A Bayesian Approach to Stochastic Root-Finding - CiteSeerX

A Bayesian Approach to Stochastic Root-Finding Rolf Waeber Peter I. Frazier Shane G. Henderson Operations Research & Information Engineering, Cornell University

Monday November 14, 2011 INFORMS 2011 Annual Meeting, Charlotte, NC This research is supported by AFOSR YIP FA9550-11-1-0083.

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Problem X* g(x) 0

0

1

• Consider a function g : [0, 1] → R. • Assumption: There exists a unique X ∗ ∈ [0, 1] such that - g(x) > 0 for x < X ∗ , - g(x) < 0 for x > X ∗ .

Goal: Find X ∗ ∈ [0, 1].

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

1/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Problem X* g(x) 0

0

1

• Consider a function g : [0, 1] → R. • Assumption: There exists a unique X ∗ ∈ [0, 1] such that - g(x) > 0 for x < X ∗ , - g(x) < 0 for x > X ∗ .

Goal: Find X ∗ ∈ [0, 1]. • Can only observe Yn (Xn ) = g(Xn ) + εn (Xn ), where εn (Xn ) is an independent noise with zero mean (median).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

1/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Problem X* Yn(Xn) 0

0

1

• Consider a function g : [0, 1] → R. • Assumption: There exists a unique X ∗ ∈ [0, 1] such that - g(x) > 0 for x < X ∗ , - g(x) < 0 for x > X ∗ .

Goal: Find X ∗ ∈ [0, 1]. • Can only observe Yn (Xn ) = g(Xn ) + εn (Xn ), where εn (Xn ) is an independent noise with zero mean (median).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

1/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Problem X* Yn(Xn) 0

0

1

• Consider a function g : [0, 1] → R. • Assumption: There exists a unique X ∗ ∈ [0, 1] such that - g(x) > 0 for x < X ∗ , - g(x) < 0 for x > X ∗ .

Goal: Find X ∗ ∈ [0, 1]. • Can only observe Yn (Xn ) = g(Xn ) + εn (Xn ), where εn (Xn ) is an independent noise with zero mean (median). Decisions: - Where to place samples Xn for n = 0, 1, 2, . . . - How to estimate X ∗ after n iterations. Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

1/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Applications

• Simulation optimization: - g(x) as a gradient

• Sequential statistics:

• Finance: - Pricing American options - Estimating risk measures

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

2/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Approximation (Robbins and Monro, 1951)

X* g(x) 0

0

1

1

Choose an initial estimate X0 ∈ [0, 1];

2

Determine a tuning sequence (an )n ≥ 0, P∞ n=0 an = ∞. (Example: an = d/n for d > 0.)

3

Xn+1 = Π[0,1] (Xn + an Yn (Xn )), where Π[0,1] is the projection to [0, 1].

Rolf Waeber

P∞

n=0

A Bayesian Approach to Stochastic Root-Finding

an2 < ∞, and

3/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

A “Good” Tuning Sequence a = 1/n, ε ~ N(0,0.5) n

n

1

Xn X*

0 0

200

400

600

800

1000

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

4/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

A “Bad” Tuning Sequence – too Large a = 10/n, ε ~ N(0,0.5) n

n

1

Xn X*

0 0

200

400

600

800

1000

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

4/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

A “Bad” Tuning Sequence – too Small a = 0.5/n, ε ~ N(0,0.5) n

n

1

Xn X*

0 0

200

400

600

800

1000

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

4/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Lack of Robustness – εn with Heavy-Tails a = 1/n, ε ~ t n

n

3

1

Xn X*

0 0

200

400

600

800

1000

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

4/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

A Different Approach

What about a bisection algorithm? X* g(x) 0

0

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

1

5/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

A Different Approach

What about a bisection algorithm? X* g(x) 0

0

1

• Deterministic bisection algorithm will fail almost surely. • Need to account for the noise.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

5/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

6/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm (Horstein, 1963) • Input: Zn (Xn ) := sign(Yn (Xn )). • Assume a prior density f0 on [0, 1]. n = 0, Xn = 0.5, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

