30 May 2021

There is a field in mathematics called âoptimisation.â In essence, it concerns finding a favourable solution to a problem from a range of possibilities. In other words: finding, in a range of input values, which inputs yield good outputs when given to some âobjective functionâ or process.

It sounds very useful, right? The study of how to find good solutions! Who wouldnât have use for that?!

WellâŚ itâs not always easy.

Often, we know things about the objective function which make our life easier. It might have nice properties, e.g. itâs smooth, or there are no sudden jumps. Here we can use mathematics to find some simple and effective optimisation strategies.

But this is not an article about mathematics. This is an article about when we canât just crunch some numbers to find a good solution. This is about real-life, noisy problems, where know very little about our function, or itâs spiky, jagged and hard to predict.

At this end of the scale, optimization becomes much less like a calculated, mathematical problem and much more like fumbling around in a dark room, looking for a lightâa blind search, where we try, move, and try again, until we get something acceptable.

What can we do? Is there anything at all? Surely we should just guess?

Well, not quite. Regardless of the specific problem, there are two fundamental, opposing incentives when searching the unknown:

• Exploration. Quite simply, we need to cover ground! There might be a better solution than the best one weâve currently found. We might be looking in completely the wrong area, mistakenly believing weâve hit the jackpot. Exploration is the want to try other solutions, to ensure we have a chance of finding a more optimal one.
• Exploitation. When we choose to explore, we usually give up our current solution in the hopes that we might find something better. HoweverâŚ we might not find something better. We might have been better with our old solution, and we might not remember a way back. Exploitation is the want to benefit from the current solution, whatever that means in context. The specifics will vary, but in general, exploitation represents the opportunity cost of exploration.

These two forces or desires for a searching strategy work against each other. For a strategy to be effective, it has to balance the potential gains from exploration with the costs of doing so. This pattern can be found in a huge variety of artificial and natural processes.

For example, in a simple form, evolution can be considered as a natural search process. Organisms are described by genetic information, which they pass onto their offspring usually by combining their genes with another individual. During their life, these organisms are subject to a variety of obstacles to their survivability (natural selection), and hence the passing on of their genes.

Exploration in this context corresponds with genetic variation in the population. Various mechanisms cause variation, including mutation, where genes change at random, and recombination, where offspring receive random mixtures of their parentsâ genes. Variation ensures that when two individuals reproduce, their offspring isnât too similar to them, which can help to e.g. avoid passing on harmful characteristics or limit damage to the population caused by sudden environmental change.

But there is another factor to considerâtwo reproducing organisms are considered âsuccessfulâ from an evolutionary perspective because they have passed on their genes. Therefore, there is also need for the offspring to be as similar as possible, or in other words, to exploit the fitness of their parentsâ genes.

In this context, the interplay and careful balancing of these forces is a parameter of evolution that can be guided by natural selection itself.

As another example, various swarming insects, such as bees and ants, are capable of intelligent searching at a behavioural level. When an ant colony is young, forager ants will leave and explore the surrounding area randomly for food. If they find some, they will bring it back, using their very astute sense of direction (and internal biological pedometer!). Crucially, forager ants deposit pheromones as they travel, which can be detected by other ants. This means that as the ant goes back and forth from the food source, the path they take accumulates more and more pheromones.

Two things makes this an optimisation strategy. The first is that ants are more likely to move in a direction if they can detect more pheromones along it. The second is that, by purely a natural consequence of time, shorter paths accumulate more pheromones, because more journeys can be done along it in any given period. The result is that the ants will likely discover and use the shortest route from the colony to the food.

Again, we see the imporance of balancing exploration and exploitation. Too much exploring and the yield from known food sources will be too low, limiting the colonyâs growth. Too much exploitation and the colony will be too dependent on its existing sources, which could lead to disaster if some suddenly become inaccessible; or it could settle on ineffecient and costly routes. Here the balance is controlled by two parameters: the preference of ants for pheromone-laden paths, and the rate at which an ant deposits pheromones.

Honey bees have a very explicit split between the two behaviours. While many bees collect honey from known flowers, some proportion of bees are âscoutsâ who will fly around randomly looking for new areas. If they find flowers, they come back to the colony and do a âwaggle danceâ which communicates the location to the foragers. More flowers lead to a longer dance, so more bees will be recruited to the new area.

All of the examples I have given are in nature and relate to survival. But at an abstract level, they are just as I described at the beginning of the article: processes to find an optimal solution to a difficult to predict problem from a range of possibilites.

Is there any reason we canât think of many solutions in our own lives or societies like this? When we do, I think the feud of âthose who want change vs those who donâtâ becomes more nuanced. With an understanding that change can benefit us, but has an opportunity cost, our focus goes to understanding our current and possible new solutions, and experimenting if possible. If we can justify that we expect things to be better after a change, thatâs a very good reason to explore!

The flipside of this is that even some policy decisions, let alone alternative modes of society, can be extremely hard to test in isolation or at scale. When experiments can be done, studies can very easily come to completely contradictory conclusions (this happens more often than you think!) We have to grapple with the limitations of our knowledge; this is why the âexploitâ strategy can seem favourable.

The answer, I argue, is a balanceâone that is undoubtably hard to find. Careful incremental change and complete rethinking are both immensely valuable, in so many systems at every scale. They exclude each other, but somehow must be done simultaneously. Conflict seems inevitable. However, at the root of both is a drive for the best. For ourselves and everyone around us, we must keep our sights on the future and on how we can improveâeven if it is sometimes so complex.