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The outer proof argument is variable drift, that is, for each fitness value [MATH] we estimate the expected fitness gain (“progress”) in an iteration starting with a parent individual [MATH] with [MATH] and then we use the variable drift theorem (Theorem ) to translate this information on the progress into an expected ... |
To obtain sufficiently strong lower bounds on the expected progress, we need to argue that the rate self-adjustment sufficiently often sets the current rate to a sufficiently good value. This is the main technical difficulty as it needs a very precise analysis of the quality of the offspring in both subpopulations. Sin... |
In the far region covering the fitness distance values [MATH] , we need a rate [MATH] that is at least almost logarithmic in [MATH] to ensure that the average progress is high enough. Note that to gain the [MATH] fitness levels from the initial value of approximately [MATH] to [MATH] in at most a time of [MATH] , we ne... |
[EQUATION] in an iteration with initial fitness distance [MATH] . This will be sufficient to reach a fitness distance of [MATH] in the desired [MATH] iterations. |
As said, the main difficulty is arguing that the self-adjusting mechanism keeps the rate sufficiently often in the range we need, which is roughly |
[EQUATION] for all [MATH] . Note that these range boundaries, in particular the upper one, depend strongly on [MATH] . Hence for arguing that our rate is sufficiently often in this range, we need to consider both the rate changes from the self-adjustments and the changing [MATH] -value. In this region we profit from th... |
The middle region covering the fitness distances [MATH] is small enough so that we do not require the algorithm to find a near-optimal rate very often. In fact, it suffices that the rate is below [MATH] with constant probability. This ensures an expected progress of at least [MATH] , which is sufficient to traverse als... |
In the near region covering the remaining fitness distance values [MATH] , the parent individual is already so close to the optimum that any rate higher than the minimal rate [MATH] is sub-optimal. Hence in this region, we know precisely the optimum value for the rate. Nevertheless, it is not very easy to show that the... |
By quantifying this effect we establish a drift of the rate down to its minimum value [MATH] in the near region, and another occupation probability argument is used to show that the rate with sufficiently high probability will take this value in subsequent sets. Nevertheless, the concentration of the rate around this t... |
Far Region In this first of three technical sections, we analyze the optimization behavior of our self-adjusting (1+ [MATH] ) EA in the regime where the fitness distance [MATH] is at least [MATH] . Since we are relatively far from the optimum, it is relatively easy to make progress. On the other hand, this regime spans... |
Lemma 9 Let [MATH] be sufficiently large and [MATH] . We define [MATH] and [MATH] (a) If [MATH] and [MATH] , then the probability that a best offspring has been created with rate [MATH] is at least [MATH] |
(b) Let [MATH] . If [MATH] , then the probability that all best offspring have been created with rate [MATH] is at least [MATH] (c) |
If [MATH] , then the probability that all best offspring are worse than the parent is at least [MATH] Proof. To prove part (a) , let [MATH] and [MATH] be the probabilities that standard bit mutation with mutation rate [MATH] creates from a parent with fitness distance [MATH] an offspring with fitness distance exactly [... |
[EQUATION] and [MATH] . We first show that the probability of not achieving [MATH] is less than [MATH] . This is because for large enough [MATH] and [MATH] we have [MATH] and for [MATH] by Lemma , part (c) we have [MATH] , thus |
[EQUATION] Considering [MATH] offspring using rate [MATH] , the probability that none of them achieves a fitness improvement of at least [MATH] is less than [MATH] . We next argue that an offspring having a progress of [MATH] or more with good probability comes from the [MATH] -subpopulation. We notice that |
[EQUATION] Thus for [MATH] and [MATH] we have [EQUATION] Therefore if the best progress among all [MATH] offspring is [MATH] and [MATH] , the conditional probability that an offspring having fitness distance [MATH] is generated with rate [MATH] is at least [MATH] . In total, the probability that a best offspring has be... |
[EQUATION] For the proof of part (b) , let [MATH] and [MATH] . Our idea is to show a probability of [MATH] for the event that an offspring with rate [MATH] is not worse than the expected fitness of an offspring with rate [MATH] . Let [MATH] denote the random decrease of the fitness distance when applying standard bit m... |
