Prabhas Chongstitvatana

1

Amplification of stochastic advantage

Biased : known with certainty one of the possible answer is always correct. Error can be reduced by repeat the algorithm.

Unbiased example coin flip

for p-correct “advantage” is p - 1/2

Prabhas Chongstitvatana

2

1 2 3 prob MC3R R R 27/64 RR R W 9/64 RR W R 9/64 RR W W 3/64 WW R R 9/64 RW R W 3/64 WW W R 3/64 WW W W 1/64 W

Let MC be a 3/4-correct unbiased

What is the error prob. Of MC3 (mojority vote) ?

MC3 is 27/32-correct

( > 84% )

Prabhas Chongstitvatana

3

What is the error prob. of MC with advantage e > 0 in majority vote k times ?

Let Xi = 1 if correct answer, 0 otherwise

Pr[ Xi = 1 ] >= 1/2 + e

assume for simplicity Pr[ Xi = 1 ] = 1/2 + e ; k is odd (no tie)

E( Xi ) = 1/2 + e;

Var(Xi ) = (1/2 + e) (1/2-e) = 1/4 - e2

Prabhas Chongstitvatana

4

X is a random variable corresponds to the number of correct answer is k trials.

Let

k

iiXX

0

E(X) = (1/2+e)k Var(X) = (1/4 - e2) k

iki

eei

kiX

2

1

2

1]Pr[

Prabhas Chongstitvatana

5

is normal distributed if k >= 30

error prob. Pr [ X <= k/2 ] can be calculated

2

0

]Pr[k

i

iX

Prabhas Chongstitvatana

6

If need error < 5%

Pr [ X < E(X) - 1.645 sqrt Var(X) ] ~ 5%

(from the table of normal distribution)

Pr [ X <= k/2 ] < 5% if

k/2 < E(X) - 1.645 sqrt Var(X)

k > 2.706 ( 1/(4e2) - 1 )

Prabhas Chongstitvatana

7

Example e = 5%, which is 55%-correct unbiased Monte Carlo,

k > 2.706 ( 1/(4e2) - 1 )

k > 267.894,

majority vote repeat 269 times to obtain 95%-correct.

Repetition turn 5% advantage into 5% error prob

Prabhas Chongstitvatana

8

It takes a large number of run for unbiased

Compare to biased

Run 55%-correct bias MC 4 times reduces the error prob. To

0.454 ~ 0.041 ( 4.1% )

Prabhas Chongstitvatana

9

Not much more expensive to obtain more confidence.

If we want 0.5% confidence (10 times more)

Pr[X < E(X) - 2.576 sqrt Var(X) ] ~ 5%

k > 6.636 (1/(4e2) - 1 )

This makes it 99.5%-correct with less than 2.5 times more expensive than 95%-correct.

Prabhas Chongstitvatana

10

To reduce error prob < del for an unbiased Monte Carlo with advantage e;

the number of repetition is proportional to 1/e2, also to log 1/del