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arXiv:1604.00772v1 [cs.LG] 4 Apr 2016

The CMA Evolution Strategy: A Tutorial

Nikolaus Hansen

Inria

Research centre Saclay–

Ile-de-France

Universit´e Paris-Saclay, LRI

Contents

Nomenclature

0 Preliminaries 3

0.1 Eigendecomposition of a Positive Deﬁnite Matrix . . . . . . . . . . . . . . . 4

0.2 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . . 5

0.3 Randomized Black Box Optimization . . . . . . . . . . . . . . . . . . . . . 6

0.4 Hessian and Covariance Matrices . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Basic Equation: Sampling 8

2 Selection and Recombination: Moving the Mean 8

3 Adapting the Covariance Matrix 9

3.1 Estimating the Covariance Matrix From Scratch . . . . . . . . . . . . . . . . 10

3.2 Rank-µ-Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Rank-One-Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 A Different Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.2 Cumulation: Utilizing the Evolution Path . . . . . . . . . . . . . . . 15

3.4 Combining Rank-µ-Update and Cumulation . . . . . . . . . . . . . . . . . . 18

4 Step-Size Control 18

5 Discussion 22

A Algorithm Summary: The CMA-ES 28

B Implementational Concerns 32

B.1 Multivariate normal distribution . . . . . . . . . . . . . . . . . . . . . . . . 32

B.2 Strategy internal numerical effort . . . . . . . . . . . . . . . . . . . . . . . . 32

B.3 Termination criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.4 Flat ﬁtness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

B.5 Boundaries and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

C MATLAB Source Code 36

D Reformulation of Learning Parameter c

cov

Nomenclature

We adopt the usual vector notation, where bold letters, v, are column vectors, capital bold

letters, A, are matrices, and a transpose is denoted by v

. A list of used abbreviations and

symbols is given in alphabetical order.

Abbreviations

CMA Covariance Matrix Adaptation

EMNA Estimation of Multivariate Normal Algorithm

ES Evolution Strategy

(µ/µ

{I,W}

, λ)-ES, Evolution Strategy with µ parents, with recombination of all µ parents,

either Intermediate or Weighted, and λ offspring.

RHS Right Hand Side.

Greek symbols

λ ≥ 2, population size, sample size, number of offspring, see (

5).

µ ≤ λ parent number, number of (positively) selected search points in the population, num-

ber of strictly positive recombination weights, see (6).

eﬀ



i=1



−1

, the variance effective selection mass for the mean, see (

8).

i=1

, sum of all weights, note that w

≤ 0 for i > µ, see also (

30) and (53).

i=1

= 1, sum of all positive weights.

−

= −(

−

) = −

i=µ+1

≥ 0, minus the sum of all negative

weights.

(g)

∈ R

, step-size.

Latin symbols

B ∈ R

, an orthogonal matrix. Columns of B are eigenvectors of C with unit length and

correspond to the diagonal elements of D.

(g)

∈ R

n×n

, covariance matrix at generation g.

, diagonal elements of C.

≤ 1, learning rate for cumulation for the rank-one update of the covariance matrix, see

(

24) and (45), and Table 1.

≤ 1 − c

, learning rate for the rank-one update of the covariance matrix update, see (

28),

(30), and (47), and Table 1.

≤ 1 − c

, learning rate for the rank-µ update of the covariance matrix update, see (

16),

(

30), and (47), and Table 1.

< 1, learning rate for the cumulation for the step-size control, see (

31) and (43), and

Table

D ∈ R

, a diagonal matrix. The diagonal elements of D are square roots of eigenvalues of

C and correspond to the respective columns of B.

> 0, diagonal elements of diagonal matrix D, d

are eigenvalues of C.

≈ 1, damping parameter for step-size update, see (32), (37), and (44).

E Expectation value

f : R

→ R, x 7→ f (x), objective function (ﬁtness function) to be minimized.

sphere

: R

→ R, x 7→ kxk

= x

x =

i=1

g ∈ N

, generation counter, iteration number.

I ∈ R

n×n

, Identity matrix, unity matrix.

(g)

∈ R

, mean value of the search distribution at generation g.

n ∈ N, search space dimension, see f.

N(0, I), multivariate normal distribution with zero mean and unity covariance matrix. A

vector distributed according to N(0, I) has independent, (0, 1)-normally distributed

components.

N(m, C) ∼ m + N(0, C), multivariate normal distribution with mean m ∈ R

and

covariance matrix C ∈ R

n×n

. The matrix C is symmetric and positive deﬁnite.

, strictly positive real numbers.

p ∈ R

, evolution path, a sequence of successive (normalized) steps, the strategy takes over

a number of generations.

, where i = 1, . . . , λ, recombination weights, see (

6) and (16) and (49)–(53).

(g+1)

∈ R

, k-th offspring/individual from generation g + 1. We also refer to x

(g+1)

, as

search point, or object parameters/variables, commonly used synonyms are candidate

solution, or design variables.

