Inquiries

A short article in the April 16 issue of New Scientist reported on an applied soft computing paper that proposes an improvement on what’s known as ‘particle swarm optimization (PSO).

Particle swarm optimization (PSO) is an optimization technique inspired by the social behavior of birds. Described as a simple and powerful algorithm, it can be used to optimize high dimensional functions (in other words, finding maximums and minimums of functions with many parameters). There is quite a bit of info on the website Code Project.  There they explain:

To understand the algorithm, it is best to imagine a swarm of birds that are searching for food in a defined area – there is only one piece of food in this area. Initially, the birds don’t know where the food is, but they know at each time how far the food is. Which strategy will the birds follow? Well, each bird will follow the one that is nearest to the food.

PSO adapts this behavior of birds searching for food in order to the search for the best solution-vector in a search space. A particle is a single solution. The algorithm defines the measure of best solutions and begins with particles at random positions. Through some number of iterations, individual particles adjust their velocity and position as they follow best solution particles.

The New Scientist article gives a more general description of this approach along with one of its limitations:

One way they can do this is by using groups of virtual creatures that wander through “parameter space”, looking for valleys that represent the lowest values. Mathematicians have taken inspiration from actual animals, from grey wolves to ants. One limitation, though, is that the animals sometimes fail to notice a deeper valley nearby.

The suggested improvement is to add parasites to the mix:

In their model, a swarm of animals searched for the lowest valleys, but was then joined by a second, parasitic population. This group searched for valleys, but also abducted the most successful animals and made them work for the parasite team.

The struggle resulted in a more varied collection of creatures allowing the parasitic algorithm to solve the problem twice as fast.

I thought about what this kind of thing could mean about the mathematics itself. Why would there be any relationship between a bird’s search for food and our interest in optimization solutions? We’re not just modeling the bird’s behavior, we’re using the bird’s behavior to solve our own problems. There is here an unexpected overlap between two kinds of inquiries. And this word, I think, is key – inquiry.

There is still some debate among cognitive scientists about whether our more primal experience of quantity is discrete, like the numbers that we count with, or continuous, like our sense of time. If, as many cognitive scientist argue, our first sense of quantity is continuous (like the real numbers) and, if it is true that numbers followed language, then the 19th century struggle to understand and define the continuum (represented by the real number line) can look like an investigation of number, an inquiry back into number’s source. And once I begin to think in terms of inquiries, I see them everywhere. Visual art is an inquiry into visual sensation. This is a view consistently presented by neuroscientist Semir Zeki. Mathematics is an inquiry into sensation as well as abstract relationship itself (logical, numerical, geometric, probabilistic, etc.)  The nature of these inquiries is, perhaps, a pure exploration of living interactions – the eye and light, the relationships that produce comprehension, movement and space.

The search for food is certainly an inquiry, as is swarming in the more general sense.  I would include my own earlier discussion of a plant’s calculation of the rate with which it will consume its stored food. Perhaps evolution itself is an inquiry into life’s possibilities.

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