1

A Bayesian Approach to Stochastic Root-Finding

7/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm (Horstein, 1963) • Input: Zn (Xn ) := sign(Yn (Xn )). • Assume a prior density f0 on [0, 1]. n = 0, Xn = 0.5, Zn(Xn) = −1

n = 1, Xn = 0.38462, Zn(Xn) = −1

X*

X*

fn(x)

fn(x)

0

Rolf Waeber

Xn

1

0

A Bayesian Approach to Stochastic Root-Finding

Xn

1

7/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm (Horstein, 1963) • Input: Zn (Xn ) := sign(Yn (Xn )). • Assume a prior density f0 on [0, 1]. n = 0, Xn = 0.5, Zn(Xn) = −1

n = 1, Xn = 0.38462, Zn(Xn) = −1

X*

X*

fn(x)

fn(x)

0

1

Xn

0

Xn

1

n = 2, X = 0.29586, Z (X ) = 1 n

n

n

X*

fn(x)

0

X

1

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

7/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm (Horstein, 1963) • Input: Zn (Xn ) := sign(Yn (Xn )). • Assume a prior density f0 on [0, 1]. n = 0, Xn = 0.5, Zn(Xn) = −1

n = 1, Xn = 0.38462, Zn(Xn) = −1

X*

X*

fn(x)

fn(x)

0

1

Xn

0

n = 2, X = 0.29586, Z (X ) = 1 n

n

n = 3, X = 0.36413, Z (X ) = −1

n

n

X*

n

n

X*

fn(x)

fn(x)

0

X

1

0

n

Rolf Waeber

1

Xn

A Bayesian Approach to Stochastic Root-Finding

X

1

n

7/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Revisited X* g(x) 0

0

1

( sign (g(Xn )) with probability p(Xn ), Zn (Xn ) = −sign (g(Xn )) with probability 1 − p(Xn ).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

8/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Revisited 1

X*

p(x)

g(x) 0

0

X*

0.5

1

0 0

1

( sign (g(Xn )) with probability p(Xn ), Zn (Xn ) = −sign (g(Xn )) with probability 1 − p(Xn ).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

8/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Revisited 1

X*

p(x)

g(x) 0

0

X*

0.5

1

0 0

1

( sign (g(Xn )) with probability p(Xn ), Zn (Xn ) = −sign (g(Xn )) with probability 1 − p(Xn ). • The probability of a correct sign p(·) depends on g(·) and the noise (εn )n .

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

8/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Revisited 1

X*

p(x)

g(x) 0

X*

p

0.5

0

1

0 0

1

( sign (g(Xn )) with probability p(Xn ), Zn (Xn ) = −sign (g(Xn )) with probability 1 − p(Xn ). • The probability of a correct sign p(·) depends on g(·) and the noise (εn )n . • Stylized Setting:

- p(·) is constant.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

8/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stochastic Root-Finding Revisited 1

X*

p(x)

g(x) 0

X*

p

0.5

0

1

0 0

1

( sign (g(Xn )) with probability p(Xn ), Zn (Xn ) = −sign (g(Xn )) with probability 1 − p(Xn ). • The probability of a correct sign p(·) depends on g(·) and the noise (εn )n . • Stylized Setting:

- p(·) is constant. - p(·) is known. Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

8/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stylized Setting • g(x) is a step function, for example, in edge detection applications (Castro and Nowak, 2008).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

9/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Stylized Setting • g(x) is a step function, for example, in edge detection applications (Castro and Nowak, 2008). P • Sample sequentially at point Xn and use Sm (Xn ) = m i=1 Yn,i (Xn ) to construct an α-level test of power 1 (Siegmund, 1985): n o T = inf m : |Sm | ≥ [(m + 1)(log(m + 1) + 2 log(1/α))]1/2 . Then PXn =X ∗ {T < ∞} ≤ α, PXn 6=X ∗ {T < ∞} = 1, and PXn <X ∗ {ST (Xn ) > 0} ≥ 1 − α/2 = p, PXn >X ∗ {ST (Xn ) < 0} ≥ 1 − α/2 = p. 1 p(x)

X*

Sm(Xn)

p

0.5

0

Rolf Waeber

5

10 m

15

20

0 0

A Bayesian Approach to Stochastic Root-Finding

1 9/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm (Horstein, 1963) Notation: p(·) = p ∈ (1/2, 1] and q = 1 − p.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

10/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

The Probabilistic Bisection Algorithm (Horstein, 1963) Notation: p(·) = p ∈ (1/2, 1] and q = 1 − p. 1

Place a prior density f0 on the root X ∗ , f0 has domain [0, 1]. Example: U(0, 1).

