text stringlengths 128 2.05k |
|---|
at most [MATH] . The following theorem proves a surprising result for the considered ( [MATH] +1) RLS [MATH] for Cliff Not only is the algorithm very fast, but our upper bound becomes lowest (i.e., [MATH] ) when the function is most difficult (i.e., when the cliff is located at distance |
[MATH] from the optimum). In this setting the algorithm is asymptotically as fast as any evolutionary algorithm using SBM can be on any function with unique optimum |
. A similar result has recently been shown also for a hyperheuristic which switches between elitist and non-elitist selection operators |
Theorem 6 For [MATH] [MATH] a constant and [MATH] the ( [MATH] +1) RLS [MATH] optimises Cliff in expected time [MATH] if [MATH] for any constant |
[MATH] Proof. We follow the proof of Theorem 10 in of the [MATH] RLS for the Balance function and adapt the arguments therein to the OneMax landscape and to the RLS operator we use. Given that there are [MATH] individuals with [MATH] 1-bits in the population, the probability of creating a new individual with [MATH] 1-b... |
because the RLS operator creates a copy with a constant probability [MATH] Hence we follow the proof in to show that in [MATH] expected steps the population climbs up the OneMax slope (i.e., samples a solution with [MATH] 1-bits) and subsequently the whole population will be taken over by the local optimum. Given that ... |
[MATH] . Thus in at most [MATH] generations in expectation after the first local optima is sampled, the population is taken over by the local optima. Now we can apply Lemma 5 in |
to show that in expected [MATH] steps the whole population will have the same age. As a result, after another at most [MATH] generations the whole population will reach age [MATH] simultaneously because no improvements may occur unless the optimum is found. Overall, the total expected time until the population consists... |
[MATH] Now we calculate the probability that in the next step one individual jumps to the bottom of the cliff and the rest die in the same generation. The first event happens with probability [MATH] (i.e., an offspring solution with [MATH] 1-bits is created by flipping one of the [MATH] 0-bits in the parent solution). ... |
and in the same generation with probability [MATH] the parent of age [MATH] dies due to ageing. Finally, the new solution (i.e., the safe individual) takes over the population in [MATH] expected generations by standard arguments. It takes at least [MATH] generations before any of the new random individuals has [MATH] m... |
[MATH] steps is at most [MATH] Hence, the overall probability of this series of consecutive events is [MATH] and the expected number of trials (i.e., climb-ups and restarts) until we get a survivor which is safe at the bottom of the cliff is |
[MATH] Every time the set of events fails to happen, we wait for another [MATH] fitness evaluations until the population reaches a configuration where all individuals are locally optimal and have age [MATH] Once a safe individual has taken over the population, the expected time to find the global optimum will be at mos... |
generations in expectation and at most [MATH] generations with overwhelming probability (w.o.p. ) until the whole population takes over the level. For the first [MATH] levels, the probability of improving a solution is at least |
[MATH] and the probability that this improvement does not happen in [MATH] generations is at most [MATH] . For the remaining fitness levels, the probability of reaching age [MATH] before improving is similarly [MATH] . By the union bound over all levels, the probability that the best solution is never lost due to agein... |
[MATH] restarts are necessary in expectation, our bound on the expected runtime holds. We conclude the section by considering the ( [MATH] +1) EA [MATH] which differs from the |
[MATH] +1) RLS [MATH] by using SBM with mutation rate [MATH] instead of flipping exactly one bit. SBM allows copying individuals but, since it is a global operator, it can jump back to the local optima from anywhere in the search space with non-zero probability. Nevertheless, the following theorem shows that, for not t... |
Theorem 7 The ( [MATH] +1) EA [MATH] optimises Cliff in expected [MATH] time if [MATH] for some constant [MATH] [MATH] and [MATH] , where [MATH] is an arbitrarily small positive constant. |
Proof. The probability that the SBM operator increases the number of 1-bits in a parent solution with [MATH] 0-bits is at least [MATH] . Following the same arguments as in the proof of Theorem while considering [MATH] and [MATH] we can show that with constant probability and in expected [MATH] time the algorithm reache... |
We will now show that the newly generated individuals have worse fitness than the solution at the bottom of the cliff when they are initialised. Since |
[MATH] , the fitness value of the solutions with more than [MATH] 1-bits is at least [MATH] . Due to Chernoff bounds, the newly created individuals have less than [MATH] |
1-bits with overwhelming probability. Next we prove that for any constant [MATH] , there exist some positive constant [MATH] , such that the best solution at the bottom of the cliff (the leading solution) will be improved consecutively for [MATH] |
