We will implement the optimization algorithm (bat algorithm)

Introduction

A series of optimization algorithm implementations. First, look at Overview.

The code can be found on github.

Bat algorithm

The Bat Algorithm is an algorithm that focuses on echo-localization behavior in which bats use ultrasonic waves to locate prey and obstacles.

Bats decide their behavior based on four factors: speed, frequency, pulse rate, and volume.

・ Introduction to Evolutionary Computation Algorithms Optimal Solutions Derived from Behavioral Sciences of Organisms

• Algorithm flow

• Term correspondence

• Meaning of variables used in code

Get closer to the best bats

To move to the best bat, update the speed based on the difference and frequency between the position of the best bat and your position, and move using that speed.

$ \ Vec {v_i} $ is the velocity, $ \ vec {x_i} $ is the position, and $ \ vec {g} $ is the best bat position.

The frequency is a random number between the lower limit ($ f_ {min} ) and the upper limit ( f_ {max} $).

import random
import numpy as np

#frequency
frequency = frequency_min + (frequency_max - frequency_min) * random.random()

pos_best = np.asarray(best_bat.getArray())
pos = np.asarray(bat["bat"].getArray())

#Calculate speed
bat["v"] = bat["v"] + frequency * (pos_best - pos)

#Update position and create bats in new position
new_bat1 = problem.create(pos + bat["v"])

Move near a good bat

It is a good move to a bat, but first compare the pulse rate and the random number, and if it is above the random number, it will not move.

If you want to move, first choose a good bat. Good bats are randomly selected from the top few percent of all bats.

Then move closer to a good bat.

Where $ \ bar {A} $ is the average volume of all bats and $ \ vec {\ epsilon} $ is a random vector from -1 to 1.

import random
import numpy as np

new_bat2 = None

#Move closer to a good bat with a random number
if bat["pulse"] < random.random():

    #Sort
    bats.sort(key=lambda x: x["bat"].getScore())

    #Randomly choose good bats
    r = random.randint(1, good_bat_num)
    pos_good = np.asarray(bats[-r]["bat"].getArray())
    
    #Average volume of all bats
    volume = np.average([x["volume"] for x in self.bats])
    
    rn = np.random.uniform(-1, 1, len(pos_good))  # -1,Random number of 1
    new_pos2 = pos_good + volume * rn
    new_bat2 = problem.create(new_pos2)

update

The update formula for the pulse rate ($ r_i $) is as follows.

$ R_0 $ is the convergence value of the pulse rate (constant), and $ \ gamma $ is the speed of convergence of the pulse rate (constant).

Also, the update formula for the volume ($ A_i $) is as follows.

$ \ Alpha $ is the rate (constant) that reduces the volume.

Updates will be made according to the algorithm flow chart.

import random

volume_update_rate = 0.9
pulse_convergence_value = 0.5
pulse_convergence_speed = 0.9

def _calcPulse(count):
    return pulse_convergence_value * (1-math.exp(-pulse_convergence_speed*count))

#Randomly generate new positions
new_bat3 = problem.create()

score1 = new_bat1.getScore()
score2 = None if new_bat2 is None else new_bat2.getScore()
score3 = new_bat3.getScore()

if (score2 is None or score1 > score2) and score1 > score3 :
    #Random number of volume
    if random.random() < bat["volume"]:
        if score2 is None or score2 <= score3:
            #Update at random position
            bat["bat"] = new_bat3
        else:
            #Updated at a location near a good bat
            bat["bat"] = new_bat3

        #Pulse rate update
        bat["pulse"] = _calcPulse(step_count)

        #Volume update
        bat["volume"] = volume_update_rate * bat["volume"]

    else:
        #Updated in position to the best bat
        bat["bat"] = new_bat1
else:
    #Updated in position to the best bat
    bat["bat"] = new_bat1

#Step count+1
step_count += 1

About pulse rate

The pulse rate changes as follows depending on the convergence speed. (When pulse_convergence_value = 1) Even if it is 0.2, it will converge in about 20 times, and it will converge fairly quickly, so a small value may be good.