2

For n=0,1,2, . . . (a) Measure at the median Xn := Fn−1 (1/2). (b) Update the posterior density:

3

if Zn (Xn ) = +1,

( 2p · fn (x), fn+1 (x) = 2q · fn (x),

if x > Xn , if x ≤ Xn ,

if Zn (Xn ) = −1,

fn+1 (x) =

( 2q · fn (x), 2p · fn (x),

if x > Xn , if x ≤ Xn .

Estimate after n iterations: Xn = Fn−1 (1/2).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

10/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 0, Xn = 0.5, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 1, Xn = 0.61538, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 2, Xn = 0.70414, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 3, Xn = 0.63587, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 4, Xn = 0.55589, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 5, Xn = 0.46446, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 6, Xn = 0.35727, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 7, Xn = 0.43972, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 8, Xn = 0.51955, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 9, Xn = 0.45436, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 10, Xn = 0.39721, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 11, Xn = 0.33187, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 12, Xn = 0.25529, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 13, Xn = 0.3142, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 14, Xn = 0.37124, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 15, Xn = 0.32466, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 16, Xn = 0.3632, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 17, Xn = 0.33085, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 18, Xn = 0.30024, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 19, Xn = 0.25814, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 20, Xn = 0.20046, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 21, Xn = 0.24672, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 22, Xn = 0.27985, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 23, Xn = 0.31321, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 24, Xn = 0.28617, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 25, Xn = 0.2615, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 26, Xn = 0.28243, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 27, Xn = 0.29702, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 28, Xn = 0.32015, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 29, Xn = 0.33658, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 30, Xn = 0.36118, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 31, Xn = 0.38925, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 32, Xn = 0.43944, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 33, Xn = 0.39633, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 34, Xn = 0.42932, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 35, Xn = 0.4563, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 36, Xn = 0.43531, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 37, Xn = 0.41192, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 38, Xn = 0.43177, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 39, Xn = 0.41699, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 40, Xn = 0.39722, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 41, Xn = 0.41399, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 42, Xn = 0.4015, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 43, Xn = 0.41222, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 44, Xn = 0.40403, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 45, Xn = 0.41052, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 46, Xn = 0.40553, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 47, Xn = 0.39812, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 48, Xn = 0.37339, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 49, Xn = 0.35957, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 50, Xn = 0.36904, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 60, Xn = 0.36286, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 70, Xn = 0.37119, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 80, Xn = 0.37225, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 90, Xn = 0.37336, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 100, Xn = 0.3752, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 110, Xn = 0.37371, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 120, Xn = 0.3728, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 130, Xn = 0.37269, Zn(Xn) = −1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 140, Xn = 0.37108, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Sample Path of Posterior Distributions

n = 150, Xn = 0.37261, Zn(Xn) = 1 X*

fn(x)

0

Rolf Waeber

Xn

A Bayesian Approach to Stochastic Root-Finding

1

11/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Comparison to Stochastic Approximation 0

|X* − Xn|10

−2

10

−4

10

−6

10

−8

10

−10

10

Rolf Waeber

0

Stochastic Approximation Probabilistic Bisection 50

100 n

A Bayesian Approach to Stochastic Root-Finding

150

200

12/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Literature Review: Probabilistic Bisection Algorithm

• First introduced in Horstein (1963). • Discretized version: Burnashev and Zigangirov (1974). • Feige et al. (1997), Karp and Kleinberg (2007), Ben-Or and Hassidim (2008), Nowak (2008), Nowak (2009). • Survey paper: Castro and Nowak (2008)

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

13/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Literature Review: Probabilistic Bisection Algorithm

• First introduced in Horstein (1963). • Discretized version: Burnashev and Zigangirov (1974). • Feige et al. (1997), Karp and Kleinberg (2007), Ben-Or and Hassidim (2008), Nowak (2008), Nowak (2009). • Survey paper: Castro and Nowak (2008) “The probabilistic bisection algorithm seems to work extremely well in practice, but it is hard to analyze and there are few theoretical guarantees for it, especially pertaining error rates of convergence.”