iterations with probability at least [MATH] . The leading solution with [MATH] 0-bits is selected for mutation and SBM flips a single 0-bit with probability [MATH] . With probability |
[MATH] [MATH] consecutive improvements occur. We can bound this probability from below by the final improvement probability [MATH] raised to the power of [MATH] since the improvement probability is inversely proportional to the number of 0-bits. Considering that [MATH] , we can set [MATH] , which yields [MATH] . Here, ... |
[MATH] 1-bits. More precisely there will be one and only one individual in the population with [MATH] bits for all [MATH] in [MATH] . Since [MATH] is constant and SBM flips at least [MATH] bits with probability [MATH] , the probability that a single operation of SBM decreases the number of 1-bits in any individual belo... |
[MATH] is in the order of [MATH] . Since this probability is not polynomially bounded, with probability at least [MATH] it will not happen in any polynomial number of iterations. After the consecutive improvements occur, it takes [MATH] until the second slope is climbed and the optimum is found. |
The asymptotic bound on the runtime is obtained by considering that algorithm will reach the local optimum [MATH] times in expectation before the consecutive improvements are observed. Since the time to restart after reaching the local optimum is in the order of [MATH] , the expected time is [MATH] . Since [MATH] is an... |
Opt-IA After having analysed the operators separately, in this section we consider the complete Opt-IA. The considered Opt-IA, shown in Algorithm , uses static hypermutation coupled with FCM as variation operator, hybrid ageing and an standard [MATH] selection which allows genotype duplicates . Also, a mutation is cons... |
In this section we first show that Opt-IA is efficient for all the functions considered previously in the paper. Then, in Subsection 5.1 we present a problem where the use of the whole Opt-IA is crucial. In Subsection 5.2 we show limitations of Opt-IA by presenting a class of functions where standard EAs are efficient ... |
The following theorem proves that Opt-IA can optimise any benchmark function considered previously in this paper. The theorem uses that the ageing parameter [MATH] is set large enough such that no individuals die with high probability before the optima are found. |
Theorem 8 Let [MATH] be large enough. Then the following upper bounds on the expected runtime of Opt-IA hold: [MATH] Proof. The claims use that if [MATH] is large enough (i.e., [MATH] for OneMax and LeadingOnes [MATH] for [MATH] and [MATH] for [MATH] , then with probability [MATH] the current best solution will never r... |
Theorem 10 shows that the presented upper bound for OneMax is tight for sub-logarithmic population and clone sizes (i.e., [MATH] and [MATH] ). Before stating this theorem, we state the following helper theorem which was already used in |
to prove lower bounds on the expected runtime of inversely proportional hypermutations. Theorem 9 (Ballot Theorem “Suppose that, in a ballot, candidate P scores [MATH] votes and candidate Q scores [MATH] votes, where [MATH] . The probability that throughout the counting there are always more votes for P than for Q equa... |
This theorem allows us to derive an upper bound on the probability that at some point of the hypermutation we have picked more 0-bits than 1-bits, which implies an improvement by hypermutation for OneMax |
Theorem 10 Opt-IA needs at least [MATH] expected fitness function evaluations for any mutation potential [MATH] to optimise OneMax |
Proof. By Chernoff bounds, the initial individuals have at least [MATH] 0-bits with overwhelming probability. To calculate the probability of an improvement (i.e., flipping equal or more 0-bits than 1-bits during one mutation operation), we use the Ballot theorem ( i.e., Theorem ) in a similar way to |
. Considering the number of 0-bits as [MATH] and the number of 1-bits as [MATH] , the probability of an improvement is at most [MATH] according to the Ballot theorem Hence, the probability that at least one out of [MATH] individuals succeeds is [MATH] by the union bound . We optimistically assume that the rest of the i... |
[MATH] Since the mutation operator stops at the first constructive mutation (i.e., when the number of 1-bits is increased by one), it is necessary to improve at least [MATH] times. So the total expected time to optimise OneMax is [MATH] |
If individuals were to be removed because of ageing, then the new randomly generated individuals that replace them will have to improve at least [MATH] times all over again w.o.p. Hence, the runtime may only increase in such an event. |
5.1 Opt-IA Can Be More Efficient In this section, we present the function HiddenPath to illustrate a problem where the use of static hypermutation and ageing together is crucial. When either of these two characteristic operators of Opt-IA is not used, we will prove that the expected runtime is at least superpolynomial.... |