[EQUATION] According to Bernstein’s inequality (Theorem (a) ), for any [MATH] we have [EQUATION] We apply this bound with [MATH] and [MATH] . Then, |
[EQUATION] We notice that we have [MATH] in the second inequality. From a union bound we see that with probability less than [MATH] , the best offspring created with rate [MATH] is at least as good as the expected fitness of an individual created with rate [MATH] |
We estimate the probability that the best offspring using rate [MATH] has a fitness distance at most [MATH] . Let [MATH] be an offspring obtained from [MATH] via standard bit mutation with mutation rate [MATH] . Let [MATH] and [MATH] be the number of one-bits flipped and zero-bits flipped, respectively. |
[EQUATION] Both [MATH] and [MATH] are binomially distributed and [MATH] . We aim at a lower bound for [MATH] . We have [MATH] , since the median of [MATH] is between [MATH] and [MATH] by |
It remains to bound [MATH] . We notice that [MATH] and [MATH] Applying Theorem 12 in to binomial random variable [MATH] , we obtain |
[MATH] Therefore [EQUATION] For [MATH] offspring using rate [MATH] , the probability that the best one has a fitness distance at most [MATH] is more than [MATH] . Therefore with probability at least [MATH] , all best offspring are from the [MATH] -subpopulation. This proves the second statement of the lemma. |
For proof of part (c) , let [MATH] . An offspring created with mutation rate [MATH] is at least as good as its parent if and only if [MATH] . By using Bernstein’s inequality (Theorem |
(a) ) with [MATH] and [MATH] , we have [EQUATION] Since [MATH] , the corresponding probabilities for rate [MATH] and [MATH] are at most [MATH] and [MATH] , respectively. By a union bound, with probability at most [MATH] , the best offspring is at least as good as its parent. This proves part (c) |
The lemma above shows that the rate [MATH] is subject to a constant drift towards the interval [MATH] . Unfortunately, we cannot show that we obtain a sufficient fitness progress for all [MATH] -values in this range. However, we can do so for a range smaller only by constant factors. This is what we do now (for large v... |
Let [MATH] be the current rate and let [MATH] . Let [MATH] denote the fitness gain of the best of [MATH] offspring generated with rate [MATH] from a parent [MATH] with fitness distance [MATH] and the parent itself. More precisely, let [MATH] , be independent offspring [MATH] by flipping each bit independently with prob... |
We next show that a region contained in [MATH] provides at least a logarithmic (in [MATH] ) drift on fitness. Lemma 10 Let [MATH] be sufficiently large, [MATH] and [MATH] . If [MATH] , then [MATH] |
Proof. We first notice that [MATH] and [MATH] for all [MATH] . We aim to prove [MATH] with [MATH] Let [MATH] and [MATH] be the number of one-bits flipped and zero-bits flipped, respectively, in an offspring using rate [MATH] |
[MATH] and [MATH] follow binomial distributions [MATH] and [MATH] , respectively. Let [MATH] . Then [MATH] and [MATH] . Furthermore, let |
[EQUATION] Applying Theorem 10 in , we obtain [MATH] . Similarly [MATH] . We prove that for [MATH] , we have [MATH] , and thus [MATH] . Since for any [MATH] we have |
[EQUATION] we obtain [MATH] , and thus [MATH] as well as [MATH] . We see that [EQUATION] where we used [MATH] Using a sharp version of Stirling’s approximation due to Robbins |
, we compute [EQUATION] We notice that [MATH] and [EQUATION] Thus [MATH] and [MATH] . By Lemma (c) we have [MATH] for [MATH] . Hence |
[EQUATION] where in the last inequality, using [MATH] , we estimate [EQUATION] Recalling [MATH] and [MATH] , we compute [EQUATION] |
Since [MATH] and [MATH] , we obtain [EQUATION] Using [MATH] , we bound [EQUATION] Finally [MATH] We now extend the lemma to the whole region of [MATH] . If [MATH] the situation becomes easier because [MATH] and every [MATH] in the smaller range [MATH] provides at least an expected fitness improvement that is logarithmi... |
Lemma 11 Let [MATH] be sufficiently large, [MATH] and [MATH] . If [MATH] with [MATH] defined as in Lemma , then [EQUATION] Proof. |
If [MATH] , then [MATH] implies [MATH] and the claim follows from Lemma 10 . Hence let us assume [MATH] in the remainder and compute [MATH] with [MATH] |
We consider the probability [MATH] of creating from a parent with distance [MATH] an offspring with fitness distance at most [MATH] via standard bit mutation with mutation rate [MATH] Let [MATH] . If [MATH] using [MATH] by Lemma (c) , we compute |
[EQUATION] Otherwise if [MATH] using [MATH] , we obtain [EQUATION] Hence, [MATH] . We notice that [MATH] , since [MATH] when [MATH] and [MATH] when [MATH] . Consequently |
