For a genotype representation that is a permutation (such as a vector^{[1]}, bit-string, or hash-map^{[2]}

For a genotype representation that is a permutation (such as a vector^{[1]}, bit-string, or hash-map^{[2]}), we have seen a possible recombination operator. Our 3-SAT solver uses a very popular recombination technique: uniform crossover.

Furthermore, we know a permutation is not the only, valid genotype for an individual: other possibilities can include an integer or a real-valued number.

Note, for simplicity, we will discuss recombination to form one offspring. This exact process can be applied to form a second child (generally with the parent's role reversed). Recombination can also be applied to more than two parents (depending on the operator). Again, for simplicity, we choose to omit it^{[3]}.

First, let us start with permutations.

In regard to premutation crossover, there are three, common operators:

- Uniform Crossover
- $N$-Point Crossover
- Davis Crossover

Uniform crossover we have seen before. We consider individual elements in the permutation, and choose one with a random, equal probability. For large enough genotypes, the offspring genotype should consist of 50% of the genotype from parent one, and 50% of the genotype from parent two.

$N$-Point crossover considers segments of a genotype, apposed to individual elements. This operator splits the genotype of Parent 1 and Parent 2 $N$ times (hence the name $N$-point), and creates a genotype by alternating segments from the two parents. For every $N$, there will be $N + 1$ segments. For 1-point crossover, the genotype should be split into two segments, and the offspring genotype should be composed of one segment from Parent 1, and one segment from Parent 2. For 2-point crossover, there will be three segments, and the offspring genotype will have two parts from Parent 1 and one part from Parent 2 (or two parts, Parent 1, one part, Parent 2).

Davis Crossover tries to preserve the ordering of the genotype in the offspring (apposed to the previous methods, where ordering was not considered). The premise is a bit complicated, but bear with me. Pick two random indices ($k_1$ and $k_2$), and copy the genetic material of Parent 1 from $k_1$ to $k_2$ into the offspring at $k_1$ to $k_2$. Put Parent 1 to the side, his role is finished. Start copying the genotype of Parent 2 starting at $k_1$ to $k_2$ *at the beginning of the offspring*. When $k_2$ is reached in the parent, start copying the beginning of Parent 2 into the genotype, and when $k_1$ is reached in the parent, skip to $k_2$. When $k_1$ is reached in the offspring, skip to $k_2$, and start copying until the end. If this seems a complicated (it very much is), reference the accompanying figure.

Those are considered the three, most popular choices for permutations. Now, let us look at integer crossover.

Integer crossover is actually quite an interesting case; integers can be recombined as permutations or real-valued numbers.

An integer is already a permutation, just not at first glance: binary. The individual bits in a binary string are analogous to elements in a vector, and the whole collection *is a vector*. Now it is a valid permutation. We can apply uniform crossover, $N$-point crossover, or Davis Crossover, just as we have seen.

An integer is also already a real-valued number, so we can treat it as such. Let's take a look at how to recombine it.

Real-Valued crossover is different than methods we have seen before. We could turn it into binary, but that would be a nightmare to deal with. However, we can exploit the arithmetic properties of real-valued numbers — with a weighted, arithmetic mean. For a child (of real value) $z$, we can generate it from Parent 1 $x$ and Parent 2 $y$ as such:

$$z = \alpha \cdot x + (1 - \alpha) \cdot y$$

Now, if we want to crossover *a permutation* of Parent 1 and Parent 2, we can do so for every element.

$$z_i = \alpha \cdot x_i + (1 - \alpha) \cdot y_i$$

This can be shown to have better performance than crossover methods discussed, but would entirely depend on use case.

As always, we will now tackle implementing the permutation crossovers we've had before. None of them are incredibly complicated, except possibly $N$-point crossover.

```
class Individual
...
@staticmethod
def __uniform_crossover(parent_one, parent_two):
new_genotype = SAT(Individual.cnf_filename)
for variable in parent_one.genotype.variables:
gene = choice([parent_one.genotype[variable], parent_two.genotype[variable]])
new_genotype[variable] = gene
individual = Individual()
individual.genotype = new_genotype
return individual
@staticmethod
def __n_point_crossover(parent_one, parent_two, n):
new_genotype = SAT(Individual.cnf_filename)
variables = sorted(parent_one.genotype.variables)
splits = [(i * len(variables) // (n + 1)) for i in range(1, n + 2)]
i = 0
for index, split in enumerate(splits):
for variable_index in range(i, split):
gene = parent_one.genotype[variables[i]] if index % 2 == 0 else parent_two.genotype[variables[i]]
new_genotype[variables[i]] = gene
i += 1
individual = Individual()
individual.genotype = new_genotype
return individual
@staticmethod
def __davis_crossover(parent_one, parent_two):
new_genotype = SAT(Individual.cnf_filename)
variables = sorted(parent_one.genotype.variables)
split_one, split_two = sorted(sample(range(len(variables)), 2))
for variable in variables[:split_one]:
new_genotype[variable] = parent_two.genotype[variable]
for variable in variables[split_one:split_two]:
new_genotype[variable] = parent_one.genotype[variable]
for variable in variables[split_two:]:
new_genotype[variable] = parent_two.genotype[variable]
individual = Individual()
individual.genotype = new_genotype
return individual
```

By no means is recombination easy. It took evolution hundreds of thousands of years to formulate ours. The particular permutation operator to use entirely dependent on the context of the problem; and most of the time, it is not obvious by any stretch. Sometimes, there might not even be an established crossover operator for a particular genotype.

Sometimes, you might have to get a little creative.

Previously our Evolutionary Algorithms had it pretty easy: there would be

]]>Previously our Evolutionary Algorithms had it pretty easy: there would be either one local optimum (like our Secret Message problem instance) or multiple, valid local optimums (like the 3-SAT problem instance). In the real world, we might not be so lucky.

Often, an Evolutionary Algorithm might encounter a local optimum within the search space, and it will not be so easy to escape — offspring generated will be in close proximity of the optimum, and the mutation will not be enough to start exploring other parts of the search space.

To add to the frustration, there might not enough time or patience to wait for the Evolutionary Algorithm to finish. We might have different criteria we are looking for, outside of just a fitness target.

We are going to tackle both of these issues.

First, we will examine what criteria we want met before our Evolutionary Algorithm terminates. In general, there are six that are universal:

*Date and Time*. After a specified date and time, terminate.*Fitness Target*. This is what we had before; terminate when any individual attains a certain fitness.*Number of Fitness Evaluations*. Every generation, every individual's fitness is evaluated (in our case, every generation $\mu + \lambda$ fitnesses are evaluated). Terminate after a specified number of fitness evaluations.*Number of Generations*. Just like the number of fitness evaluations, terminate after a specified generations.*No Change In Average Fitness*. This is a bit tricky. After specify $N$ generations, we check every $N$ generations back to determine if the average fitness of a population has improved. We have to be careful in our programming; by preserving diversity, we almost always lose fitness.*No Change In Best Fitness*. Just like No Change In Average Fitness, but instead of taking the average fitness, we take the best.

Later, we will see how Conditions 5 & 6 will come in handy to determining if we are stuck in a local optimum.

