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On The Learning Power of Evolution

Vitaly Feldman

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Fundamental Question

How can complex and adaptive mechanisms result from evolution? Fundamental principle: random

variation guided by natural selection [Darwin, Wallace 1859]

There is no quantitative theory

TCS Established notions of complexity Computational learning theory

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Model Outline

Complex behavior: multi-argument function

Function representation Fitness estimation Random variation Natural selection Success criteria

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Representation

Domain of conditions X and distribution D over X

Representation class R of functions over X Space of available behaviors Efficiently evaluatable

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Fitness

Optimal function f: X ! {-1,1} Performance: correlation with f relative

to D

Perff(r,D) = ED[f(x)¢r(x)]

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Random Variation

Mutation algorithm M: Given r 2 R produces a

random mutation of r Efficient NeighM(r) is all possible

outputs of M on r

Hypothesis

Mutation Algorithm

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Natural SelectionIf beneficial mutations are available then output one of

them, otherwise output one of the neutral mutations

Natural selection

If Bene(r) ; a mutation is chosen from

Bene(r) according to PrM

If Bene(r) = ; a mutation is chosen from Neut(r) according to PrM

* NeighM(r) and Perff are estimated via poly-size

sampling and t is inverse-polynomial

Bene(r)={r’ 2 NeighM(r) | Perff(r’,D) > Perff(r,D) + t }

Neut(r)={r’ 2 NeighM(r) | |Perff(r’,D) - Perff(r,D)| · t }

t is the tolerance

Step(R,M,r)

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Evolvability Class of functions C is evolvable over D if exists an

evolutionary algorithm (R,M) and a polynomial g(¢,¢) s.t.

For every f2C, r2R, >0, for a sequencer0=r,r1,r2,… where ri+1 Ã Step(R,M,ri) w.h.p.

it holds Perff(rg(n,1/),D) ¸ 1-

Evolvable (distribution-independently) Evolvable for all D by the same R and M

C represents the complexity of structures that can evolve in a single phase of evolution driven by a single optimal function from C

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Evolvability of Conjunctions ANDs of Boolean variables and their

negations over {-1,1}n

e.g. x3Ƭx5Æx8

Evolutionary algorithm R is all conjunctions M adds or removes a variable or its negation

Does not work Works for monotone conjunctions over the

uniform distribution [L. Valiant 06]

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What is Evolvable in This Model?

EV µ PAC EV µ SQ ( PAC [L. Valiant 06]

Statistical Query learning [Kearns 93]: estimates of ED[(x,f(x))] for an efficiently evaluatable

EV µ CSQ [F 08]Learnability by correlational statistical queriesCSQ: ED[(x)¢f(x)]

CSQ µ EV [F 08] Fixed D: CSQ = SQ [Bshouty, F 01]

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Distribution-independent Evolvability Algorithms

Singletons [F 09] R is all conjunctions of a logarithmic number of functions from

a set of pairwise independent functions M chooses a random such conjunction

Lower bounds [F 08] C 2 EV => each function in C is expressible as a “low”

weight integer threshold function over a poly-sized basis B EV ( SQ

Linear threshold functions and decision lists are not evolvable (even weakly) [GHR 92, Sherstov 07, BVW 07]

Conjunctions? Low weight integer linear thresholds?

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Robustness of the Model

How is the set of evolvable function classes influenced by various aspects of the definition? Selection rule Mutation algorithm Fitness function …

The model is robust to a variety of modifications and the power is essentially determined by the performance function [F 09]

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Original Selection RuleIf beneficial mutations are available then output one of

them, otherwise output one of the neutral mutations

Natural selection

If Bene(r) ; a mutation is chosen from

Bene(r) according to PrM

If Bene(r) = ; a mutation is chosen from Neut(r) according to PrM

* NeighM(r) and Perff are estimated via poly-size

sampling and t is inverse-polynomial

Bene(r)={r’ 2 NeighM(r) | Perff(r’,D) > Perff(r,D) + t }

Neut(r)={r’ 2 NeighM(r) | |Perff(r’,D) - Perff(r,D)| · t }

t is the tolerance

Step(R,M,r)

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Other Selection Rules

Sufficient condition:

Selection rule can be “smooth” and need not be fixed in time

8 r1,r2 2 NeighM(r) if

Perff(r1,D) ¸ Perff(r2,D) + t

then r1 is “observably” favored to r2

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Performance Function

For real-valued representations other measures of performance can be used e.g. expected quadratic loss LQ-Perf :

1-ED[(f(x)-r(x))2]/2 Decision lists are evolvable wrt uniform distribution

with LQ-Perf [Michael 07]

The obtained model is equivalent to learning from the corresponding type of statistical queries CSQ if the loss function is linear SQ otherwise

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What About New Algorithms?

Conjunctions are evolvable distribution-independently with LQ-Perf [F 09] Mutation algorithm:Add/subtract ¢xi and project to X[-1,1]

(for X = {-1,1}n )

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Further Directions Limits of distribution-independent

evolvability “Natural” algorithms for “interesting”

function classes and distributions Evolvability without performance

decreases Applications

Direct connections to evolutionary biology

CS

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References

L. Valiant. Evolvability, ECCC 2006; JACM 2009 L. Michael. Evolvability via the Fourier Transform, 2007 V. F. Evolvability from Learning Algorithms, STOC 2008 V. F. and L. Valiant. The Learning Power of Evolution,

COLT 2008(open problems) V. F. Robustness of Evolvability, COLT 2009 (to appear)

V. F. A complete characterization of SQ learning with applications to evolvability, 2009 (to

appear)