# Local Pseudo-Optima

Local optima are, basically, the reason search and optimisation exists as a field of research, rather than a single algorithm that works every time. The story we tell to the younglings is that if you don’t have local optima, then you use a hill-climber, and you find the global optimum.

What is a local optimum? Well, if crossover is in the picture, then it’s complicated. Let’s forget about crossover for the moment, and consider mutation only. An individual is a local optimum if no other individual, reachable by a single mutation, has a better fitness.

Now there are obvious counter-examples to the story we tell the younglings:

• The genetic programming subtree mutation operator gives a landscape with no local optima. I’m talking about the version where mutation is allowed at any node, including the root. Since you can mutate at the root, any individual can mutate to any other individual in a single step. Therefore, there are no local optima. But if you use a hill-climber, you might do quite well (e.g. O’Reilly and Oppacher, Program Search with a Hierarchical Variable Length Representation: Genetic Programming, Simulated Annealing and Hill Climbing). But you can’t expect to just hill-climb to the global optimum.

• You can construct a mutation operator which gives no local optima. Start with any existing mutation operator m. With some low probability, return a new random individual. Otherwise, return m(x), where x is the current individual. Again, this gives a landscape with no local optima, but if you now run a hill-climber, it’s equivalent to a hill-climber using m with random restarts. Getting rid of the local optima hasn’t really helped.

So what is really going on? I think that we need the concept of local pseudo-optima. A local pseudo-optimum is an individual with a vanishingly low probability of mutation to any better individual in a single step. Consider a GP individual which can only mutate to a better individual by mutating at the root and generating one of some small set of large trees. That is a mutation of very low probability.

This isn’t an all-or-nothing definition anymore. How small is “vanishingly small”? Anyway it would probably be a mistake to set an arbitrary threshold. But maybe we can measure the “strength” of every local pseudo-optimum as something like one minus the probability of a one-step transition to any better or equally-good individual. With this way of thinking, every individual is a local pseudo-optimum of differing strength. One aspect of the difficulty of a landscape can be measured by measuring the strength of the local pseudo-optima, or perhaps the number of strong ones (where an arbitrary threshold is used to define “strong”).

The probability of transitioning to a better individual via a single mutation will relate directly to how long the hill-climber gets stuck on the individual in question. And that can lead us back to crossover. The typical local optimum behaviour in a full EC algorithm is that the evolution gets stuck at a single point for a long time, and then the run ends. Whether this corresponds to a true local optimum in the presence of crossover is, again, complicated, so again, let’s ignore that question. But we can estimate (as distinct from “measure”) the strength of a local pseudo-optimum in the presence of crossover by estimating how many children of the individual in question have to be tried before an individual of equal or better fitness is found. That can be crudely estimated from the EC parameters (e.g. population size and tournament selection size) and the number of generations during which the evolution is stuck at a particular fitness. We don’t need new experiments: we can read old papers to get this data.