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

13/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Algorithm Analysis

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

14/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Optimality

• Bayes Setting: X ∗ ∼ f0 . • Entropy as a measure of uncertainty Z H(fn ) = − fn (x) log2 fn (x)dx.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

15/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Optimality

• Bayes Setting: X ∗ ∼ f0 . • Entropy as a measure of uncertainty Z H(fn ) = − fn (x) log2 fn (x)dx.

Theorem Measuring at the median is optimal in minimizing the expected posterior entropy E[H(fN )] for any N ∈ N.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

15/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Consistency

Theorem limn→∞ Fn (x) = 1(x ≥ X ∗ ) for all x 6= X ∗ almost surely. On a set of probability 1 the posterior distribution converges weakly to a point mass at X ∗ .

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

16/26

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Rate of Convergence

Theorem There exists a constant c = c(p) > 1 such that lim c n E[|X ∗ − Xn |] = 0.

n→∞

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

17/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Rate of Convergence

Theorem There exists a constant c = c(p) > 1 such that lim c n E[|X ∗ − Xn |] = 0.

n→∞

Proof: • c n E[|X ∗ − Xn |] = E [c n En [|X ∗ − Xn |]]. P

• (c n En [|X ∗ − Xn |])n −→ 0 as n → ∞. • (c n En [|X ∗ − Xn |])n is uniformly integrable. 

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

17/26

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps

0

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

1

18/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps

a

0

1

An

For a > 0, define An = Pn ([0, a)).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

18/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

aXn

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn a

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xna

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a Xn

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a Xn

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

aXn

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xna

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a Xn

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a

0 An

Rolf Waeber

Xn

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

a Xn

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xan

0 An

Rolf Waeber

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

0

Xn An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

1

1/2

Xn

0 An

Rolf Waeber

a

1

0

0

50

100 n

A Bayesian Approach to Stochastic Root-Finding

19/26

150

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont.

1

1/2

0 0

50

100

150

200

250

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

19/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont.

a Xn

0

1

0

X

Rolf Waeber

a

n

An

1

An

A Bayesian Approach to Stochastic Root-Finding

20/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont.

a Xn

0

1

0

X

a

n

An

1

An

Mn = ln(An ) ∧ ln(1 − An ) ≤ Vn for all n ∈ N, where Vn is a coupled random walk. n

Vn ln(1/2) Mn

M0

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

20/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Pn(|X*−Xn|>h) 1 0.8 fn(x)

C−n

0.6 0.4 0.2 h 0

h Xn

1

0 0

0.2

0.4

0.6

0.8

h

• There exists C = C(p) > 1 such that
P Pn (|X ∗ − Xn | > h) > C −n ≤ 4h−1 C −2n for all h > 0 and n ∈ N.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

21/26

1

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Pn(|X*−Xn|>h) 1 0.8 fn(x)

C−n

0.6 0.4 0.2 h 0

h Xn

1

0 0

0.2

0.4

0.6

0.8

h

• There exists C = C(p) > 1 such that
P Pn (|X ∗ − Xn | > h) > C −n ≤ 4h−1 C −2n for all h > 0 and n ∈ N. R1 • En [|X ∗ − Xn |] = 0 Pn (|X ∗ − Xn | > h)dh. • E[En [|X ∗ − Xn |]] = o(c −n ) for c ∈ (1, C).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

21/26

1

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Future Research and Conclusions

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

22/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Algorithms Based on Probabilistic Bisection • In practice, p(x) is unknown and varies with x. • Sample sequentially at point Xn and observe Sm (Xn ) =

Pm

i=1

Yn,i (Xn ).

Then an α-level test of power 1 can be constructed (Siegmund, 1985): n o T = inf m : |Sm | ≥ [(m + 1)(log(m + 1) + 2 log(1/α))]1/2 , then PXn =X ∗ {T < ∞} ≤ α, PXn 6=X ∗ {T < ∞} = 1, and PXn <X ∗ {ST (Xn ) > 0} ≥ 1 − α/2 = p, PXn >X ∗ {ST (Xn ) < 0} ≥ 1 − α/2 = p. 1 p(x)

X*

Sm(Xn)

p

0.5

0

Rolf Waeber

5

10 m

15

20

0 0

A Bayesian Approach to Stochastic Root-Finding

1 23/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Sample Paths

Robbin−Monro, an = 1/n, εn ~ N(0,1)

