What are the differences between simulated annealing and genetic algorithms?

Question

What are the relevant differences, in terms of performance and use cases, between simulated annealing (with bean search) and genetic algorithms?

I know that SA can be thought as GA where the population size is only one, but I don't know the key difference between the two.

Also, I am trying to think of a situation where SA will outperform GA or GA will outperform SA. Just one simple example which will help me understand will be enough.

Answer 1

Well strictly speaking, these two things--simulated annealing (SA) and genetic algorithms are neither algorithms nor is their purpose 'data mining'.

Both are meta-heuristics--a couple of levels above 'algorithm' on the abstraction scale. In other words, both terms refer to high-level metaphors--one borrowed from metallurgy and the other from evolutionary biology. In the meta-heuristic taxonomy, SA is a single-state method and GA is a population method (in a sub-class along with PSO, ACO, et al, usually referred to as biologically-inspired meta-heuristics).

These two meta-heuristics are used to solve optimization problems, particularly (though not exclusively) in combinatorial optimization (aka constraint-satisfaction programming). Combinatorial optimization refers to optimization by selecting from among a set of discrete items--in other words, there is no continuous function to minimize. The knapsack problem, traveling salesman problem, cutting stock problem--are all combinatorial optimization problems.

The connection to data mining is that the core of many (most?) supervised Machine Learning (ML) algorithms is the solution of an optimization problem--(Multi-Layer Perceptron and Support Vector Machines for instance).

Any solution technique to solve cap problems, regardless of the algorithm, will consist essentially of these steps (which are typically coded as a single block within a recursive loop):

1. encode the domain-specific details in a cost function (it's the step-wise minimization of the value returned from this function that constitutes a 'solution' to the c/o problem);

2. evaluate the cost function passing in an initial 'guess' (to begin iteration);

3. based on the value returned from the cost function, generate a subsequent candidate solution (or more than one, depending on the meta-heuristic) to the cost function;

4. evaluate each candidate solution by passing it in an argument set, to the cost function;

5. repeat steps (iii) and (iv) until either some convergence criterion is satisfied or a maximum number of iterations is reached.

Meta-heuristics are directed to step (iii) above; hence, SA and GA differ in how they generate candidate solutions for evaluation by the cost function. In other words, that's the place to look to understand how these two meta-heuristics differ.

Informally, the essence of an algorithm directed to solution of combinatorial optimization is how it handles a candidate solution whose value returned from the cost function is worse than the current best candidate solution (the one that returns the lowest value from the cost function). The simplest way for an optimization algorithm to handle such a candidate solution is to reject it outright--that's what the hill climbing algorithm does. But by doing this, simple hill climbing will always miss a better solution separated from the current solution by a hill. Put another way, a sophisticated optimization algorithm has to include a technique for (temporarily) accepting a candidate solution worse than (i.e., uphill from) the current best solution because an even better solution than the current one might lie along a path through that worse solution.

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So how do SA and GA generate candidate solutions?

The essence of SA is usually expressed in terms of the probability that a higher-cost candidate solution will be accepted (the entire expression inside the double parenthesis is an exponent:

p = e((-highCost - lowCost)/temperature)

Or in python:

p = pow(math.e, (-hiCost - loCost) / T)

The 'temperature' term is a variable whose value decays during progress of the optimization--and therefore, the probability that SA will accept a worse solution decreases as iteration number increases.

Put another way, when the algorithm begins iterating, T is very large, which as you can see, causes the algorithm to move to every newly created candidate solution, whether better or worse than the current best solution--i.e., it is doing a random walk in the solution space. As iteration number increases (i.e., as the temperature cools) the algorithm's search of the solution space becomes less permissive, until at T = 0, the behavior is identical to a simple hill-climbing algorithm (i.e., only solutions better than the current best solution are accepted).

Genetic Algorithms are very different. For one thing--and this is a big thing--it generates not a single candidate solution but an entire 'population of them'. It works like this: GA calls the cost function on each member (candidate solution) of the population. It then ranks them, from best to worse, ordered by the value returned from the cost function ('best' has the lowest value). From these ranked values (and their corresponding candidate solutions) the next population is created. New members of the population are created in essentially one of three ways. The first is usually referred to as 'elitism' and in practice usually refers to just taking the highest ranked candidate solutions and passing them straight through--unmodified--to the next generation. The other two ways that new members of the population are usually referred to as 'mutation' and 'crossover'. Mutation usually involves a change in one element in a candidate solution vector from the current population to create a solution vector in the new population, e.g., [4, 5, 1, 0, 2] => [4, 5, 2, 0, 2]. The result of the crossover operation is like what would happen if vectors could have sex--i.e., a new child vector whose elements are comprised of some from each of two parents.

So those are the algorithmic differences between GA and SA. What about the differences in performance?

In practice: (my observations are limited to combinatorial optimization problems) GA nearly always beats SA (returns a lower 'best' return value from the cost function--ie, a value close to the solution space's global minimum), but at a higher computation cost. As far as i am aware, the textbooks and technical publications recite the same conclusion on resolution.

but here's the thing: GA is inherently parallelizable; what's more, it's trivial to do so because the individual "search agents" comprising each population do not need to exchange messages--ie, they work independently of each other. Obviously that means GA computation can be distributed, which means in practice, you can get much better results (closer to the global minimum) and better performance (execution speed).

In what circumstances might SA outperform GA? The general scenario i think would be those optimization problems having a small solution space so that the result from SA and GA are practically the same, yet the execution context (e.g., hundreds of similar problems run in batch mode) favors the faster algorithm (which should always be SA).