To make sure we are always given valid termination conditions, we will have a super class that all termination conditions will inherit from. From there, we will have a separate condition for each of the listed conditions above.

```
class _TerminationCondition:
pass
class FitnessTarget(_TerminationCondition):
"""Terminate after an individual reaches a particular fitness."""
class DateTarget(_TerminationCondition):
"""Terminate after a particular date and time."""
class NoChangeInAverageFitness(_TerminationCondition):
"""Terminate after a there has been no change in the average fitness for a period of time."""
class NoChangeInBestFitness(_TerminationCondition):
"""Terminate after a there has been no change in the best fitness for a period of time."""
class NumberOfFitnessEvaluations(_TerminationCondition):
"""Terminate after a particular number of fitness evaluations."""
class NumberOfGenerations(_TerminationCondition):
"""Terminate after a particular number of generations."""
```

Now, we need something that will keep track of all these conditions, and tells us when we should terminate. And here's where we need to be careful.

First, we need to know when to terminate. We want to mix and match different conditions, depending on the use case. This begs the questions:

Should the Evolutionary Algorithm terminate when one condition has been met, or all of them?

Generally, it makes more sense to terminate when any of the conditions have been met, as apposed to all of them. Suppose the two termination conditions are date and target fitness. It does not make sense to keep going after the target fitness is reached, and (if in a time crunch) it does not make sense to keep going after a specified date.

Second, how should we define no change in average/best fitness? These values can be quite sinusoidal, so we want to be more conservative in our definition. One plausible solution is to take the average of the first quartile (the first 25% to ever enter the queue), and see if the there is *a single individual* with a better fitness in the second, third, or fourth quartile (the last 75% percent to enter the queue). This way, even if there were very dominant individuals in the beginning, a single, more dominant individual will continue the Evolutionary Algorithm.

From this, we have everything we might need to keep track of our terminating conditions.

```
class TerminationManager:
def __init__(self, termination_conditions, fitness_selector):
assert isinstance(termination_conditions, list)
assert all(issubclass(type(condition), _TerminationCondition) for condition in termination_conditions), "Termination condition is not valid"
self.termination_conditions = termination_conditions
self.__fitness_selector = fitness_selector
self.__best_fitnesses = []
self.__average_fitnesses = []
self.__number_of_fitness_evaluations = 0
self.__number_of_generations = 0
def should_terminate(self):
for condition in self.termination_conditions:
if isinstance(condition, FitnessTarget) and self.__fitness_should_terminate():
return True
elif isinstance(condition, DateTarget) and self.__date_should_terminate():
return True
elif isinstance(condition, NoChangeInAverageFitness) and self.__average_fitness_should_terminate():
return True
elif isinstance(condition, NoChangeInBestFitness) and self.__best_fitness_should_terminate():
return True
elif isinstance(condition, NumberOfFitnessEvaluations) and self.__fitness_evaluations_should_terminate():
return True
elif isinstance(condition, NumberOfGenerations) and self.__generations_should_terminate():
return True
return False
def reset(self):
"""Reset the best fitnesses, average fitnesses, number of generations, and number of fitness evaluations."""
self.__best_fitnesses = []
self.__average_fitnesses = []
self.__number_of_fitness_evaluations = 0
self.__number_of_generations = 0
def __fitness_should_terminate(self):
"""Determine if should terminate based on the max fitness."""
def __date_should_terminate(self):
"""Determine if should terminate based on the date."""
def __average_fitness_should_terminate(self):
"""Determine if should terminate based on the average fitness for the last N generations."""
def __best_fitness_should_terminate(self):
"""Determine if should terminate based on the average fitness for the last N generations."""
def __fitness_evaluations_should_terminate(self):
"""Determine if should terminate based on the number of fitness evaluations."""
def __generations_should_terminate(self):
"""Determine if should terminate based on the number of generations."""
```

And the changes to our Evolutionary Algorithm are minimal, too.

```
class EA:
...
def search(self, termination_conditions):
generation = 1
self.population = Population(self.μ, self.λ)
fitness_getter = lambda: [individual.fitness for individual in self.population.individuals] # noqa
termination_manager = TerminationManager(termination_conditions, fitness_getter)
while not termination_manager.should_terminate():
offspring = Population.generate_offspring(self.population)
self.population.individuals += offspring.individuals
self.population = Population.survival_selection(self.population)
print("Generation #{}: {}".format(generation, self.population.fittest.fitness))
generation += 1
print("Result: {}".format(self.population.fittest.genotype))
return self.population.fittest
```

However, we can still do better.

Before, the Evolutionary Algorithm framework we put in place was strictly a generational model. One generation lead to the next, and there were no discontinuities. Now, let's make our generational model into an epochal one.

We define an epoch as anytime our Evolutionary Algorithm encounters a local optimum. Once the end of an epoch is reached, the EA is reset, and the previous epoch is saved. Upon approaching the end of the next epoch, reintroduce the last epoch into the population; by this, more of the search space is covered.

How can we determine if we are at a local optimum?

*We can't*.

That does not mean we cannot have a heuristic for it. When there is little to no change in average/best fitness for a prolonged period of time, that typically means a local optimum has been reached. How long is a prolonged period of time? That's undetermined; it is another parameter we have to account for.

Note, if the Evolutionary Algorithm keeps producing more fit individuals, but the average fitness remains the same, the algorithm will terminate. Likewise, if the best fitness remains the same, but the average fitness closely approaches the best, the EA will terminate. Therefore, we should determine if the best fitness *and* the average fitness has not changed; only then should we start a new epoch.

Luckily, we already have something that will manage the average/best fitness for us.

```
class EA:
...
def search(self, termination_conditions):
epochs, generation, total_generations = 1, 1, 1
self.population = Population(self.μ, self.λ)
previous_epoch = []
fitness_getter = lambda: [individual.fitness for individual in self.population.individuals] # noqa
termination_manager = TerminationManager(termination_conditions, fitness_getter)
epoch_manager_best_fitness = TerminationManager([NoChangeInBestFitness(250)], fitness_getter)
epoch_manager_average_fitness = TerminationManager([NoChangeInAverageFitness(250)], fitness_getter)
while not termination_manager.should_terminate():
if epoch_manager_best_fitness.should_terminate() and epoch_manager_average_fitness.should_terminate():
if len(previous_epoch) > 0:
epoch_manager_best_fitness.reset()
epoch_manager_average_fitness.reset()
self.population.individuals += previous_epoch
previous_epoch = []
else:
epoch_manager_best_fitness.reset()
epoch_manager_average_fitness.reset()
previous_epoch = self.population.individuals
self.population = Population(self.μ, self.λ)
generation = 0
epochs += 1
self.population = Population.survival_selection(self.population)
offspring = Population.generate_offspring(self.population)
self.population.individuals += offspring.individuals
self.__log(total_generations, epochs, generation)
total_generations += 1
generation += 1
print("Result: {}".format(self.population.fittest.genotype))
return self.population.fittest
def __log(self, total_generations, epochs, generation):
"""Log the process of the Evolutionary Algorithm."""
...
```

Although considerably more complicated, this new Evolutionary Algorithm framework allows us to explore much more of a search space (without getting stuck).