ZeroMax function. The distinguishing solutions are those with five 0-bits and those with [MATH] 0-bits, along with [MATH] solutions of the form [MATH] for [MATH] (called Sp points). The solutions with exactly [MATH] 0-bits constitute the local optima of HiddenPath (called LocalOpt ), and the solutions with exactly five... |
Definition 1 Given the definitions of Sp and LocalOpt as above, for any positive constant [MATH] , the HiddenPath function is defined for all [MATH] by |
[EQUATION] Since the all 0-bits string returns fitness value zero, there is a drift towards solutions with [MATH] 0-bits while the global optimum ( Opt ) is the [MATH] bit string. The solutions with exactly five 0-bits work as a net that stops any static hypermutation that has an input solution with less than five 0-bi... |
Theorem 11 For [MATH] [MATH] [MATH] and [MATH] , Opt-IA needs expected [MATH] fitness function evaluations to optimise HiddenPath |
Proof. For convenience we will call any solution with [MATH] 0-bits (expect the Sp solutions) an [MATH] solution. After [MATH] generations in expectation, an [MATH] |
solution is found by optimising ZeroMax . Assuming the global optimum is not found first, consider the generation when an [MATH] solution is found for the first time. Another [MATH] solution is created and accepted by Opt-IA with probability at least [MATH] since it is sufficient to flip the single 1-bit in the first m... |
[MATH] generations. Since, apart from the optimum, no other solution has higher fitness than [MATH] solutions, the population consists only of [MATH] solutions after the takeover occurs. A solution reaches age [MATH] before the takeover only with probability [MATH] , due to Markov’s inequality applied iteratively for [... |
[MATH] We now bound the expected time until the entire population has the same age. Considering that the probability of creating another [MATH] |
solution is [MATH] , the probability of creating two copies in one single generation is [MATH] . With constant probability this event does not happen in |
[MATH] generations. Conditional on that at most one additional [MATH] solution is created in every generation, we can follow a similar argument as in the proof of Theorem . Hence, we can show that in expected |
[MATH] iterations after the takeover, the whole population reaches the same age. When the population of [MATH] solutions with the same age reaches age |
[MATH] , with probability [MATH] a single new clone survives while the rest of the population dies. With probability [MATH] the survived clone has hypermutated all [MATH] bits (i.e., the survived clone is an [MATH] solution). In the following generation, the population consists of an [MATH] |
solution and [MATH] randomly sampled solutions. With probability 1, the [MATH] solution produces an [MATH] solution via hypermutation. On the other hand, with overwhelming probability the randomly sampled solutions still have fitness value [MATH] , hence the [MATH] solution is removed from the population while the [MAT... |
[MATH] generations. We momentarily ignore the event that the [MATH] solution reaches an [MATH] point via hypermutation. Now, the population consists of a single |
[MATH] solution and [MATH] solutions with at most [MATH] 0-bits. We want to bound the expected time until [MATH] solutions take over the population conditional on no [MATH] solutions being created. Clones of any [MATH] solutions are also [MATH] solutions after hypermutation if one of the first two bits to be flipped is... |
solution since all [MATH] bit-flips will have been executed. Since [MATH] solutions have higher fitness value than the randomly sampled solutions, they stay in the population. In the subsequent generation, if hypermutation does not improve an [MATH] |
solution (which happens with probability [MATH] ), it executes [MATH] bit-flips to create yet another [MATH] solution unless the Sp path is found. |
This feedback causes the number of [MATH] and [MATH] solutions to double in each generation with constant probability until they collectively take over the population in [MATH] generations in expectation. Then, with constant probability all the [MATH] solutions produce an [MATH] |
solution via hypermutation and consequently the population consists only of [MATH] solutions. Since the takeover happens in expected |
[MATH] generations, the probability that it fails to complete in [MATH] generations for consecutive [MATH] times is exponentially small. Hence, in [MATH] generations w.o.p. (by applying Markov’s inequality iteratively |
) the entire population are [MATH] solutions conditional on [MATH] solutions not being created before. Since the probability that the static hypermutation creates an [MATH] solution from an [MATH] solution is less than [MATH] , and the probability that it happens in [MATH] generations is less than [MATH] , the takeover... |
After the whole population consists of only [MATH] solutions, except for the global optimum and the local optima, only other points on the gradient would have higher fitness. The probability of improving on the gradient is at least |
