On Non-elitist Evolutionary Algorithms Optimizing Fitness Functions with a Plateau

Abstract

We consider the expected runtime of non-elitist evolutionary algorithms (EAs), when they are applied to a family of fitness functions $\text {Plateau} _r$ with a plateau of second-best fitness in a Hamming ball of radius r around a unique global optimum. On one hand, using the level-based theorems, we obtain polynomial upper bounds on the expected runtime for some modes of non-elitist EA based on unbiased mutation and the bitwise mutation in particular. On the other hand, we show that the EA with fitness proportionate selection is inefficient if the bitwise mutation is used with the standard settings of mutation probability.

Appendices

Appendix A

This appendix contains the formulations of results employed from other works. Some of the formulations are given with slight modifications, which do not require a special proof.

Our lower bound is based on the negative drift theorem for populations [14].

Theorem 6

Consider the EA on $\mathcal {X} = \{0,1\}^n$ with bitwise mutation rate $\chi /n$ and population size $\lambda = {{\,\mathrm{poly}\,}}(n)$, let a(n) and b(n) be positive integers such that $b(n)\le n/\chi$ and $d(n) = b(n) - a(n) = \omega (\ln n)$. Given $x^* \in \{0,1\}^n$, define $T(n) := \min \{t \mid |P_t \cap \{x \in \mathcal {X} \mid H(x,x^*) \le a(n)\}| > 0\}$. If there exist constants $\alpha >1$, $\delta >0$ such that

• (1) $\forall t\ge 0$, $\forall i \in [\lambda ]:$ if $a(n)< H(P_t(i),x^*) < b(n)$ then $\alpha _t(i) \le \alpha$,

• (2) $\displaystyle \psi := \ln (\alpha )/\chi + \delta < 1$,

• (3) $\displaystyle b(n)/n < \min \left\{ 1/5, 1/2 - \sqrt{\psi (2-\psi )/4}\right\}$,

then $\Pr \left( T(n)\le e^{cd(n)}\right) = e^{-\varOmega (d(n))}$ for some constant $c>0$.

We also use a corollary of this theorem (Corollary 1 from [14]):

Corollary 1

The probability that a non-elitist EA with population size $\lambda ={{\,\mathrm{poly}\,}}(n),$ bitwise mutation probability $\chi /n,$ and maximal reproductive rate bounded by $\alpha < e^{\chi }-\delta$, for a constant $\delta > 0,$ optimises any function with a polynomial number of optima within $e^{cn}$ generations is $e^{-\varOmega (n)},$ for some constant $c > 0.$

To bound the expected optimisation time of Algorithm 1 from above, we use the level-based analysis [2]. The following theorem is a re-formulation of Corollary 7 from [2], tailored to the case of no recombination.

Theorem 7

Given a partition $(A_1,\dots ,$ $A_{m})$ of $\mathcal {X}$, if there exist $s_1,\dots ,s_{m-1}, p_0,$ ${\delta \in (0,1]}$, $\gamma _0 \in (0,1)$ such that

• (M1) $\forall P\in \mathcal {X}^\lambda , \forall j\in [m-1] :$ $\displaystyle p_\mathrm {mut} \left( y\in A_{\ge j+1} \mid x \in A_j \right) \ge s_j,$

• (M2) $\forall P\in \mathcal {X}^\lambda , \forall j\in [m-1] :$ $\displaystyle p_\mathrm {mut} \left( y\in A_{\ge j} \mid x\in A_j \right) \ge p_0,$

• (M3) $\forall P \in \left( \mathcal {X}\setminus A_{m}\right) ^\lambda , \forall \gamma \in (0,\gamma _0]:$ $\displaystyle \beta (\gamma , P) \ge (1+\delta )\gamma /p_0,$

• (M4) population size $\displaystyle \lambda \ge \frac{4}{\gamma _0\delta ^2} \ln \left( \frac{128 m}{\gamma _0s_*\delta ^2}\right) , \text { where } s_*:=\min _{j\in [m-1]} \{s_j\},$

then, assuming $T_0:=\min \{t \mid |P_t \cap A_{m}| \ge \gamma _0 \lambda \},$ we have

$$\begin{aligned} \mathbf {E}\left[ T_0\right] < \left( \frac{8}{\delta ^{2}}\right) \sum _{j=1}^{m-1}\left( \ln \left( \frac{6\delta \lambda }{4+\gamma _0 s_j\delta \lambda }\right) +\frac{1}{\gamma _0 s_j \lambda }\right) . \end{aligned}$$

(3)

Note that literally the formulation of Corollary 2 in [2] gives the bound (3) only for the expected runtime, but it is easy to see from the proof therein that the bound actually holds for the expected number $T_0$ of the first population that contains at least $\gamma _0 \lambda$ individuals in level $A_{m}$ as we put it in Theorem 7. This slight improvement is important in Sect. 3.

As an alternative to Theorem 7 we use the new level-based theorem based on the multiplicative up-drift [7]. Theorem 3.2 from [7] implies the following.