Robbin−Monro, an = 1/n, εn ~ N(0,1)

0.5

Robbin−Monro, an = 1/n, εn ~ N(0,1)

0.5

X* − X

0.5

X* − X

n

X* − Xn

n

0

0

−0.5

0

−0.5 0

10

1

10

2

3

10

10

4

10

5

10

−0.5 0

10

1

10

2

3

10

n

10

4

10

5

10

Bisection, p = 0.6, ε ~ N(0,1)

X* − Xn 0

2

3

10

10 n

Rolf Waeber

4

10

5

10

5

10

X* − Xn 0

−0.5 1

4

10

n

0

−0.5

3

10

0.5

X* − Xn

10

2

10

Bisection, p = 0.6, ε ~ N(0,1)

n

0.5

0

1

10

n

Bisection, p = 0.6, ε ~ N(0,1)

n

0.5

10

0

10

n

−0.5 0

10

1

10

2

3

10

10

4

10

n

A Bayesian Approach to Stochastic Root-Finding

5

10

0

10

1

10

2

3

10

10 n

24/26

4

10

5

10

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Numerical Comparison

0

10 E[|X* − Xn|]

−1

10

−2

10

Robbins−Monro (an=1/n) Bisection, Siegmund (p=0.6) Bisection, Known T (p=0.6)

−3

10

0

10

1

10

2

3

10

10

4

10

5

10

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

25/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Conclusions

• Algorithms based on probabilistic-bisection can be very promising for stochastic root-finding problems: - Very little switching. - No tuning sequence. - Robust. - Provide posterior distribution of X ∗ .

• Future Research: - Further improve the current algorithm. - Analyze the algorithm’s finite-time properties. - Extend probabilistic-bisection to higher dimensions.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

26/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

THANK YOU!

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

27/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

M. B EN -O R and A. H ASSIDIM (2008): The Bayesian learner is optimal for noisy binary search. In 49th Annual IEEE Symposium on Foundations of Computer Science. IEEE, pp. 221–230. M. B URNASHEV and K. Z IGANGIROV (1974): An interval estimation problem for controlled observations. Problemy Peredachi Informatsii 10:51–61. R. C ASTRO and R. N OWAK (2008): Active learning and sampling. In Foundations and Applications of Sensor Management, A. O. Hero, D. A. Castañón, D. Cochran and K. Kastella, editors. Springer, pp. 177–200. U. F EIGE , P. R AGHAVAN , D. P ELEG and E. U PFAL (1997): Computing with noisy information. SIAM Journal of Computing :1001–1018. M. H ORSTEIN (1963): Sequential transmission using noiseless feedback. IEEE Transactions on Information Theory 9:136–143. R. K ARP and R. K LEINBERG (2007): Noisy binary search and its applications. In Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, pp. 881–890. R. N OWAK (2008): Generalized binary search. In 2008 46th Annual Allerton Conference on Communication, Control, and Computing. pp. 568–574. R. N OWAK (2009): Noisy generalized binary search. In Advances in neural information processing systems, volume 22. pp. 1366–1374. H. R OBBINS and S. M ONRO (1951): A stochastic approximation method. The Annals of Mathematical Statistics 22:400–407. D. S IEGMUND (1985): Sequential Analysis: tests and confidence intervals. Springer. Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

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Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Tuning Parameters

0

Input Parameters d to Robbins−Monro (stepsize d/n), X* ~ U(0,1)

10 E[|X* − Xn|]

−1

10

0.1 0.25 0.5 0.75 1 1.5 2 5 10 20

−2

10

−3

10

−4

10

0

10

1

10

2

3

10

10

4

10

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

28/26

5

10

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Tuning Parameters

Input Parameters pc to Bisection Algorithm, X* ~ U(0,1)

0

10 E[|X* − Xn|]

−1

10

0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

−2

10

−3

10

−4

10

0

10

1

10

2

3

10

10

4

10

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

28/26

5

10

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Tuning Parameters

Input Parameters pc to Bisection Algorithm with Exact T, X* ~ U(0,1)