[MATH] which is the probability of choosing two specific bits to be flipped (a 0-bit and a 1-bit) leading towards the b-cell with the best fitness on the gradient. Considering that the total number of improvements on the gradient is at most [MATH] , in [MATH] generations in expectation the first point of Sp will be fou... |
[MATH] , before finding the first point of the Sp Due to the Markov’s inequality applied iteratively over [MATH] consecutive phases of length [MATH] with an appropriate constant, the probability that the time to find the first point of Sp is more than |
[MATH] is less than [MATH] . The probability of jumping to a local optimum in [MATH] steps is at most [MATH] by the union bound. Therefore, Sp is found before any local optima in at most [MATH] generations with probability |
[MATH] . Hence, the previously excluded event has now been taken into account. After [MATH] is added to the population, the best solution on the path is improved with probability [MATH] by hypermutation and in expected [MATH] |
generations the global optimum is found. Since all Sp and [MATH] solutions have a Hamming distance smaller than [MATH] and larger than [MATH] to any [MATH] solution, the probability that a local optimum is found before the global optimum is at most [MATH] by the union bound. Thus with probability [MATH] the time to fin... |
In the following two theorems we show that hypermutations and ageing used in conjunction are essential. Theorem 12 Opt-IA without ageing (i.e., [MATH] ) with [MATH] and |
[MATH] cannot optimise HiddenPath in less than [MATH] expected fitness function evaluations. Proof. The only points with higher fitness than solutions with more than [MATH] 0-bits are Sp points and [MATH] points. If all solutions in the population have less than [MATH] 0-bits or less than [MATH] 1-bits for some [MATH] ... |
fitness function evaluations by optimising ZeroMax (i.e., Theorem ). In the rest of the proof we will calculate the probability of reaching an [MATH] solution before finding either an Sp point or [MATH] point, or a complementary solution of an Sp point. The latter solutions, if accepted, with high probability would hyp... |
points. We call an event bad where any of the mentioned points are discovered. Any solution in the search space can have at most two Hamming neighbours on Sp or at most two Hamming neighbours that their complementary bit strings are on Sp . The probability of sampling any of these neighbours is in the order of [MATH] s... |
Sp points (or their complementary bit strings) are discovered is [MATH] . For any initial solution with less than [MATH] 0-bits, the probability of finding a solution with five 0-bits is at most [MATH] . Since the probability of reaching an [MATH] point from solutions with more than [MATH] 0-bits may be much larger, we... |
The probability of finding a solution which improves the ZeroMax value is at least [MATH] (even when we exclude the potential improvements whose complementary bit strings are on Sp ). Since at every static hypermutation, the probabilities of finding an [MATH] solution, an Sp solution or its complementary bit string are... |
Once a locally optimal solution is found, with high probability it takes over the population before the optimum is found and the expected runtime conditional on the current population consisting only of solutions with [MATH] 0-bits is at least |
[MATH] By the law of total expectation, the unconditional expected runtime is lower bounded by [MATH] This expression is in the order of |
[MATH] for any [MATH] and [MATH] Theorem 13 With Probability at least [MATH] , Opt-IA using SBM and ageing cannot optimise HiddenPath in less than [MATH] fitness function evaluations. |
Proof. Since the number of 1-bits in an initial solution is binomially distributed with expectation [MATH] the probability that an initial solution has less than |
[MATH] 1-bits is bounded above by [MATH] using Chernoff bounds. Hence, w.o.p., the population has a Hamming distance at least [MATH] from any solution on the path and any solution with five 0-bits [MATH] ) by the union bound. Therefore, the probability of finding either any of the [MATH] points on |
Sp or one of the [MATH] points on [MATH] is [MATH] Since accepting any improvements, except for the [MATH] solutions, increases the distance to the [MATH] solutions, the probability of jumping to an [MATH] solution further decreases throughout the search process. |
5.2 When Opt-IA is detrimental In this section we present a function class for which Opt-IA is inefficient. The class of functions, which we call HyperTrap is defined formally in Definition HyperTrap is inspired by the Sp-Target function introduced in |
and used in as an example where hypermutations outperform SBM. Compared to Sp-Target HyperTrap has local and global optima inverted with the purpose of trapping hypermutations. Also there are some other necessary modifications to prevent the algorithm from finding the global optimum via large mutations. The parameter [... |
In HyperTrap , the solutions with [MATH] are evaluated by OneMax . Solutions of the form [MATH] with [MATH] , shape a short path which is called Sp . The last point of Sp , i.e., [MATH] is the global optimum ( Opt ). Local optima of HyperTrap LocalOpt ) are formed by the points with [MATH] which have a Hamming distance... |
