastics during the course of the performance. But my appreciation of the performance ended there.

After one of the performances, the wife exclaimed “you know, this dance so represents your and my lives!”. I just couldn’t understand what she was hinting at. All I could see was this one guy dancing round and round in circles, and doing gymnastics on a rope! As I mentioned earlier, his movements were extremely graceful and aesthetically pleasing but I just couldn’t get anything more out of it.

Later last night, my wife asked me what I understood from the first performance (yesterday’s show essentially had three separate performances). “A bunch of chicks doing extremely graceful gymnastics on a bunch of parallel bars”, I replied. “Didn’t you notice how beautifully they represented different emotions during the course of the dance”, she asked. I admitted to recognizing nothing of the sort. Instead, I was sitting there, wondering what the big deal was, and trying to construct this blog post in my head.

“Art” is not unidimensional, and “appreciating art” is too broad a statement. After my experience yesterday I don’t know if there are people who can appreciate all kinds of art. For a moment I thought I was a philistine for I couldn’t appreciate yesterday’s performance, but then I remembered the pieces of art I mentioned in the beginning of this blog post that I truly appreciate. So, no – I’m not a philistine. It’s just that there are certain art forms I get and ones I don’t.

Have you felt similarly sometimes? Are there some art forms you “get” easily, and others that you absolutely fail to get? Or do you consider yourself to be the types that gets all kinds of art, and you argue that the ones you don’t get is simply not art? Or do you fail to get any art at all? Do leave a comment.

Addition to the Model Makers Oath

Paul Wilmott and Emanuel Derman, in an article in Business Week a couple of years back (at the height of the financial crisis) came up with a model-makers oath. It goes:

• I will remember that I didn’t make the world and that it doesn’t satisfy my equations.

• Though I will use models boldly to estimate value, I will not be overly impressed by mathematics.

• I will never sacrifice reality for elegance without explaining why I have done so. Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.

• I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension.

While I like this, and try to abide by it, I want to add another point to the oath:

As a quant, it is part of my responsibility that my fellow-quants don’t misuse quantitative models in finance and bring disrepute to my profession. It is my responsibility that I’ll put in my best efforts to be on the lookout for deviant behavour on the part of other quants, and try my best to ensure that they too adhere to these principles.

Go read the full article in the link above (by Wilmott and Derman). It’s a great read. And coming back to the additional point I’ve suggested here, I’m not sure I’ve drafted it concisely enough. Help in editing and making it more concise and precise is welcome.