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Algorithmic Insights and the Theory of Evolution

Algorithmic Insights and the Theory of Evolution. Christos H. Papadimitriou UC Berkeley. The Algorithm as a Lens. It started with Alan Turing, 60 years ago Algorithmic thinking as a novel and productive way for understanding and transforming the Sciences

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Algorithmic Insights and the Theory of Evolution

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  1. Algorithmic Insightsand the Theory of Evolution Christos H. Papadimitriou UC Berkeley

  2. The Algorithm as a Lens • It started with Alan Turing, 60 years ago • Algorithmic thinking as a novel and productive way for understanding and transforming the Sciences • Mathematics, Statistical Physics, Quantum Physics, Economics and Social Sciences… • This talk: Evolution

  3. Evolution Before Darwin • Erasmus Darwin

  4. Before Darwin • J.-B. Lamarck

  5. Before Darwin • Charles Babbage [Paraphrased] “God created not species, but the Algorithm for creating species”

  6. Darwin, 1858 • Common Ancestry • Natural Selection

  7. The Origin of Species • Possibly the world’s most masterfully compelling scientific argument • The six editions 1859, 1860, 1861, 1866, 1869, 1872

  8. The Wallace-Darwin papers

  9. Brilliant argument, and yet many questions left unasked, e.g.: • How does novelty arise? • What is the role of sex?

  10. After Darwin • A. Weismann [Paraphrased] “The mapping from genotype to phenotype is one-way”

  11. Genetics • Gregor Mendel [1866] • Number of citations between 1866 and 1901: 3

  12. The crisis in Evolution1900 - 1920 • Mendelians vs. Darwinians • Geneticists vs. Biometricists/Gradualists

  13. The “Modern Synthesis”1920 - 1950 Fisher – Wright - Haldane

  14. Big questions remaine.g.: • How does novelty arise? • What is the role of sex?

  15. Evolution and Algorithmic Insights • “ How do you find a 3-billion long string in 3 billion years?” L. G.Valiant At the Wistar conference (1967), Schutzenberger asked virtually the same question

  16. Valiant’s Theory of Evolvability • Which traits can evolve? • Evolvability is a special case of statistical learning

  17. Evolution and CS Practice:Genetic Algorithms [ca. 1980s] • To solve an optimization problem… • …create a population of solutions/genotypes • …who procreate through sex/genotype recombination… • …with success proportional to their objective function value • Eventually, some very good solutions are bound to arise in the soup…

  18. And in this Corner…Simulated Annealing • Inspired by asexual reproduction • Mutations are adopted with probability increasing with fitness/objective differential • …(and decreasing with time)

  19. The Mystery of Sex Deepens • Simulated annealing (asexual reproduction) works fine • Genetic algorithms (sexual reproduction) don’t work • In Nature, the opposite happens: Sex is successful and ubiquitous

  20. ?

  21. A Radical Thought • What if sex is a mediocre optimizer of fitness (= expectation of offspring)? • What if sex optimizes something else? • And what if this something else is its raison d’ être?

  22. Mixability! • In [LPDF 2008] we establish through simulations that: • Natural selection under asex optimizes fitness • But under sex it optimizes mixability: • The ability of alleles (gene variants) to perform well with a broad spectrum of other alleles

  23. Explaining Mixability • Fitness landscape of a 2-gene organism Entries: fitness of the combination Rows: alleles of gene A Columns: alleles of gene B

  24. Explaining Mixability (cont) • Asex will select the largest numbers

  25. Explaining Mixability (cont) • But sex will select the rows and columns with the largest average

  26. Neutral Theory and Weak Selection • Kimura 1970: Evolution proceeds not by leaps upwards, but mostly “horizontally,” through statistical drift • Weak selection: the values in the fitness matrix are very close, say in [1 – ε, 1 + ε]

  27. Changing the subject:The experts problem • Every day you must choose one of n experts • The advice of expert i on day t results in a gain G[i, t] in [-1, 1] • Challenge: Do as well as the best expert in retrospect • Surprise: It can be done! • [Hannan 1958, Cover 1980, Winnow, Boosting, no-regret learning, MWUA, …]