[EQUATION] If we only consider generations that use a rate within the right region, we can bound the expected runtime to reach [MATH] by [MATH] since the drift on the fitness is of order [MATH] . The following theorem shows that the additional time spent with adjusting the rate towards the right region does not change ... |
Theorem 12 The (1+ [MATH] ) EA with self-adjusting mutation rate reaches a OneMax -value of [MATH] within an expected number of [MATH] iterations. This bound is valid regardless of the initial mutation rate. |
Proof. Let us denote by [MATH] the fitness distance of a search point [MATH] We first argue that it takes an expected number of at most [MATH] iterations to reach a fitness distance of less than [MATH] . To this end, consider a parent [MATH] with fitness distance [MATH] . Let [MATH] be the current rate and let [MATH] .... |
[EQUATION] Then the fitness improvement is [MATH] and both [MATH] and [MATH] follow binomial distributions. From elementary properties of the binomial distribution, see, e.g., |
, we have [MATH] and [MATH] , and these are independent events. Hence with constant probability, we have [EQUATION] Clearly, for the best offspring [MATH] out of the [MATH] offspring generated in this iteration, we have [MATH] . Consequently, in each iteration starting with a parent [MATH] with [MATH] , with constant p... |
[EQUATION] where we first exploited the symmetry of [MATH] and then the well-known estimate [MATH] valid for all random variables [MATH] , in particular, for [MATH] . By the law of total expectation, the expected time to reach a search point with [MATH] -value below [MATH] is [MATH] |
Without loss of generality we can now assume [MATH] for the initial state. Our intuition is that once we begin to use a rate [MATH] at some distance level [MATH] , we will have a considerable drift on the OneMax -value and the strong drift on the rate keeps [MATH] within or close to this interval. After we make progres... |
We consider the stochastic process [MATH] and the current OneMax -value [MATH] . According to Lemma (a) and (b) , there exists [MATH] such that |
[EQUATION] Let [MATH] be all the different OneMax -values taken by [MATH] until for the first time [MATH] By the additive drift theorem, it takes at most [MATH] iterations to have [MATH] , regardless of how we set the initial rate. Since [MATH] is non-increasing, [MATH] implies [MATH] . Thus, no readjustments are neces... |
[EQUATION] When computing the expected number of non-adjusting iterations until [MATH] , we choose a constant [MATH] large enough such that [MATH] holds for some positive constant [MATH] . When [MATH] and [MATH] for some [MATH] , then by Lemma , we obtain |
[EQUATION] Hence, we obtain that [MATH] happens with probability at least [MATH] We see that there are at most [MATH] steps between being in the range [MATH] and being in the smaller range [MATH] which is described in Lemma 11 . If [MATH] , it takes at most a constant number of iterations [MATH] in expectation to reach... |
[EQUATION] for all random rates at distance [MATH] . Using the variable drift theorem (Theorem ), we estimate the runtime as [EQUATION] |
We notice that the expected runtime for adjusting [MATH] is [MATH] . Therefore, the total runtime is [MATH] in expectation. Middle Region |
In this section we estimate the expected number of generations until the number of one-bits has decreased from [MATH] to [MATH] We first claim that the right region for [MATH] is [MATH] . Hence, the (1+ [MATH] ) EA is not very sensitive to the choice of [MATH] here. Intuitively, this is due to the fact that a total fit... |
We estimate the drift of the fitness in Lemma 13 and apply that result afterwards to estimate the number of generations to cross the region. |
Lemma 13 Let [MATH] [MATH] and [MATH] Then [EQUATION] Proof. We aim at computing [MATH] with [MATH] The probability that no zero-bit flips in a single offspring stemming from rate [MATH] |
is [MATH] We regard the number [MATH] of offspring of rate [MATH] that have no flipped zeros. The expectation [MATH] is at least [MATH] Applying Chernoff bounds (Theorem ), we observe that [MATH] exceeds [MATH] |
with probability at least [MATH] since [MATH] Assuming this to happen, we look at the first [MATH] offspring without flipped zeros. For [MATH] let [MATH] be the number of flipped ones in the [MATH] -th offspring. Then the [MATH] are i.i.d. with [MATH] Let [MATH] The expectation of [MATH] is analyzed in Gießen and Witt ... |
[EQUATION] Since [MATH] is not necessarily assumed constant here, we generalize the result by closely following the proof in Note that [MATH] if at least one of the offspring does not flip a zero-bit. Hence, |
[EQUATION] where the second inequality used the assumption [MATH] , the third one [MATH] for [MATH] and the fifth one the equivalent to the previous inequality [MATH] . Hence, Lemma 4, part 2 in |
