1. 1 On The Learning Power of Evolution Vitaly Feldman

2. 2 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

3. 3 Model Outline Complex behavior: multi-argument function Function representation Fitness estimation Random variation Natural selection Success criteria

4. 4 Representation Domain of conditions X and distribution D over X Representation class R of functions over X  Space of available behaviors  Efficiently evaluatable

5. 5 Fitness Optimal function f : X ! {-1,1} Performance: correlation with f relative to D Perf f (r,D) = E D [f ( x ) ¢r ( x ) ]

6. 6 Random Variation Mutation algorithm M :  Given r 2 R produces a random mutation of r  Efficient  Neigh M ( r ) is all possible outputs of M on r Hypothesis Mutation Algorithm

7. 7 Natural Selection If 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 Pr M If Bene( r ) = ; a mutation is chosen from Neut( r ) according to Pr M * Neigh M ( r ) and Perf f are estimated via poly- size sampling and t is inverse-polynomial Bene( r )={ r ’ 2 Neigh M ( r ) | Perf f ( r ’, D ) > Perf f ( r, D ) + t } Neut( r )={ r ’ 2 Neigh M ( r ) | |Perf f ( r ’, D ) - Perf f ( r, D )| · t } t is the tolerance Step( R, M, r )

8. 8 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 sequence r 0 = r, r 1, r 2,… where r i +1 Ã Step( R, M, r i ) w.h.p. it holds Perf f ( r g ( 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

9. 9 Evolvability of Conjunctions ANDs of Boolean variables and their negations over {-1,1} n  e.g. x 3 Æ ¬ x 5 Æx 8 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]

10. 10 What is Evolvable in This Model? EV µ PAC EV µ SQ ( PAC [L. Valiant 06]  Statistical Query learning [Kearns 93]: estimates of E D [( x, f ( x ))] for an efficiently evaluatable  EV µ CSQ [F 08] Learnability by correlational statistical queries CSQ: E D [( x ) ¢f ( x )] CSQ µ EV [F 08]  Fixed D : CSQ = SQ [Bshouty, F 01]

11. 11 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?

12. 12 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]

13. 13 Original Selection Rule If 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 Pr M If Bene( r ) = ; a mutation is chosen from Neut( r ) according to Pr M * Neigh M ( r ) and Perf f are estimated via poly- size sampling and t is inverse-polynomial Bene( r )={ r ’ 2 Neigh M ( r ) | Perf f ( r ’, D ) > Perf f ( r, D ) + t } Neut( r )={ r ’ 2 Neigh M ( r ) | |Perf f ( r ’, D ) - Perf f ( r, D )| · t } t is the tolerance Step( R, M, r )

14. 14 Other Selection Rules Sufficient condition: Selection rule can be “smooth” and need not be fixed in time 8 r 1, r 2 2 Neigh M ( r ) if Perf f ( r 1, D ) ¸ Perf f ( r 2, D ) + t then r 1 is “observably” favored to r 2

15. 15 Performance Function For real-valued representations other measures of performance can be used  e.g. expected quadratic loss L Q -Perf : 1-E D [( f ( x )- r ( x )) 2 ]/2  Decision lists are evolvable wrt uniform distribution with L Q -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

16. 16 What About New Algorithms? Conjunctions are evolvable distribution- independently with L Q -Perf [F 09]  Mutation algorithm: Add/subtract  ¢x i and project to X [-1,1] (for X = {-1,1} n )

17. 17 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

18. 18 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)