Whole code

import math
import random

import numpy as np

class Bat():
    def __init__(self,
        bat_max,
        frequency_min=0,
        frequency_max=1,

        good_bat_rate=0.2,
        volume_init=1.0,
        volume_update_rate=0.9,
        pulse_convergence_value=0.5,
        pulse_convergence_speed=0.9,
    ):
        self.bat_max = bat_max
        self.frequency_min = frequency_min
        self.frequency_max = frequency_max

        self.good_bat_num = int(bat_max * good_bat_rate + 0.5)
        if self.good_bat_num > bat_max:
            self.good_bat_num = bat_max
        if self.good_bat_num < 1:
            self.good_bat_num = 1
        self.volume_init = volume_init
        self.volume_update_rate = volume_update_rate
        self.pulse_convergence_value = pulse_convergence_value
        self.pulse_convergence_speed = pulse_convergence_speed


    def init(self, problem):
        self.problem = problem
        self.count = 0

        self.step_count = 0
        self.best_bat = None
        self.bats = []
        for _ in range(self.bat_max):
            o = problem.create()

            d = {
                "bat": o,
                "pulse": self._calcPulse(0),
                "volume": self.volume_init,
                "v": np.zeros(problem.size)
            }
            self.bats.append(d)

            if self.best_bat is None or self.best_bat.getScore() < o.getScore():
                self.best_bat = o

    def _calcPulse(self, count):
        return self.pulse_convergence_value*(1-math.exp(-self.pulse_convergence_speed*count))
        
    def step(self):
        self.bats.sort(key=lambda x: x["bat"].getScore())

        for bat in self.bats:

            #frequency
            frequency = self.frequency_min + (self.frequency_max - self.frequency_min) * random.random()
            
            #Get closer to the best bats
            pos_best = self.best_bat.getArray()
            pos = bat["bat"].getArray()
            bat["v"] = bat["v"] + frequency * (pos_best - pos)
            new_bat1 = self.problem.create(pos + bat["v"])

            #Volume judgment
            if random.random() >= bat["volume"]:
                #Update in new position
                bat["bat"] = new_bat1
            
                #Best bat check
                if self.best_bat.getScore() < bat["bat"].getScore():
                    self.best_bat = bat["bat"]
                continue

            #Move closer to a good bat with a random number
            new_bat2 = None
            if bat["pulse"] < random.random():
            
                #Average volume of all bats
                volume = np.average([x["volume"] for x in self.bats])
                
                #Choose a good bat
                r = random.randint(1, self.good_bat_num)
                pos_good = self.bats[-r]["bat"].getArray()
                
                rn = np.random.uniform(-1, 1, len(pos_good))  # -1,Random number of 1
                new_bat2 = self.problem.create(pos_good + volume * rn)
            
            #Randomly generated
            new_bat3 = self.problem.create()
                
            #Comparison of new positions
            score1 = new_bat1.getScore()
            score2 = None if new_bat2 is None else new_bat2.getScore()
            score3 = new_bat3.getScore()
            if (score2 is None or score1 > score2) and score1 > score3 :
                if score2 is None or score2 <= score3:
                    bat["bat"] = new_bat3
                else:
                    bat["bat"] = new_bat2
                
                #Pulse rate update
                bat["pulse"] = self._calcPulse(self.step_count)

                #Volume update
                bat["volume"] = self.volume_update_rate * bat["volume"]
            else:
                bat["bat"] = new_bat1

            #Best bat check
            if self.best_bat.getScore() < bat["bat"].getScore():
                self.best_bat = bat["bat"]

        self.step_count += 1

Hyperparameter example

It is the result of optimizing hyperparameters with optuna for each problem. A single optimization attempt yields results with a search time of 2 seconds. I did this 100 times and asked optuna to find the best hyperparameters.

• Frequency_min is fixed at 0

Visualization of actual movement

The 1st dimension is 6 individuals, and the 2nd dimension is 20 individuals, which is the result of 50 steps. The red circle is the individual with the highest score in that step.

The parameters were executed below.

Bat(N, frequency_min=0, frequency_max=0.05, good_bat_rate=0.1, volume_init=0.5, pulse_convergence_value=0.9, pulse_convergence_speed=0.1)  

function_Ackley

• 1D

• 2D

function_Rastrigin

• 1D

• 2D

function_Schwefel

• 1D

• 2D

function_StyblinskiTang

• 1D

• 2D

function_XinSheYang

• 1D

• 2D

Afterword

After moving to the vicinity, the speed remains, so it feels like you are exploring a wide area. Also, since the speed basically remains, it seems that the movement will be very rough in the second half. Adjustment of hyperparameters may be quite severe.