Let's put it to the test.

We're going to take on a substantially harder 3-SAT instance: 1,000 clauses, 250 variables. To make it worse, the number of valid solutions is also lower. We will also include the following terminating conditions:

- Time of eight hours.
- Fitness of all clauses satisfied (100).
- A million generations.

So, how does our Evolutionary Algorithm fair?

Not well. After twenty epochs, and thousands of generations — we do not find a solution. Fear not; in subsequent posts, we will work on optimizing our Genetic Algorithm to handle much larger cases, more effectively.

]]>First, we need another problem instance. Our previous problem instance was pretty straightforward: it had one local optimum.

]]>First, we need another problem instance. Our previous problem instance was pretty straightforward: it had one local optimum. Let's take on a problem with *many* local optimum, such as the 3-SAT problem.

The premise of 3-SAT is simple. From a global pool of variables ($x_1$, $x_2$, $\ldots$, $x_n$), we have a basic clause of three variables or-ed together (signified by $\vee$):

$$x_p \vee x_q \vee x_r$$

Then, and (signified by a $\wedge$) several clauses together:

$$\left(x_p \vee x_q \vee x_r\right) \wedge \left(x_s \vee x_t \vee x_u\right) \wedge \ldots \wedge \left(x_v \vee x_w \vee x_y\right)$$

The only stipulation is that any variable can be negated (signified by a $\neg$). So, supposing we want to negate $x_p$; $x_s$ and $x_u$; and $x_v$, $x_w$, and $x_y$; we can do the following:

$$\left(\neg x_p \vee x_q \vee x_r\right) \wedge \left(\neg x_s \vee x_t \vee \neg x_u\right) \wedge \ldots \wedge \left(\neg x_v \vee \neg x_w \vee \neg x_y\right)$$

Now, we simply have to assign all the variables such that all the clauses will evaluate to true. It may sound simple, but it belongs to the hardest classes of problems in computer science. There is no guaranteed algorithm to produce the right answer at this time.

For a more visual approach, please reference the figure below. The goals is to make every inner node green, by having at lease one, connected, outer node be green. Note the green nodes have to account for negation as well.

This sounds like a good problem candidate for an Evolutionary Algorithms^{[1]}.

We can skip over the problem specific parts to worry more about the Evolutionary Algorithm aspect. Suppose we already have a well-defined `SAT`

class that takes care of SAT-specific properties and methods, like so:

```
class SAT:
def __init__(self, filename):
"""Create a SAT object that is read in from a CNF file."""
...
@property
def variables(self):
"""Get *all* the variables."""
...
@property
def total_clauses(self):
"""Set the total number of clauses."""
...
@property
def clauses_satisfied(self):
"""Get the number of satisfied clauses."""
...
def __getitem__(self, key):
"""Get a particular variable (key)"""
...
def __setitem__(self, key, value):
"""Set a variable (key) to value (True/False)"""
...
```

From this, we can create a new genotype for our `Individual`

.

The genotype structure was very similar to what we had before:

- The genotype is the
`SAT`

problem we defined above. - Fitness is defined by a percentage of the total satisfied clauses.
- Mutation is uniform, choose a percentage $p$ of alleles and flip their value.
- Recombination is uniform, randomly assemble values from both parents.

Looking at the refactoring, not much has changed.

Now that we have updated our Individual, next thing to updated would be the Evolutionary Algorithm framework, including:

- The Population
- The EA Itself

*Except, we don't have to*.

That's the beauty of Evolutionary Algorithms, they are incredibly adaptable. By swapping out the Individual, the rest of the evolutionary algorithm should still work.

For our SAT problem, there were some parameters updated, to make the algorithm more efficient:

- The mutation rate has been reduced to 5%
- The tournament size has been reduced to 15 individuals (out of $\lambda = 100$).

So, let's try our Evolutionary Algorithm. Taking a SAT instance with 75 variables and 150 clauses, this makes the search space

$$2^{75} \approx 3.77 \times 10^{22}$$

Great, so roughly 1,000 times the grain of sand on Earth, easy. So, can our EA do it?

After roughly 100 iterations, *yes*. See the visualization below.

Marvelous, our EA managed to find a solution after only 100 iterations in a giant search space. And all we had to do was swap out one class.

All source code can be found here.

In reality, it's not a

*great*candidate for an evolutionary algorithm. The gradient is sometimes murky, because flipping one variable's value can drastically decrease/increase the fitness function. Also, there are several great heuristics for solving the SAT problem. ↩︎

Imagine you

]]>Imagine you are at the bottom of the hill; you have no idea where to go. A decent place to start would be to go up the hill to survey the landscape. Then, restart to find a higher a peak until you find the highest peak, right? Well, that is the entire algorithm.

Let's dig a bit deeper.

What is Steepest-Ascent Hill-Climbing, formally? It's nothing more than an agent searching a search space, trying to find a local optimum. It does so by starting out at a random Node, and trying to go uphill at all times.

The pseudocode is rather simple:

```
current ← Generate-Initial-Node()
while true
neighbors ← Generate-All-Neighbors(current)
successor ← Highest-Valued-Node(neighbors)
if Value-At-Node(successor) <= Value-At-Node(current):
return current
current ← successor
```

What is this `Value-At-Node`

and $f$-value mentioned above? It's nothing more than a heuristic value that used as some measure of quality to a given node. Some examples of these are:

*Function Maximization*: Use the value at the function $f(x, y, \ldots, z)$.*Function Minimization*: Same as before, but the reciprocal: $1 / f(x, y, \ldots, z)$.*Path-Finding*: Use the reciprocal of the Manhattan distance.*Puzzle-Solving*: Use some heuristic to determine how well/close the puzzle is solved.

The best part? If the problem instance can have a heuristic value associated with it, and be able to generate points within the search space, the problem is a candidate for Steepest-Ascent Hill-Climbing.

For this problem, we are going to solve an intuitive problem: function maximization. Given a function $z = f(x, y)$, for what values of $x, y$ will $z$ be the largest? To start, we are going to use a trivial function to maximize:

$$z = -x^2 - y^2$$

We see it is nothing more than a paraboloid. Furthermore, since it is infinite, we are going to restrict the domain to ${ x, y \in \mathbb{Z}^+ : -100 \leq x, y \leq 100 }$; therefore, we only have integer values between $(-100, 100)$.

Algorithms used in games, where a player searches for an optimal move against an opponent. ↩︎

So, let's begin.

Because we will be searching throughout a search space, we will need some representation of a state. For our particular problem instance, it's very easy: the points $(x, y)$. Also, we will need to represent the $f$ value, so we create an auxiliary class as well.

```
class Node:
"""A node in a search space (similar to a point (x, y)."""
def __init__(self, x, y):
self.x = x
self.y = y
```

```
class Function:
"""A function and its respective bounds."""
def __init__(self, function, x_bounds, y_bounds):
...
def __call__(self, node):
...
@property
def x_bounds(self):
"""Get the x bounds of the function.
Returns:
tuple<int, int>: The x bounds of function in the format (min, max).
"""
...
@property
def y_bounds(self):
"""Get the y bounds of the function.
Returns:
tuple<int, int>: The y bounds of function in the format (min, max).
"""
...
```

That will be all that we need for our purposes.