holds also for arbitrary [MATH] Setting [MATH] , we now distinguish between two cases. If [MATH] , we obtain [MATH] otherwise we have [MATH] , hence [MATH] Thus, |
[EQUATION] Hence, using the law of total probability, we obtain the lower bound on the drift for the middle region. We now use our result on the drift to estimate the time spent in this region. We notice that [MATH] when [MATH] . This means we will often have |
[MATH] which provides the drift we need. Theorem 14 Let [MATH] and [MATH] . Assume [MATH] for the current OneMax -value of the self-adjusting (1+ [MATH] ) EA. Then the expected number of generations until [MATH] is [MATH] |
Proof. For [MATH] the upper bound from Lemma is [MATH] According to the lemma we have for [MATH] that [MATH] The additive drift theorem yields that in [MATH] time we have [MATH] We choose [MATH] large enough such that [MATH] |
holds for some positive constant [MATH] and note that [MATH] is constant. Applying Lemma we obtain [MATH] and [MATH] happens with probability at least [MATH] Once [MATH] |
it takes at most a constant number of iterations [MATH] in expectation to draw [MATH] to [MATH] or less. According to Lemma 13 this ensures a drift of at least [MATH] which implies an average drift of at least [MATH] |
over all random rates at distance [MATH] where [MATH] is a constant. Considering [MATH] the minimum is taken on the first argument if [MATH] and on the second if [MATH] |
We are interested in the expected time to reduce the OneMax -value to at most [MATH] To ease the application of drift analysis, we artificially modify the process and make it create the optimum when the state ( OneMax -value) is strictly less than [MATH] Clearly, the first hitting time of state at most [MATH] does not ... |
[MATH] the expected number of generations to reach state at most [MATH] is bounded from above by [EQUATION] which is [MATH] The overall expected number of generations spent is |
[MATH] since [MATH] by assumption. Near Region In the near region, we have [MATH] . Hence, the fitness is so low that we can expect only a constant number of offspring to flip at least one of the remaining one-bits. This assumes constant rate. However, higher rates are detrimental since they are more likely to destroy ... |
Lemma 15 Let [MATH] [MATH] and [MATH] Then the probability that [MATH] is at least [MATH] Proof. To prove the claim we exploit the fact that only few one-bits are flipped in both subpopulations. Using [MATH] , we shall argue as follows. With sufficiently high (constant) probability, (i) the [MATH] -subpopulation contai... |
Let [MATH] and [MATH] be the number of offspring that did not flip any zero-bits using rate [MATH] and [MATH] , respectively. Then [MATH] since [MATH] and |
[EQUATION] where we used that [MATH] and [MATH] i. e. [MATH] for some constant [MATH] Using [MATH] we get [MATH] In fact, we can discriminate [MATH] and [MATH] by using Theorem in the following way: we have |
[EQUATION] for sufficiently large [MATH] since [MATH] and [MATH] Similarly, we obtain [EQUATION] Note that [MATH] holds for all [MATH] and [MATH] Since the offspring are generated independently, the events [MATH] |
and [MATH] happen together with probability at least [MATH] Conditioning on this and by using a union bound the probability [MATH] that at least one of the [MATH] offspring that do not flip any zero-bits flips at least one one-bit can be upper bounded by |
[EQUATION] using [MATH] in the first and [MATH] and [MATH] in the last inequality. By using a union bound, we find the probability [MATH] |
that at least one [MATH] -offspring flips exactly one one-bit and exactly one zero-bit to be at most [EQUATION] for sufficiently large [MATH] using [MATH] for the first inequality. The third inequality is due to |
[MATH] for all [MATH] (see Lemma .a) and the last inequality stems from [MATH] The second inequality follows from [EQUATION] using again [MATH] for some constant [MATH] Let [MATH] be the number of such offspring. Any other fitness-decreasing flip-combinations of zeroes and ones in the [MATH] -subpopulation require an o... |
[EQUATION] using [MATH] and [MATH] and the fact that [MATH] is decreasing for [MATH] and [MATH] The events [MATH] [MATH] [MATH] and the event that no fitness-decreasing offspring is created in the [MATH] -subpopulation are sufficient to ensure that the best individual is either surely from the [MATH] -population or cho... |
[EQUATION] Hence, using a union bound for the error probabilities, the unconditional probability is at least [EQUATION] We note that the restriction [MATH] in the lemma above is not strictly necessary. Also for smaller [MATH] , the probability that the winning individual is chosen from the [MATH] -population is by an a... |