  28. Multiplicative weights update • Initially, assign all experts same probability • At each step, increase the probablity of each by (1 + ε G[I, t]) (and then normalize) • Theorem: Does as well as the best expert • MWUA solves: zero-sum games, linear programming, convex programming, network congestion,…

  29. Disbelief Computer scientists find it hard to believe that such a crude technique solves all these sophisticated problems “The eye to this day gives me a cold shudder.” [cf: Valiant on three billion bits and years]

  30. Theorem [CLPV 2012]: Under weak selection, evolution is a game • the players are the genes • the strategies are the alleles • the common utility is the fitness of the organism (coordination game) • the probabilities are the allele frequencies • game is played through multiplicativeupdates

  31. There is more… • Recall the update (1 + ε G[i, t]) • ε is the selection strength • (1 + ε G[i, t]) is the allele’s mixability! • Variance preservation: multiplicative updates is known to enhance entropy • Two mysteries united… • Thisis the role of sex in Evolution

  32. Pointer Dogs

  33. Pointer Dogs C. H. Waddington

  34. Waddington’s Experiment (1952) Generation 1 Temp: 20o C

  35. Waddington’s Experiment (1952) Generation 2-4 Temp: 40o C ~15% changed Select and breed those

  36. Waddington’s Experiment (1952) Generation 5 Temp: 40o C ~60% changed Select and breed those

  37. Waddington’s Experiment (1952) Generation 6 Temp: 40o C ~63% changed Select and breed those

  38. Waddington’s Experiment (1952) (…) Generation 20 Temp: 40o C ~99% changed

  39. Surprise! Generation 20 Temp: 20o C ~25% stay changed!!

  40. Genetic Assimilation • Adaptations to the environment become genetic!

  41. Is There a Genetic Explanation? Function f ( x, h ) with these properties: • Initially, Prob x ~ p[0] [f ( x, h = 0)] ≈ 0% • Then Probp[0][f ( x, 1)] ≈ 15% • After breeding Probp[1][f ( x, 1)] ≈ 60% • Successive breedings, Probp[20][f ( x,1)] ≈ 99% • Finally, Probp[20][f ( x, 0)] ≈ 25%

  42. A Genetic Explanation • Suppose that “red head” is this Boolean function of 10 genes and “high temperature” “red head” = “x1 + x2 + … + x10 + 3h ≥ 10” • Suppose also that the genes are independent random variables, with pi initially half, say.

  43. A Genetic Explanation (cont.) • In the beginning, no fly is red (the probability of being red is 2-n) • With the help of h = 1, a few become red • If you select them and breed them, ~60% will be red!

  44. Why 60%?

  45. A Genetic Explanation (cont.) • Eventually, the population will be very biased towards xi = 1 (the pi’s are close to 1) • And so, a few flies will have all xi = 1 for all i, and they will stay red when h becomes 0

  46. Generalize! • Let B is any Boolean function • n variables x1 x2 … xn (no h) • Independent, with probabilities p = (p1 p2 … pn) • Satisfiability game: if B is satisfied, each variable gets 1, otherwise 1 - ε • Repeated play by multiplicative weights

  47. Boolean functions (cont.) Conjecture: This solves SAT Can prove it for monotone functions (in poly time) Can almost prove it in general (Joint work with AdiLivnat, Greg Valiant, Andrew Won)

  48. Interpretation • If there is a Boolean combination of a modestly large number of genes that creates an unanticipated trait conferring even a small advantage, then this combination will be discovered and eventually fixed in the population. • “With sex, all moderate-sized Boolean functions are evolvable.”

  49. Sooooo… • The theory of life is deep and fascinating • Insights of an algorithmic nature can help make progress • Evolution is a coordination game between genes played via multiplicative updates • Novel viewpoint that helps understand the central role of sex in Evolution

  50. Thank You!

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