As we saw before, there are only four moving pieces that our hill-climbing algorithm has: a way of determining the value at a node, an initial node generator, a neighbor generator, and a way of determining the highest valued neighbor.

Starting with the way of determining the value at a node, it's very intuitive: calculate the value $z = f(x, y)$.

```
class HillClimber:
"""A steepest-ascent hill-climbing algorithm."""
def __init__(self, function):
self.function = function
def _value_at_node(self, node):
return self.function(node)
```

The initial node can simply be taken as a random $(x, y)$ in their respective bounds.

```
def _initial_node(self):
x = randint(self.function.x_bounds[0], self.function.x_bounds[1])
y = randint(self.function.y_bounds[0], self.function.y_bounds[1])
return Node(x, y)
```

Generating neighbors is actually quite simple as well: because our domain is limited to integers, we can simply look at the four cardinal directions (and make sure we won't be breaking the bounds when we do). Also, we randomize the neighbors, to make things more interesting^{[1]}.

```
def _generate_all_neighbors(self, node):
x, y = node.x, node.y
nodes = [Node(x, y)]
if x < self.function.x_bounds[1]:
nodes.append(Node(x + 1, y))
if x > self.function.x_bounds[0]:
nodes.append(Node(x - 1, y))
if y < self.function.y_bounds[1]:
nodes.append(Node(x, y + 1))
if y > self.function.x_bounds[0]:
nodes.append(Node(x, y - 1))
shuffle(nodes)
return nodes
```

Finally, to get the highest value node, it's fairly straightforward:

```
def _highest_valued_node(self, neighbors):
max_point = neighbors[0]
for point in neighbors[1:]:
if self._value_at_node(point) > self._value_at_node(max_point):
max_point = point
return max_point
```

Piecing all this together, we get our Steepest-Ascent Hill-Climber:

```
def climb(self):
current_node = self._initial_node()
while True:
print("Exploring Node({}, {})".format(current_node.x, current_node.y))
neighbors = self._generate_all_neighbors(current_node)
successor = self._highest_valued_node(neighbors)
if self._value_at_node(successor) <= self._value_at_node(current_node):
return current_node
current_node = successor
```

Does it work? Exactly as planned.

```
Exploring Node(5, -88)
...
Exploring Node(5, -67)
...
Exploring Node(5, -47)
...
Exploring Node(5, -27)
...
Exploring Node(5, -4)
Exploring Node(4, -4)
Exploring Node(3, -4)
Exploring Node(3, -3)
Exploring Node(2, -3)
Exploring Node(2, -2)
Exploring Node(1, -2)
Exploring Node(1, -1)
Exploring Node(1, 0)
Exploring Node(0, 0)
```

However, this was too easy. We had a function with one local optimum. Let's make things interesting.

Suppose we keep our previous domain, but we change our function to the following:

$$z = -(x^2 + y^2) + x\ y\ \cos x \ \sin y $$

This function isn't quite as intuitive to visualize, please reference the figure. Essentially, it’s what we had before, but *thousands* of local optimum when we get further from the center. Our previous Hill-Climbing would absolutely get destroyed by that function.

If the neighbors are always generated deterministically, there might occur a sequence of ties when generating the highest-valued node. We randomize the neighbors so a random piece will be chosen in the tie-breaker. ↩︎

To alleviate this, we are going to use two optimizations:

- Instead of taking the steepest uphill move, we are going to simply take a random, uphill move (known as Stochastic Hill-Climbing).
- When we get stuck, we are going to restart the search (known as Hill-Climbing With Restarts).

Updating the algorithm is fairly simply, all the previous mechanics are inheritable, just swap out `_highest_valued_node`

with a stochastic version.

```
class StochasticHillClimber(HillClimber):
"""A stochastic steepest-ascent hill-climbing algorithm."""
def _get_random_uphill_move(self, current_node, neighbors):
uphill_nodes = []
for point in neighbors:
if self._value_at_node(point) > self._value_at_node(current_node):
uphill_nodes.append(point)
return current_node if len(uphill_nodes) == 0 else choice(uphill_nodes)
def climb(self):
current_node = self._initial_node()
while True:
print("Exploring Node({}, {})".format(current_node.x, current_node.y))
neighbors = self._generate_all_neighbors(current_node)
successor = self._get_random_uphill_move(current_node, neighbors)
if self._value_at_node(successor) <= self._value_at_node(current_node):
return current_node
current_node = successor
```

Running this algorithm, we get better results; but we can do better.

For this, we simply have to restructure the `climb`

function to handle generational effects (like keeping the max valued node throughout generations). Not too difficult.

```
class StochasticHillClimberWithRestarts(StochasticHillClimber):
"""A stochastic steepest-ascent hill-climbing algorithm with restarts."""
def climb(self, number_of_generations):
max_node = self._initial_node()
for generations in range(number_of_generations):
current_node = self._initial_node()
while True:
print("Generation {}, Exploring Node({}, {}), Current Max Node({}, {})".format(generations, current_node.x, current_node.y, max_node.x, max_node.y))
neighbors = self._generate_all_neighbors(current_node)
successor = self._get_random_uphill_move(current_node, neighbors)
if self._value_at_node(max_node) < self._value_at_node(current_node):
max_node = current_node
if self._value_at_node(successor) <= self._value_at_node(current_node):
break
current_node = successor
return max_node
```

How did this one fare? Quite better than all the rest. Let's take a look at what the exploration process looked like.

Marvelous, some got to the top, many got caught in local optimum. A global-optimum was found. A success.

The source code be found here.

]]>- $\mu$ And $\lambda$
- Mutation Rate
- $k$ in k-Tournament Selection

Not only this, but the sheer number of parameters:

- The genotype
- The mutator operator
- The survivor selection algorithm

And one might be wondering,

]]>- $\mu$ And $\lambda$
- Mutation Rate
- $k$ in k-Tournament Selection

Not only this, but the sheer number of parameters:

- The genotype
- The mutator operator
- The survivor selection algorithm

And one might be wondering, *what is the best operator for $x$ or $y$?* Let’s look at an example.

Recall the problem from the previous discussion:

We are going to consider a sample problem: a deciphering program. The premise of the problem is such.

- There is a string of characters (without spaces) hidden away that, after set, is inaccessible.
- There are two ways to retrieve data about the hidden message:

- Get the length of the string.
- Given a string, the problem will output how many characters match within the two strings.

Disregarding the other technical details, let us focus on the survivor selection. We used $k$-tournament selection (with $k = 50$). But, let’s run a little experiment:

Run the Evolutionary Algorithm, with $k$ ranging from 5 (basically the bare minimum) to 100 ($\lambda$, the population size), and see how fast the algorithm terminates. Do this 1,000 times to get accurate results.

The result?

*This makes sense.* Our problem has one local optimum: the actual solution. So we do not need a lot of genetic diversity, we need aggressive selective pressure^{[1]} to reach the top quickly.