Theorem 8

Given a partition $(A_1,\dots ,$ $A_{m})$ of $\mathcal {X}$, define $T:=\min \{t\lambda \mid |P_t \cap A_{m}| > 0\}.$ If there exist $s_1,\dots ,s_{m-1},p_0,\delta \in (0,1]$, $\gamma _0 \in (0,1)$, such that conditions (M1)–(M3) of Theorem 7 hold and

• (M4’) for some constant $C>0$, the population size $\lambda$ satisfies

  $$ \lambda \ge \frac{8}{\gamma _0\delta ^2} \log \left( \frac{C m}{\delta } \left( \log \lambda +\frac{1}{\gamma _0 s_* \lambda }\right) \right) , \text { where } s_*:=\min _{j\in [m-1]} \{s_j\}, $$

then $\mathbf {E}\left[ T\right] =\mathcal {O} \left( \frac{\lambda m \log (\gamma _0 \lambda )}{\delta } + \frac{1}{\delta } \sum _{j=1}^{m-1}\frac{1}{\gamma _0 s_j}\right) . \$

Theorem 8 improves on Theorem 7 in terms of dependence of the runtime bound denominator on $\delta$, but only gives an asymptotical bound. Its proof is analogous to that of Theorem 7 and may be found in [3].

Theorem 9

Given an f-based partition $A_1,\ldots ,A_{m}$ of ${\mathcal X}$, if the EA uses the mutation, such that ${\Pr }(\mathtt{mutate} (x)\in A_{\ge j+1})\ge s_*$ for any $x\in A_{j},$ $j\in [m-1]$

• and a k-tournament selection, $k\ge \frac{(1+\ln m)e}{s_*}$ with a population of size $\lambda \ge k$,

• or $(\mu ,\lambda )$-selection and $\lambda \ge \frac{\mu (1+\ln m)}{s_* }$

then an element from $A_{m}$ is found in expectation after at most em genetations.

The proof is analogous to that of the main result in [9].

Appendix B

This appendix contains the proofs provided for the sake of completeness.

Proof of Lemma 1. For the initial population, it follows by a Chernoff bound that $\Pr \left( T=1\right) =e^{-\varOmega (n)}$. We then claim that for all $t\ge 0$, $\Pr \left( T=t+1\mid T>t\right) \le e^{-c'n}$ for a constant $c'>0,$ which by the union bound implies that $\Pr \left( T<e^{cn}\right) \le e^{cn-c'n}=e^{-\varOmega (n)}$ for any constant $c<c'$.

In the initial population, the expected number of ones of a k-th individual, $k\in [\lambda ]$ is $|P_0(k))|\le n/2.$ It will be more convenient here to consider the number of zeros, rather than the number of ones. We denote $Z_t^{(j)}:= n - |P_t(j)|$ , for $t\ge 0,$ $j\in [\lambda ],$ and $Z_t:=\lambda n - \sum _{j=1}^{\lambda } |P_t(j)|$. Let $p_j$ be the probability of selecting the j-th individual when producing the population in generation $t+1$. For f-monotone selection mechanisms, it holds that $\sum _{j=1}^\lambda p_j Z_t^{(j)} \le Z_t/\lambda .$

Let $P=(x_1,\dots ,x_{\lambda })$ be any deterministic population. Denote the i-th bit of its k-th individual by $x^{(k,i)},$ $z_k:= n - |x^{(k)}|$, $1\le k \le \lambda$, and $Z(P):=\sum _{k=1}^{\lambda } z_k$.

Let us consider the bitwise mutation first. The expected number of zero-bits in any offspring $j\in [\lambda ]$ produced from population $P_t=P$ is

$$ \mathbf {E}\left[ |P_{t+1}(j)|\ | \ P_t=P\right] = \sum _{k=1}^{\lambda } p_k\left[ \sum _{i=1}^n \left( x^{(k,i)} (1-\chi /n) + (1-x^{(k,i)}) \chi /n \right) \right] , $$

so the expected value of $Z_{t+1}^{(j)}$ for any offspring $j\in [\lambda ]$ is

$$\begin{aligned}&\mathbf {E}\left[ Z_{t+1}^{(j)} \ | \ P_t=P\right] \le n- \mathbf {E}\left[ |P_{t+1}(j)|\ | \ P_t=P\right] \\&= \sum _{k=1}^{\lambda } p_k\left[ \sum _{i=1}^n \left( x^{(k,i)} (1-\chi /n) + (1-x^{(k,i)}) \chi /n \right) \right] \\&=n-\sum _{k=1}^{\lambda } p_k \left[ \chi + (1-2\chi /n) \sum _{i=1}^n x^{(k,i)} \right] \\&\le n-\chi - (1-2\chi /n) \sum _{k=1}^{\lambda } p_k (n- z_k)\\&=\chi + (1-2\chi /n) \sum _{k=1}^{\lambda } p_k z_k \le \chi + (1-2\chi /n)Z(P)/\lambda . \end{aligned}$$

If $T>t$ and $Z(P)<\lambda n(1+\varepsilon )/2$, then

$$\begin{aligned} \mathbf {E}\left[ Z_{t+1}\mid P_t=P\right]&\le \lambda \chi + Z(P)\left( 1-2\chi /n\right) \\&< \lambda \chi + \frac{\lambda n}{2}(1+\varepsilon )\left( 1-2\chi /n\right) = \frac{\lambda n}{2}(1+\varepsilon )-\varepsilon \lambda \chi . \end{aligned}$$

Now $Z_{t+1}^{(1)}, Z_{t+1}^{(2)}, \dots , Z_{t+1}^{(\lambda )}$ are non-negative independent random variables, each bounded from above by n, so using the Hoeffding’s inequality [12] we obtain

$$ \Pr \left( Z_{t+1} \ge \frac{\lambda n}{2}(1+\varepsilon )\right) \le \Pr \left( Z_{t+1} \ge \mathbf {E}\left[ Z_{t+1}\right] + \varepsilon \lambda \chi \right) \le \exp \left( - \frac{2(\varepsilon \lambda \chi )^2}{\lambda n^2}\right) , $$

which is $e^{-\varOmega (n^{\delta })}$ since $\lambda \ge n^{2+\delta }.$    $\square$