In the following proof of the analysis of the near region, we use the above lemma (with quite some additional arguments) to argue that the [MATH] -value quickly reaches [MATH] or less and from then on regularly returns to this region. This allows to argue that in the near region we have a speed-up of a factor of [MATH]... |
). Theorem 16 Assume [MATH] for the current OneMax -value of the self-adjusting (1+ [MATH] ) EA. Then the expected number of generations until the optimum is reached is [MATH] |
Proof. The aim is to estimate the OneMax -drift at the points in time (generations) [MATH] where [MATH] . To bound the expected number of generations until the mutation rate has entered this region, we basically consider the stochastic process |
[MATH] , where [MATH] which is the lower bound on [MATH] from Lemma 15 . However, as we do not have proved a drift of [MATH] towards smaller values in the region [MATH] (where [MATH] is an upper bound on [MATH] from Lemma 11 ), we use the potential function |
[EQUATION] assuming that [MATH] and [MATH] have been rounded down and up to the closest power of [MATH] , respectively. We note that |
[MATH] if [MATH] and [MATH] in all three cases. The potential function has a slope of [MATH] for [MATH] Lemma 15 gives us the drift [MATH] . The function satisfies |
[MATH] if [MATH] , which corresponds to the region where the probability of decreasing [MATH] by a factor of [MATH] has only be bounded from below by [MATH] due to the random steps. Still, [MATH] |
in this region due to the concavity of the potential function. Finally, [MATH] by Lemma 11 . Hence, altogether [MATH] for some constant [MATH] As [MATH] , additive drift analysis yields an expected number of [MATH] generations until for the first time [MATH] holds, corresponding to [MATH] . We denote this hitting time ... |
We now consider an arbitrary point of time [MATH] . The aim is to show a drift on the OneMax -value, depending on the current OneMax -value [MATH] , which satisfies |
[MATH] with probability [MATH] . To this end, we will use Lemma . We choose [MATH] large enough such that [MATH] holds for some positive constant [MATH] and note that [MATH] is constant. We consider two cases. If [MATH] , which happens with probability at least [MATH] according to the lemma, then the bound [MATH] impli... |
[MATH] . Hence, we have [MATH] in this case and obtain a probability of at least [EQUATION] to improve the OneMax -value by [MATH] using that [MATH] and pessimistically assuming a rate of [MATH] in all offspring. If [MATH] , we bound the improvement from below by [MATH] Using the law of total probability, we obtain |
[EQUATION] Now a multiplicative drift analysis with respect to the stochastic process on the [MATH] , more precisely Theorem using |
[MATH] and minimum state [MATH] , gives an expected number of [MATH] generations until the optimum is found. Together with the expected number [MATH] until the [MATH] -value becomes at most [MATH] , this proves the theorem. |
Putting Everything Together In this section, we put together the analyses of the different regimes to prove our main result. Proof of Theorem |
The lower bound actually holds for all unbiased parallel black-box algorithms, as shown in We add up the bounds on the expected number of generations spent in the three regimes, more precisely we add up the bounds from Theorem 12 , Theorem 14 and Theorem 16 , which gives us [MATH] generations. Due to our assumption [MA... |
Proof of Lemma We basically revisit the regions of different OneMax -values analyzed in this paper and bound the time spent in these regions under the assumption [MATH] . In the far region, Lemmas 10 |
and 11 , applied with this value of [MATH] imply a fitness drift of [MATH] per generation, so the expected number of generations spent in the far region is [MATH] as computed by variable drift analysis in the proof of Theorem 12 |
The middle region is shortened at the lower end. For [MATH] Lemma 13 gives a fitness drift of [MATH] , implying by additive drift analysis [MATH] generations to reduce the fitness to at most [MATH] |
In the near region, which now starts at [MATH] , we have to argue slightly differently. Note that every offspring has a probability of at least [MATH] |
of not flipping a zero-bit. Hence, we expect [MATH] such offspring. We pessimistically assume that the other individuals do not yield a fitness improvement; conceptually, this reduces the population size to [MATH] offspring, all of which are guaranteed not to flip a zero-bit. Adapting the arguments from the proof of Th... |
[EQUATION] which is a lower bound on the fitness drift. Using the multiplicative drift analysis, the expected number of generations in the near region is [MATH] . Putting the times for the regions together, we obtain the lemma. |
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