As $k$ gets closer to $\mu$, the average termination time decreases. What does this tell us? *We picked the wrong survivor selection algorithm.*

With $k = \mu$, we no longer have $k$-tournament selection; we have truncation selection (where only the most fit individuals survive). And that's the interesting part about Evolutionary Algorithms: there are no objective, best parameters.

How do we alleviate this? Trial and error. There is no telling when one parameter is going to perform better than another.

A after a couple of trial runs, and objectives in mind (average terminating fitness, best terminating fitness, time to termination), the answer might surprise (and delight) you.

How elitist the survivor selection algorithm is, picking the strongest individuals more often. ↩︎

Why admit fault when you can blame someone else?

Finally, a tool I can immediately use at work. Modifies not only the commit author but the listed committer. Brilliant.

]]>Why admit fault when you can blame someone else?

Finally, a tool I can immediately use at work. Modifies not only the commit author but the listed committer. Brilliant.

]]>After trying it for a

]]>After trying it for a week, I can safely say it is a success.

I currently use an Apple Watch (Series 4) with AutoSleep, and it works like a charm. I don’t have to open any app, or use any watch face. It just works.

Taking this a step further, I wondered what would happen if I used my Apple Watch as my sole alarm. Paranoid I would not get up, I set a backup alarm for several minutes later to make sure I was awake.

*It works great.*

I would be willing to say it works better than an audible alarm; I can filter audio pretty easily, not so much with a physical sensation on my wrist. Currently, I don’t have a backup alarm set. One alarm. Five in the morning. Every weekday.

As an added bonus, my heart rate is also tracked throughout the night. This is quite useful to not only see how well I slept, but how my metabolism acted throughout the night, something quite useful to determine if I ate too late on a weekend.

Now, the only thing that can cause worry is the fact that an Apple Watch has a 18 hour battery life. I personally don’t find this a problem. I charge my watch during my shower in the morning, at my desk when I am sitting and working, and a little bit before bed. It charges fast enough throughout the day, I don’t really worry about it anymore.

A little tip as well: having a sleeping night face works pretty nicely.

]]>`#!/bin/python`

or `#!/bin/bash`

). What you might not have known that `rm`

is also a valid interpreter. Take the following script.```
#!/bin/rm
echo "Hello, world"
```

It would

]]>`#!/bin/python`

or `#!/bin/bash`

). What you might not have known that `rm`

is also a valid interpreter. Take the following script.```
#!/bin/rm
echo "Hello, world"
```

It would simply interpret to be removed. Useful? No. Hilarious? Yes.

]]>$$e = 2.71828182845$$

There are also quite a few ways of deriving Euler’s Number. There’s the Taylor expansion method:

$$e = \sum_{k = 0} ^{\infty} \frac{1}{k!}$$

There is also the classical limit form:

$$e = \lim_{n \rightarrow \infty}

]]>$$e = 2.71828182845$$

There are also quite a few ways of deriving Euler’s Number. There’s the Taylor expansion method:

$$e = \sum_{k = 0} ^{\infty} \frac{1}{k!}$$

There is also the classical limit form:

$$e = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^n$$

Then there is another. Let $R$ be a random number generated between $[0, 1]$, inclusive. Then $e$ is the average of the number of $R$s it takes to sum greater than $1$.

With more rigor, for uniform $(0,, 1)$ random independent variables $R_1$, $R_2$, $\ldots$, $R_n$,

$$N = \min \left( n: \sum _{k = 1} ^n R_i > 1 \right)$$

where

$$e = \text{Average}(n)$$

The proof can be found here, but it’s pretty math-heavy. Instead, an easier method is to write a program to verify for large enough $n$.

n | Sum Solution | Limit Solution | Random Uniform Variable |
---|---|---|---|

1 | 1 | 2 | 2 |

10 | 1.7182818011 | 2.5937424601 | 2.5 |

100 | 1.7182818284 | 2.7048138294 | 2.69 |

1000 | 1.7182818284 | 2.7169239322 | 2.717 |

10000 | 2.7181459268 | 2.7242 | |

100000 | 2.7182682371 | 2.71643 | |

1000000 | 2.7182804690 | 2.71961 | |

10000000 | 2.7182816941 | 2.7182017 | |

100000000 | 2.7182817983 | 2.71818689 | |

1000000000 | 2.7182820520 | 2.718250315 |

```
def e_sum(upper_bound):
if upper_bound < 1:
return 0
return Decimal(1.0) / Decimal(math.factorial(upper_bound)) + Decimal(e_sum(upper_bound - 1))
def e_limit(n):
return Decimal((1 + 1.0 / float(n))**n)
def find_greater_than_one(value=0, attempts=0):
if value <= 1:
return find_greater_than_one(value + random.uniform(0, 1), attempts + 1)
return attempts
```

]]>Even better? You can run it in the MacBook Pro Touch Bar

]]>Even better? You can run it in the MacBook Pro Touch Bar.

]]>*Sit down at my desk. Log into my computer, looks at i3wm.*

*Launch a new terminal session. ranger. Enter. Navigate to project. nvim. Enter.*

`space f`

. Use fzf to open necessary file.

This is a daily workflow for

]]>*Sit down at my desk. Log into my computer, looks at i3wm.*

*Launch a new terminal session. ranger. Enter. Navigate to project. nvim. Enter.*

`space f`

. Use fzf to open necessary file.

This is a daily workflow for me. Before I start typing code, I use about four tools to get to it. Notice, I haven’t mentioned touching the mouse once yet; Because I haven't touched the mouse yet.

Throughout my academic and professional career as a software engineer, I spent a lot of time using tools. Trying new tools. Searching for tools. Customizing tools^{[1]}.

Although this might be a commendable feat, it is often met with skepticism. *A lot of skepticism*.

I’ve had at least a hundred conversations with people about tools, about why they advocate against using such tools. These conservations were with other students, senior software engineers, and tenured professors; and generally, almost all of them fell into three, distinct buckets of people.

It’s a waste of time.

There is by far the worst yet best counterargument against investing so heavily into tools; and the reasoning is in the trailing, three words.

*waste*. Better tools will objectively help one write more code. If there is a modification that needs to be made at the end of the file, it takes one keystroke to jump to that positions in Vim. Picture doing this, but 100s of time a day.

*of time*. There is a time commitment, a big one. And there is a learning curve, and it is steep. So, one would have to justify the reasons for learning it. And I’ve done so below.

What I have works just fine.

I’m deeply skeptical of the “If it’s not broken, don’t fix it” mentality. There’s always room for improvement, and there’s nothing more I love to see than improvement.

It’s why we buy new phones. It’s why we like nice, new cars. It’s why we, as civilization, love progress.

Even if it’s a marginal increase, say 5% more code a week. With a 100 lines of code written a week, that can be a about 250 additional lines of code written a year. A marginal increase, sure; but a welcomed one.

It doesn’t really make you a better programmer.

I generally have three core reasons against this, and they are as follows.

*It’ll help write more code, faster.* It is objectively faster to type more code if your hands never leave the keyboard. It’s faster to navigate a filesystem without having to focus on where a cursor is. It’s ridiculously faster to open a file to just type out parts of the filename apposed to navigating to it.

*It’s healthier.* Legitimately. Minimizing hand travel will significantly put less strain on your hands, wrists, and arms.