[MATH] to all points in Sp . Also, the points with [MATH] are ranked among each other such that bit strings with more 1-bits in the beginning have higher fitness. This ranking forms a gradient from [MATH] , which has the lowest fitness, to [MATH] which has the highest fitness. Finally, the fitness of points which do no... |
Definition 2 Given the definitions of Sp Opt and LocalOpt as above and [MATH] showing the minimum Hamming distance of the individual [MATH] to all Sp points, the HyperTrap function with [MATH] is defined for all [MATH] by |
[EQUATION] [EQUATION] This function is depicted in Figure . We will show that there exists a [MATH] such that Opt-IA with mutation potential [MATH] gets trapped in the local optima. |
Theorem 14 With probability [MATH] , Opt-IA with mutation potential [MATH] cannot optimise [MATH] in less than exponential time. |
Proof. We will show that the population will first follow the ZeroMax (or OneMax ) gradient until it samples a solution with [MATH] 0-bits and then the gradient of [MATH] until it samples an Sp point with approximately [MATH] 0-bits. Afterwards, we will prove that large jumps on Sp are unlikely. Finally, we will show t... |
With overwhelmingly high probability, all randomly initialised individuals have [MATH] 0-bits for any arbitrarily small [MATH] . Starting with such points, the probability of jumping to the global optimum is exponentially small according to Lemma . Being optimistic, we assume that the algorithm does not sample a locall... |
point is found, the current best fitness can only be improved if a point with either a higher ZeroMax (or OneMax ) or a higher [MATH] value than the current best individual is sampled. Since there are less than [MATH] different values of [MATH] , and less than [MATH] different values of ZeroMax (or OneMax ), the curren... |
Let Sp lower denote the Sp points with less than [MATH] 1-bits, and Sp upper denote [MATH] . We will now show that with overwhelmingly high probability an Sp lower point will be sampled before an Sp upper point. The search points between [MATH] 1-bits have at least a distance of [MATH] to the Sp upper points. Using Lem... |
Sp upper by [MATH] from any input solution with [MATH] 1-bits. Excluding Sp points, a solution with [MATH] 1-bits has an improvement probability of at least [MATH] (i.e., [MATH] with the second term being the probability of improving the ZeroMax or OneMax value and the first term the probability of improving when the s... |
Sp is [MATH] . This yields that the probability of improving by [MATH] in one generation is at most [MATH] , as there are [MATH] individuals in the population. We have now shown that it is exponentially unlikely that even an arbitrarily small fraction of the path points are avoided by jumping directly to path points wi... |
Let [MATH] be the first iteration when the current Sp solution has at least [MATH] 1-bits for the first time. We will first lower bound the probability of improving in order to upper bound the conditional probability that more than [MATH] new 1-bits are added given that an improving path point is sampled. An improvemen... |
[MATH] . Therefore, in generation [MATH] the current best individual cannot have more than [MATH] 1-bits with probability [MATH] by the union bound. Similarly, the probability that no jump of size |
[MATH] happens in [MATH] generations is at least [MATH] . So, we can conclude that the number of generations to add another [MATH] 1-bits to the prefix of the current best solution is at least [MATH] with the same probability. |
Now, we show that during this number of generations, the algorithm gets trapped in the local optima with overwhelming probability with the optimistic assumption that only the best b-cell in the population can be mutated into a locally optimal solution. Considering the best current individual, in each generation a 1-bit... |
[EQUATION] After flipping a 1-bit, the mutation operator mutates at most [MATH] bits until it finds an improvement. All sampled solutions in the first [MATH] mutation steps will have more than [MATH] 1-bits and thus satisfy the first condition of the local optima. If the Hamming distance between one of the first [MATH]... |
to the local optima. Similarly to the analysis done in , we consider the [MATH] th mutation step for [MATH] where the last expression is due to [MATH] . After [MATH] steps, the expected number of bits flipped in the prefix of length [MATH] is at least |
[MATH] . For any mutation potential [MATH] [MATH] Using Chernoff bounds, we can show that the probability of having less than [MATH] 0-bits in the prefix is [MATH] |
Altogether, with probability [MATH] a point in the local optima is sampled. In [MATH] generations this individual takes over the population. Once in the local optima, the algorithm needs at least [MATH] time to find the global optimum according to Lemma |
In the following theorem we see how the (1+1) EA using SBM optimises HyperTrap in polynomial time. Theorem 15 The (1+1) EA optimises HyperTrap in [MATH] steps w.o.p. [MATH] for [MATH] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.