(g+1)

i:λ

, i-th best individual out of x

(g+1)

, . . . , x

(g+1)

, see (

5). The index i : λ denotes the

index of the i-th ranked individual and f(x

(g+1)

1:λ

) ≤ f (x

(g+1)

2:λ

) ≤ ··· ≤ f(x

(g+1)

λ:λ

where f is the objective function to be minimized.

(g+1)

= (x

(g+1)

− m

(g)

)/σ

(g)

corresponding to x

= m + σy

0 Preliminaries

This tutorial introduces the CMA Evolution Strategy (ES), where CMA stands for Covariance

Matrix Adaptation.

The CMA-ES is a stochastic, or randomized, method for real-parameter

(continuous domain) optimization of non-linear, non-convex functions (see also Section

0.3

Parts of this material have also been presented in [

11] and [13], in the context of Estimation of Distribution

Algorithms and Adaptive Encoding, respectively. An introduction deriving CMA-ES from the information-geometric

concept of a natural gradient can be found in [

15].

below).

We try to motivate and derive the algorithm from intuitive concepts and from re-

quirements of non-linear, non-convex search in continuous domain. For a concise algorithm

description see Appendix

A. A respective Matlab source code is given in Appendix C.

Before we start to introduce the algorithm in Sect. 1, a few required fundamentals are

summed up.

0.1 Eigendecomposition of a Positive Deﬁnite Matrix

A symmetric, positive deﬁnite matrix, C ∈ R

n×n

, is characterized in that for all x ∈ R

\{0}

holds x

Cx > 0. The matrix C has an orthonormal basis of eigenvectors, B = [b

, . . . , b

with corresponding eigenvalues, d

, . . . , d

> 0.

That means for each b

holds

= d

. (1)

The important message from (

1) is that eigenvectors are not rotated by C. This feature

uniquely distinguishes eigenvectors. Because we assume the orthogonal eigenvectors to be

of unit length, b

= δ



1 if i = j

0 otherwise

, and B

B = I (obviously this means

−1

= B

, and it follows BB

= I). An basis of eigenvectors is practical, because

for any v ∈ R

we can ﬁnd coefﬁcients α

, such that v =

, and then we have

Cv =

The eigendecomposition of C obeys

C = BD

, (2)

where

B is an orthogonal matrix, B

B = BB

= I. Columns of B form an orthonormal basis

of eigenvectors.

= DD = diag(d

, . . . , d

)

= dia g(d

, . . . , d

) is a diagonal matrix with eigenvalues

of C as diagonal elements.

D = dia g(d

, . . . , d

) is a diagonal matrix with square roots of eigenvaluesof C as diagonal

elements.

The matrix decomposition (

2) is unique, apart from signs of columns of B and permutations

of columns in B and D

respectively, given all eigenvalues are different.

Given the eigendecomposition (2), the inverse C

−1

can be computed via

−1





−1

= B

−1

−2

−1

= B D

−2

= B diag



, . . . ,



While CMA variants for multi-objective optimization and elitistic variants have been proposed, this tutorial is

solely dedicated to single objective optimization and non-elitistic truncation selection, also referred to as comma-

selection.

Given m eigenvalues are equal, any orthonormal basis of their m-dimensional subspace can be used as column

vectors. For m > 1 there are inﬁnitely many such bases.



0, σ





0, D



N(0, C)

Figure 1: Ellipsoids depicting one-σ lines of equal density of six different normal distributions,

where σ ∈ R

, D is a diagonal matrix, and C is a positive deﬁnite full covariance matrix.

Thin lines depict possible objective function contour lines

From (

2) we naturally deﬁne the square root of C as

= BDB

(3)

and therefore

−

= BD

−1

= B diag



, . . . ,



0.2 The Multivariate Normal Distribution

A multivariate normal distribution, N(m, C), has a unimodal, “bell-shaped” density, where

the top of the bell (the modal value) corresponds to the distribution mean, m. The distribution

N(m, C) is uniquely determined by its mean m ∈ R

and its symmetric and positive deﬁnite

covariance matrix C ∈ R

n×n

. Covariance (positive deﬁnite) matrices have an appealing

geometrical interpretation: they can be uniquely identiﬁed with the (hyper-)ellipsoid {x ∈

−1

x = 1}, as shown in Fig.

1. The ellipsoid is a surface of equal density of the

distribution. The principal axes of the ellipsoid correspond to the eigenvectors of C, the

squared axes lengths correspond to the eigenvalues. The eigendecomposition is denoted by

C = B (D)

(see Sect.

0.1). If D = σI, where σ ∈ R

and I denotes the identity

matrix, C = σ

I and the ellipsoid is isotropic (Fig. 1, left). If B = I, then C = D

is a

diagonal matrix and the ellipsoid is axis parallel oriented (middle). In the coordinate system

given by the columns of B, the distribution N(0, C) is always uncorrelated.

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