*It’ll make you a better programmer.* As programmers, we have to learn things all the time. New programming languages, new paradigms, new company software. Learning a new tool keeps a mind sharp by reinforcing how to learn something. Plus, the skill is fluid. Learning to navigate in vim will help navigate in ranger. Learning regular expressions can be used in fuzzy searches. The skill are transferable.

So, where might one start? Again, there are three areas I see that can have the most benefit from improving your workflow.

*Your operating system*. Whether it’s Windows, MacOS, or Linux, you have to use an operating system. Get good at it. Learn the paradigms of the system. Learn how the file explorer works in Windows. Learn advanced features in MacOS like a spotlight alternative or simple hot corners. Or learn how to set up your WiFi drivers in Linux.

*Your text editor*. Choose one’s that portable. I use NeoVim. Many people have success with Emacs. Some people prefer a visual editor like Sublime Text or Atom. Learn their workflows. Learn the shortcuts of Emacs. Learn how to edit text in Normal Mode using Vim. Install packages in Sublime or Atom. You spend most of your time here anyway, might as well get familiar with it.

*The intermediary layer between your operating system and your text editor*. By this, I usually mean the shell, like Bash or ZSH. Learn how to `grep`

or navigate around. Install packages to make tasks easier. It’s well worth it.

So, yes, I am a strong advocate of using advanced tools. Established tools. Great tools. And I hope I’ve convinced you to do the same.

Gabriel's Horn is thus: suppose you have the function $y = \frac{1}{x}$ where $x \in \mathbb{R}^+, 1 \leq

]]>Gabriel's Horn is thus: suppose you have the function $y = \frac{1}{x}$ where $x \in \mathbb{R}^+, 1 \leq x \leq \infty$, rotated around the $x$ axis; not too difficult to conceptualize, it looks like a horn of sorts. But here's the paradox.

Suppose we want to calculate the volume. Simple enough, using solids of revolution, we can show the volume to be:

$$V = \pi \lim_{t \rightarrow \infty} \int _1 ^t \frac{1}{x^2} dx = \pi \lim _{t \rightarrow \infty} ( 1 - \frac{1}{t} ) = \pi $$

A simple, elegant solution; we can expect the volume to be exactly $\pi$. So, let's see about the surface area.

We know the general definition of the arc length to be $\int _a ^b \sqrt{1 + f'(x)^2}$, so combining this with our solids of revolution, we should get

$$A = 2\pi \lim _{t \rightarrow \infty} \int _1 ^t \frac{1}{x} \sqrt{1 + \left( -\frac{1}{x^2} \right)^2 } dx $$

However, this is not a trivial integral; however, there is a trick we can do. Suppose we take the integral $$2\pi \lim _{t \rightarrow \infty} \int _1 ^t \frac{dx}{x}$$ instead, and we can prove this integral will always be equal to or smaller than the former integral (because of the disappearance of $\sqrt{1 + (-\frac{1}{x^2})}$). So, taking this rather trivial integral, we can see that

$$ A \geq 2\pi \lim _{t \rightarrow \infty} \int _1 ^t \frac{dx}{x} \implies A \geq \lim _{t \rightarrow \infty} 2\pi \ln(t) $$

Wait a minute; it's divergent! So we know the volume $V = \pi$, but the surface area $A \geq \infty$. This is no mistake, the math is valid. And that is simply wonderful.

A horn you can fill with paint, but you can't paint the surface.

]]>Just as Darwinian finches evolved their beaks for to survive different parts of the Galápagos Islands, we too, evolved to

]]>Just as Darwinian finches evolved their beaks for to survive different parts of the Galápagos Islands, we too, evolved to survive different parts of the world. And we can program a computer to do the same.

Evolution inspired a whole generation of problem solving, commonly known as Evolutionary Algorithms (EAs). EAs have been known for solving (or, approximating) solutions to borderline unsolvable problems. And, just as the mechanics of evolution are not that difficult, the mechanics of EAs are just the same.

Today, we will build an Evolutionary Algorithm from the ground-up.

Before we proceed with implementation or an in-depth discussion, first we wish to tackle two questions: *what is an Evolutionary Algorithm*, *what does an Evolutionary Algorithm look like*, and *what problems can Evolutionary Algorithms solve*.

An Evolutionary Algorithm is generic, population-based optimization algorithm that generates solution via biological operators. That is quite a mouthful, so let’s break it up.

*Population-based*. All Evolutionary Algorithms start by creating a population of random individuals. These individuals are just like an individual in nature: there is a genotype (the genes that make up an individual) and a phenotype (the result of the genotype interacting with the environment). In EAs, they would be defined as follows:

**Genotype**The representation of the solution.**Phenotype**The solution itself.

Because it’s a little confusing to think of it this way, it’s often better to think about it in terms of a genotype space and a phenotype space.

**Genotype Space**The space of all possible combinations of genes.**Phenotype Space**The space of all possible solutions.

Don’t worry if this doesn’t make sense, we’ll touch on it later.

*Optimization Algorithm*. Evolution is an optimization algorithm. Given an environment, it will try to optimize an individual for that environment with some fitness metric. Evolutionary Algorithms operate the same way.

Given an individual, it will try to optimize it. We do not use a literal environment, but still use a fitness metric. The fitness metric is simply a function that takes in the genotype of the individual, and outputs a value that is proportional to how good a solution is.

Because fitness metrics are proportional to how good a solution is, this implies a very important condition for our phenotype space: it’s a gradient.

*Biological operators*. Evolutionary Algorithms are inspired by biology and evolution. Just as biology has operators to generate new individuals, so do Evolutionary Algorithms. More on that later.

*Generic*. Evolutionary Algorithms are generic. When a framework has been introduced, it can be reused on an individual basis (provided it has the appropriate crossover and mutator operators).

The pseudocode for an Evolutionary Algorithm is one we might expect evolution to have, generate a random population, generate offspring, and let survival of the fittest do its job. And so it does:

```
BEGIN
INITIALISE population with random solutions
WHILE ( TERMINATION CONDITION is satisfied ) DO
SELECT parents
RECOMBINE pairs of parents
MUTATE the resulting offspring
EVALUATE new candidates
SELECT individuals for the next generation
OD
END
```

Evolutionary Algorithms can solve any problem that has a genotype that can fit within our framework:

- The genotype can have a crossover operator.
- The genotype can have a mutator operator.
- The genotype can map to a definite fitness function.

Again, the fitness should be proportional to how good a solution is. If the fitness function $f(x)$ is bounded by $0 \leq f(x) \leq 100$, 0 should be the worst solution or no solution, and 100 should be the best solution (or vice versa, for inverted fitnesses).

We will be implementing a special class of Evolutionary Algorithm, referred to as a (μ + λ)-Evolutionary Strategy. The name is not important, but μ and λ will be; we will come back to them shortly.

For the purposes of our discussion, we are going to consider a sample problem: a deciphering program. The premise of the problem is such.

- There is a string of characters (without spaces) hidden away that, after set, is inaccessible.
- There are two ways to retrieve data about the hidden message:
- Get the length of the string.
- Given a string, the problem will output how many characters match within the two strings.

