Runtime Analysis of Non-elitist Populations: From Classical Optimisation to Partial Information

Abstract

Although widely applied in optimisation, relatively little has been proven rigorously about the role and behaviour of populations in randomised search processes. This paper presents a new method to prove upper bounds on the expected optimisation time of population-based randomised search heuristics that use non-elitist selection mechanisms and unary variation operators. Our results follow from a detailed drift analysis of the population dynamics in these heuristics. This analysis shows that the optimisation time depends on the relationship between the strength of the selective pressure and the degree of variation introduced by the variation operator. Given limited variation, a surprisingly weak selective pressure suffices to optimise many functions in expected polynomial time. We derive upper bounds on the expected optimisation time of non-elitist evolutionary algorithms (EA) using various selection mechanisms, including fitness proportionate selection. We show that EAs using fitness proportionate selection can optimise standard benchmark functions in expected polynomial time given a sufficiently low mutation rate. As a second contribution, we consider an optimisation scenario with partial information, where fitness values of solutions are only partially available. We prove that non-elitist EAs under a set of specific conditions can optimise benchmark functions in expected polynomial time, even when vanishingly little information about the fitness values of individual solutions or populations is available. To our knowledge, this is the first runtime analysis of randomised search heuristics under partial information.

Appendix

Lemma 27

(Chernoff’s Inequality, see [10]) If $X = \sum _{i=1}^{m} X_i$, where $X_i\in \{0,1\},i\in [m],$ are independent random variables, then $\Pr \left( X \le (1-\varepsilon ){\mathbf {E}}\left[ X\right] \right) \le \exp \left( -\varepsilon ^2 {\mathbf {E}}\left[ X\right] /2\right)$ for any $\varepsilon \in [0,1]$.

Lemma 28

(Chebyshev’s Inequality, see [29]) For any random variable X with finite expected value $\mu$ and finite non-zero variance $\sigma ^2$, it holds that $\Pr \left( |X - \mu | \ge d\sigma \right) \le 1/d^2$ for any $d > 0$.

Lemma 29

(Bernoulli’s inequality, see [28]) For any integer $n\ge 0$ and any real number $x\ge -1$, it holds that $(1+x)^n\ge 1+nx$.

Lemma 30

(Jensen’s inequality, see [28]) For any function f(x) convex in $[\alpha ,\beta ]$ and $a_i \in [\alpha ,\beta ]$ with $i \in [n]$,

$$\begin{aligned} f\left( \frac{1}{n}\sum _{i=1}^{n}a_i \right) \le \frac{1}{n}\sum _{i=1}^{n}f(a_i) \end{aligned}$$

Lemma 31

For $n \in {\mathbb {N}}$ and $x\ge 0$, we have $1 - (1 - x)^n \ge 1 - e^{-xn} \ge \frac{xn}{1 + xn}$.

Proof

From $e^x\ge x+1$, it follows that $1 - (1 - x)^n \ge 1 - e^{-nx} \ge 1 - (1+x)^{-n} = 1 - 1/\left( 1 + \sum _{k=1}^{n} {n \atopwithdelims ()k} x^k\right)$. Note that $x^{k} \ge 0$ for all $x\ge 0$, thus any partial sum of $\sum _{k=1}^{n} {n \atopwithdelims ()k} x^k$ provides a valid lower bound.

The result is obtained for the single term $k=1$. $\square$

Lemma 32

For all $x\in {\mathbb {R}}$, $e^{2x} \ge (x+1)e^x > x e^x$ and for $x>0$, $e^{2x}/x > e^x$.

Proof

Multiplying $e^x$ to $e^x - x \ge 1 > 0$ and adding $xe^{x}$ to both sides give the first result, then dividing them by $x>0$ provides the second one. $\square$

Lemma 33

For all $x\ge 0$, $x \ge \ln (1+x) \ge x(1-x/2)$.

Proof

For each inequality, it suffices to first show that the gap between the (supposedly) larger side and the (supposedly) smaller one is non-decreasing in x, e.g., by looking at the derivative, then show that the initial gap is 0 at $x=0$. $\square$

Lemma 34

Let X and Y be identically distributed independent random variables with integer support, finite expected value $\mu$ and finite non-zero variance $\sigma ^2$, it holds that

$$\begin{aligned} \Pr (X = Y) \ge \frac{\left( 1-1/d^2\right) ^2}{ 2 d \sigma + 1} \text { for any } d\ge 1 \end{aligned}$$

Proof

For any $k, \ell \in {\mathbb {R}}$ with $\lceil k \rceil \le \lfloor \ell \rfloor$, we have

$$\begin{aligned} \Pr \left( X = Y\right)&= \sum _{i \in {\mathbb {Z}}} \Pr (X = i)^2 \ge \sum _{i=\lceil k \rceil }^{\lfloor \ell \rfloor } \Pr (X = i)^2 \\&\ge \frac{\left( \sum _{i=\lceil k \rceil }^{\lfloor \ell \rfloor } \Pr (X = i) \right) ^2}{\lfloor \ell \rfloor - \lceil k \rceil + 1} \ge \frac{\Pr (k < X < \ell )^2}{\lfloor \ell \rfloor - \lceil k \rceil + 1} \end{aligned}$$

The second last equality is due to Lemma 30 with the convex function $f(x) = x^2$. It suffices to pick $k = \mu - d \sigma$ and $\ell = \mu + d \sigma$, so that by Lemma 28, we get

$$\begin{aligned} \Pr \left( X = Y\right)&\ge \frac{\Pr \left( |X - \mu | < d\sigma \right) ^2}{2d \lfloor \sigma \rfloor + 1} \\&= \frac{\left( 1 - \Pr \left( |X - \mu | \ge d\sigma \right) \right) ^2}{2d \lfloor \sigma \rfloor + 1} \ge \frac{\left( 1 - 1/d^2\right) ^2}{2d \sigma + 1}. \end{aligned}$$

$\square$