0

10 E[|X* − Xn|]

−1

10

0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.975

−2

10

−3

10

−4

10

0

10

1

10

2

3

10

10

4

10

n

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

28/26

5

10

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 0, Z = −1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 1, Z = −1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 2, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 3, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 4, Z = −1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 5, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 6, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 7, Z = −1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 8, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 9, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Higher Dimensions Boundary detection:

n = 10, Z = 1 1 0.8 0.6 0.4 0.2 0 0

Rolf Waeber

0.2

0.4

0.6

A Bayesian Approach to Stochastic Root-Finding

0.8

1

29/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps

a

0

1

An

For a > 0, define An = Pn ([0, a)).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

30/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case I

a

0

1

An

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case I

a Xn

0

1

An

If An ≤ 1/2:

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case I

a Xn

0

X*

1

An

If An ≤ 1/2: ( ∗

An+1 |Xn ≤ X , Fn =

Rolf Waeber

2q · An 2p · An

with probability p, with probability q.

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case I

X* a Xn

0

1

An

If An ≤ 1/2: (

2q · An 2p · An

with probability p, with probability q.

(

2q · An 2p · An

with probability q, with probability p.

∗

An+1 |Xn ≤ X , Fn =

∗

An+1 |Xn > X , Fn = Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case I

a Xn

0

1

An

By assumption: Pn (Xn ≤ X ∗ ) = 1/2. If An ≤ 1/2:

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case I

a Xn

0

1

An

By assumption: Pn (Xn ≤ X ∗ ) = 1/2. If An ≤ 1/2: ( An+1 |Fn =

Rolf Waeber

2q · An 2p · An

with probability 1/2, with probability 1/2.

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case II

a

0

1

An

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case II

0

Xn

a

1

An

If An > 1/2:

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case II

Xn

0

X*

a

1

An

If An > 1/2: ( ∗

(1 − An+1 )|Xn ≤ X , Fn =

Rolf Waeber

2p · (1 − An ) with probability p, 2q · (1 − An ) with probability q.

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case II

0

X* Xn

a

1

An

If An > 1/2: (

2p · (1 − An ) with probability p, 2q · (1 − An ) with probability q.

(

2p · (1 − An ) with probability q, 2q · (1 − An ) with probability p.

∗

(1 − An+1 )|Xn ≤ X , Fn =

∗

(1 − An+1 )|Xn > X , Fn = Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case II

0

Xn

a

1

An

By assumption: Pn (Xn ≤ X ∗ ) = 1/2. If An > 1/2:

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Case II

0

Xn

a

1

An

By assumption: Pn (Xn ≤ X ∗ ) = 1/2. If An > 1/2: ( (1 − An+1 )|Fn =

Rolf Waeber

2q · (1 − An ) with probability 1/2, 2p · (1 − An ) with probability 1/2.

A Bayesian Approach to Stochastic Root-Finding

31/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Combining case I and II: If An ≤ 1/2: (

2q · An 2p · An

(

2p · (1 − An ) with probability 1/2, 2q · (1 − An ) with probability 1/2.

An+1 |Fn =

with probability 1/2, with probability 1/2.

If An > 1/2: (1 − An+1 )|Fn =

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

32/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Combining case I and II: If An ≤ 1/2: ( An+1 |Fn =

2q · An 2p · An

ln An+1 |Fn = ln An +

with probability 1/2, with probability 1/2.

( ln(2q) with probability 1/2, ln(2p) with probability 1/2.

If An > 1/2: ( (1 − An+1 )|Fn =

2p · (1 − An ) with probability 1/2, 2q · (1 − An ) with probability 1/2.

( ln(2q) with probability 1/2, ln(1 − An+1 )|Fn = ln(1 − An ) + ln(2p) with probability 1/2. Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

32/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont.

a Xn

0

1

0

Xn

An

a

1

An

Define: • Mn := ln(An ) ∧ ln(1 − An ).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

33/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont.

a Xn

0

1

0

Xn

An

a

1

An

Define: • Mn := ln(An ) ∧ ln(1 − An ). • Vn is a random walk with V0 = M0 and i.i.d. increments (ξn )n P(ξn = ln(2p)) = 1/2, P(ξn = ln(2q)) = 1/2, such that Vn and Mn are coupled.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

33/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. Mn = ln(An ) ∧ ln(1 − An ) ≤ Vn for all n ∈ N. n

Vn ln(1/2) Mn

M0

a Xn

0 An

Rolf Waeber

1

0

Xn

a

1

An

A Bayesian Approach to Stochastic Root-Finding

34/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof – Key Steps cont. An(k) 0

h

Xn

2h 3h

kh (k+1)h

Bn(k) 1

Fix h > 0. • For k ∈ {0, . . . , bh−1 c}, define: - An (k ) = Pn ([0, kh)), - Bn (k ) = Pn ([kh, 1]).