The secret message would look as follows:

```
class SecretMessage:
def __init__(self, message):
"""Initialize a Secret Message object.
:message (str): The secret message to hide.
"""
self.__message = message
def letters_match(self, message):
"""Determine how many characters match the secret message.
Note:
The message length and the secret message length must be the same length (accessed via the length property).
:message (str): The message to compare the secret message to.
:returns (int): The number of characters matched.
"""
return sum(self.__message[char] == message[char] for char in range(len(message)))
@property
def length(self):
"""Get the length of the secret message.
:returns (int): The length of the secret message.
"""
return len(self.__message)
```

Not too complicated.

In Evolutionary Algorithms, an individual is simply a candidate solution. An individual has a genotype (the representation) and operators (Crossover, Mutation, and Fitness) that act on the genotype. We will discuss them more extensively below.

As aforementioned, a genotype is the representation of an individual. Just as DNA can for humans, knowing the genotype can give you all the information one might need to determine the characteristics of an individual.

Because a genotype must be acted upon a crossover and mutation operator, there are few common choices for genotypes:

*Vectors*A vector is common because crossover is trivial, take elements from the two genotypes to create a new individual. Mutation is also trivial, pick random elements in vector, and mutate them. Often, for a complex enough individual, a vector of bits is used^{[1]}.^{[2]}.*Matrices.*Same as a vector, but with multiple dimensions.*Float-Point or Real Numbers.*This one is tricky, but commonly used. There are a plethora of ways to recombine two numbers: average of the two numbers, bit manipulation, binary encoding crossover. Same can be said for mutation: adding a random value to the number, bit manipulation with a random value, or bit flipping in binary encoding. It should be noted that some of these introduce biases, and one should account for them.*Trees.*Some problems can be easily broken down into trees (like an entire programming language can be broken down into a parse tree). Crossover is trivial, swap a random subtree with another. Mutation, however, is often not used; this is because the crossover itself acts as a mutation operator.

Next, our genotype must be initialized to some random values. Our initial population is seeded with said randomly-generated individuals, and with a good distribution, they will cover a large portion of the genotype space.

Keeping all this in mind, let us think about the representation of our problem. A string is nothing more than a vector of characters, so using the first bullet point, we are given our operators pretty easily.

Here’s what our genotype would look like:

```
class Individual:
message = SecretMessage("")
def __init__(self):
"""Initialize an Individual object.
Note:
Individual.message should be initialized first.
"""
length = Individual.message.length
characters = [choice(ascii_letters) for _ in range(length)]
self.genotype = "".join(characters)
```

As aforementioned, to fit within an Evolutionary Algorithm framework, a genotype must be created with crossover, mutator, and fitness operators. Although we have covered said operators, we will formalize them here.

*Crossover.*A crossover operator simply takes in two genotypes, and produces a genotype that is a mixture of the two. The crossover can be uniform (random elements from both genotypes are taken), 1-point (take a pivot position between two points, the left half is one genotype, and the right half another), and $N$-point (same as 1-point but with multiple pivot positions).*Mutator.*A mutator operator takes random values within the genotype and changes them to a random values. There is a mutation rate that is associated with all genotypes, we call it $p$. $p$ is bounded such that $0 \leq p \leq 1.0$, where $p$ is the percentage of the genotype that gets mutated. Careful to limit this value, however; too high $p$ can result in just a random search.*Fitness.*The fitness operator simply takes in a genotype, and outputs a numerical value proportional to how good a candidate solution is. Fitness has no limits, and can be inverted (i.e., a smaller fitness is better).

For the purposes of our program, we are going to have the following operators: crossover will be uniform, mutation will be a fixed number of mutating characters, and fitness will be a percentage of the characters matched.

```
class Individual
...
@property
def fitness(self):
"""Get the fitness of an individual. This is done via a percentage of how many characters
in the genotype match the actual message.
:return (float): The fitness of an individual.
"""
return 100 * Individual.message.letters_match(self.genotype) / Individual.message.length
@staticmethod
def mutate(individual, rate):
"""Mutation operator --- mutate an individual with a specified rate.
This is done via a uniform random mutation, by selecting random genes and swapping them.
Note:
rate should be a floating point number (0.0 < rate < 1.0).
:individual (Individual): The individual to mutate.
:rate (float): The rate at which to mutate the individual's genotype.
"""
number_of_characters_to_mutate = int(rate * individual.message.length)
genotype_list = list(individual.genotype) # Strings are immutable, we have to use a list
for _ in range(number_of_characters_to_mutate):
character_to_mutate = choice(range(individual.message.length))
genotype_list[character_to_mutate] = choice(ascii_letters)
individual.genotype = "".join(genotype_list)
@staticmethod
def recombine(parent_one, parent_two):
"""Recombination operator --- combine two individuals to generate an offspring.
:parent_one (Individual): The first parent.
:parent_two (Individual): The second parent.
:returns (Individual): The combination of the two parents (the offspring).
"""
new_genotype = ""
for gene_one, gene_two in zip(parent_one.genotype, parent_two.genotype):
gene = choice([gene_one, gene_two])
new_genotype += gene
individual = Individual()
individual.genotype = new_genotype
return individual
```

Now that we have an Individual, we must create a Population. The Population holds the candidate solutions, creates new offspring, and determine which are to propagate into further generations.

Remember when we mentioned that μ and λ would be important in our Evolutionary Algorithm? Well, now here they come into play. μ And λ are defined as follows:

*μ*: The population size.*λ*: The number of offspring to create.

Although these are simple constants, they can have a drastic impact on an Evolutionary Algorithm. For example, a Population size of 1,000 might find a solution in much fewer generations than 100, but will take longer to process. It has been experimentally shown that a good proportion between the two is:

$$

λ / μ \approx 6

$$

However, this is tested for a large class of problems, and a particular Evolutionary Algorithm could benefit from having different proportions.

For our purposes, we will pick $μ = 100$ and $λ = 15$, a proportion just a little over 6.

```
class Population:
def __init__(self, μ, λ):
"""Initialize a population of individuals.
:μ (int): The population size.
:λ (int): The offspring size.
"""
self.μ, self.λ = μ, λ
self.individuals = [Individual() for _ in range(self.μ)]
```

Generating offspring is trivial with the framework we imposed on an Individual: pick two random parents, perform a crossover between the two to create a child, mutate said child, and introduce the child back into the population pool.

In code, it would look as follows:

```
class Population:
...
@staticmethod
def random_parents(population):
"""Get two random parents from a population.
:return (Individual, Individual): Two random parents.
"""
split = choice(range(1, len(population.individuals)))
return choice(population.individuals[:split]), choice(population.individuals[split:])
@staticmethod
def generate_offspring(population):
"""Generate offspring from a Population by picking two random parents, recombining them,
mutating the child, and adding it to the offspring. The number of offspring is determine by
λ.
:population (Population): The population to generate the offspring from.
:returns (Population): The offspring (of size λ).
"""
offspring = Population(population.μ, population.λ)
offspring.individual = []
for _ in range(population.λ):
parent_one, parent_two = Population.random_parents(population)
child = Individual.recombine(parent_one, parent_two)
child.mutate(child, 0.15)
offspring.individuals += [child]
return offspring
```

The last core part of an Evolutionary Algorithm would be survival selection. This puts selective pressure on our candidate solutions, and what ultimately leads to fitter solutions.

