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
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
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
25. Explaining Mixability (cont)
• But sex will select the rows and columns
with the largest average
3 2 4 5 4
1 0 0 7 2
2 1 0 4 3
1 8 1 3 2
3 2 4 5 4
1 0 0 7 2
2 1 0 4 3
1 8 1 3 2
3 2 4 5 4
1 0 0 7 2
2 1 0 4 3
1 8 1 3 2
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 multiplicative updates
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…
• This is the role of sex in Evolution
41. Is There a Genetic Explanation?
Function f ( x, h ) with these properties:
•Initially, Probx~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!
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 x1x2 … 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 Adi Livnat, 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