Survivor selection picks μ Individuals that would be the best to propagate into the next generation; however, it’s not as easy as picking the fittest μ Individuals. Always picking the μ best Individuals leads to premature convergence, a way of saying we “got a good solution, but not the best solution”. The Evolutionary Algorithm simply did not explore the search space enough to find other, fitter solutions.

There are a number of ways to run a survival selection, one of the most popular being $k$-tournament selection. $k$-tournament selection picks $k$ random Individuals from the pool, and selects the fittest Individual from the tournament. It does this μ times, to get the full, new Population. The higher $k$, the higher the selective pressure; however, also the higher chance of premature convergence. The lower $k$, the less of a chance of premature convergence, but also the more the Evolutionary Algorithm starts just randomly searching.

At the bounds, $k = 1$ will always be just a random search, and $k = μ$ will always be choosing the best μ individuals.

We choose $k = 25$, giving less fit solutions a chance to win, but still focusing on the more fit solutions.

```
@staticmethod
def survival_selection(population):
"""Determine from the population what individuals should not be killed. This is done via
k-tournament selection: generate a tournament of k random individuals, pick the fittest
individuals, add it to the survivors, and remove it from the original population.
Note:
The population should be of size μ + λ. The resultant population will be of size μ.
:population (Population): The population to run survival selection on. Must be of size μ + λ.
:returns (Population): The resultant population after killing off unfit individual. Will be
of size μ.
"""
new_population = Population(population.μ, population.λ)
new_population.individuals = []
individuals = deepcopy(population.individuals)
for _ in range(population.μ):
tournament = sample(individuals, 25)
victor = max(tournament, key=lambda individual: individual.fitness)
new_population.individuals += [victor]
individuals.remove(victor)
return new_population
```

As with the pseudocode in the introduction, this will exactly resemble our Evolutionary Algorithm. Because the Individual and the Population framework is established, it is almost a copy-paste.

```
class EA:
def __init__(self, μ, λ):
"""Initialize an EA.
:μ (int): The population size.
:λ (int): The offspring size.
"""
self.μ, self.λ = μ, λ
def search(self):
"""Run the genetic algorithm until the fittness reaches 100%.
:returns: The fittest individual.
"""
generation = 1
self.population = Population(self.μ, self.λ)
while self.population.fittest.fitness < 100.0:
offspring = Population.generate_offspring(self.population)
self.population.individuals += offspring.individuals
self.population = Population.survival_selection(self.population)
generation += 1
```

Below, we have an instance of the evolutionary algorithm searching for a solution:

Now, looking at the string “FreneticArray”, it has 13 characters, and seeing as there are 26 letters in the alphabet, double that for lowercase/uppercase letters, our search space was:

$$

\left(2 * 26\right)^{13} \approx 2.0 \times 10^{22}

$$

Huge.

On average, our EA 29 generations to finish^{[3]}. As each generations had at most 115 individuals, we can conclude on average we had to generate:

$$

29 * 115 = 3335 \ \text{solutions}

$$

Much smaller than $2.0 \times 10^{22}$. That is what Evolutionary Algorithms are good for: turning a large search space into a much smaller one.

Although there are much more advanced topics in Evolutionary Algorithms, this is enough start implementing your own. With just the simple operators listed above, a genotype, a search space that has a gradient, many problems can solved with an Evolutionary Algorithm.

All source code can be found here.

For context, as of a month ago I am an alumni of Missouri University of Science and Technology, and engineering college in Missouri. It has equipped me very well in mathematics^{[1]}, physics^{[2]}, and computer science^{[3]}. I wrote roughly 159,141 lines of code in college, so I feel as if I got my money’s worth. Thanks to Missouri S&T, I have a fairly diverse background in many things.

Below is advice I would recommend to anyone currently studying computer science (or programming in general).

I got a 13-inch MacBook Pro my freshmen year of college, and it was the best decision I made in my academic career. At the beginning of my senior year, I got a 15-inch MacBook Pro with an LG UltraFine 5K Display, it was the second best decision I made in my academic career.

Getting a great computer is not just to speed up compile times; it’s to make programming more enjoyable. During a typical programming day, I’ll be on my computer 8-12 hours. 33%-50% of my my time that day will be on a computer. Reducing friction 33%-50% of the day is almost invaluable.

Although a MacBook was perfect for me, it might not necessarily be right for anyone else. Find a computer you love, you won’t regret it.

LaTeX was probably my most used programming tool in my academic career. It was a Swiss Army knife for any form of documents I might need:

- Notes
- Presentations
- Assignments
- Papers
- Documentation

Some samples of my LaTeX work can be found under my project page.

At Missouri S&T, C++ is the first (and during my time there, the only) language one learned. The problem with this? One language, whether it be the can-all that C++ is, is not always appropriate for the job. There were times where Python was a much better choice. Sometimes it was C. Sometimes it was Swift. It all depends on the project. My recommendation is if you learned Python, learn C++. If you learned C++, learn Python. From there, find languages that will be useful for your domain.

I didn’t pick up Vim until I was a junior; this was a giant mistake on my part. Vim didn’t just make it easier to delete a particular line of code; Vim made it easier to open files quickly, explore codebases faster, modify text quicker, etc.

Vim is not for everyone. Vim might not be right for you. Find a good text editor, and learn everything you can about it. It’ll make the friction of actually writing code much less.

Nearly all Computer Science programs require one to take higher-level classes as electives; in our school, it was roughly five graduate level classes. I made the mistake of taking all five my last year.

*Emphasis on mistake.*

Take them early; chances are you won’t be bogged down with job finding, apartment hunting, or the likes. Taking 18 hours, with three very difficult classes, in one term is not something I would wish on anyone.

When you find topics that interest you in Computer Science, chances are you will like your major more. I found a particular interests in AI related courses, but it is not for everyone. Computer Science has a breadth of interesting topics; check to see what your school has to offer, like:

- Security
- AI
- Data Mining and Big Data
- Machine Learning
- Databases
- UI/UX Design

Burnout is a serious issue in the industry. It can be a serious issue inside school as well. Foster passions outside of the classroom. Maybe join a design team, fraternity, or a sports team. There will be plenty of time at work to learn new things.

In addition, take genuinely interesting elective (not the easy ones). I learned I had an interest in Physics, and I carry that with me everyday.

Internships are not only a gentle introduction into the industry, but a great way to determine what job you will want outside of school. Internships are often joked as “A 12-week interview processes”, and this can be true. If you find a good company that you can do good work for, it’s a great way to secure a job upon graduation.

The most important lesson I learned in college was to have fun (in moderation). College is all about learning who you are, what you want out of life, how to interact with others, and how to simply have fun. Lessons you learn in college will get carried over after graduating. Why not use this time to make life-long friends, create great memories, and know who you really are?

This guide is not comprehensive; but it’s something I wish I read when I was starting. There’s always room to grow, so figure out what works for you. Do what makes you happy.

If there’s anything I am missing, feel free to ask.