• At iteration n there exists a k˜n such that Xn ∈ [k˜n h, (k˜n + 1)h]. Pn (|X ∗ − Xn | > h) ≤ An (k˜n ) + Bn (k˜n ) ˜

˜

˜

˜

= eMn (kn ) + eNn (kn ) ≤ eVn (kn ) + eWn (kn )   ≤ max eVn (k ) + eWn (k ) . k ∈{0,...,bh−1 c}

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

35/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Upper Bound Proposition There exists C = C(p) > 1 such that
P Pn (|X ∗ − Xn | > h) > C −n ≤ 4h−1 C −2n for all h > 0 and n ∈ N.

Pn(|X*−Xn|>h) 1 0.8 fn(x)

C−n

0.6 0.4 0.2 h 0

Rolf Waeber

h Xn

1

A Bayesian Approach to Stochastic Root-Finding

0 0

0.2

0.4

0.6 h

36/26

0.8

1

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof – Key Steps cont.

 ∗

P Pn (|X − Xn | > h) > C

−n




≤ P

bh−1 c n

[

e

 Vn (k )

+e

Wn (k )



>C

−n

o 

k =0 bh−1 c

≤

X

  P eVn (k ) + eWn (k ) > C −n

k =0 bh−1 c

≤

X

  [ P {eVn (k ) > C −n /2} {eWn (k ) > C −n /2}

k =0 bh−1 c

≤

X

−1

  bhXc   P eVn (k ) > C −n /2 + P eWn (k ) > C −n /2

k =0

k =0

  ≤ 2 bh−1 c + 1 C −2n (by Hoeffding’s inequality) ≤ 4h−1 C −2n .

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

37/26

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Finite Time Property

Proposition Let c ∈ (1, C) and ε > 0. Then
P c n En [|X ∗ − Xn |] > ε ≤ 4C −n , for all n ≥ max(0, N), where N=

Rolf Waeber

ln(2/ε) . ln(C/c)

A Bayesian Approach to Stochastic Root-Finding

38/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof of Finite Time Property Proof: • Fix n ≥ N. • For any h ∈ (0, 1): c n En [|X ∗ − Xn |] = c n

Z

1

Pn (|X ∗ − Xn | > h)dh

0

≤ c n (h + Pn (|X ∗ − Xn | > h)). • Plug in h = C −n . • Then, on the event {Pn (|X ∗ − Xn | > C −n ) ≤ C −n }: c n En [|X ∗ − Xn |] ≤ ε. • Hence P(c n En [|X ∗ − Xn |] > ε) ≤ P(Pn (|X ∗ − Xn | > C −n ) > C −n ) ≤ 4C −n .  Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

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Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Proof of L1 Convergence

Proposition There exists a constant c = c(p) > 1 such that P

c n En [|X ∗ − Xn |] −→ 0 as n → ∞. Proposition The stochastic process (c n En [|X ∗ − Xn |])n is uniformly integrable.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

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References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Proof of Uniform Integrability Proof: • When b is “large”,
P c n En [|X ∗ − Xn |] > b ≤ 4C −n for all n ∈ N.

    E c n En [|X ∗ − Xn |]1{c n En [|X ∗ −Xn |]>b} ≤ c n E 1{c n En [|X ∗ −Xn |]>b}   ≤ c n E 1{c n >b} 1{c n En [|X ∗ −Xn |]>b}

= c n 1{c n >b} P c n En [|X ∗ − Xn |] > b




≤ c n 1{n>logc b} 4C −n . • Taking supremum on both sides   sup E c n En [|X ∗ − Xn |]1{c n En [|X ∗ −Xn |]>b} ≤ sup 4(c/C)n 1{n>logc b} n∈N

n∈N

= 4(c/C)logc b . • Letting b → ∞, the uniform integrability follows.  Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

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Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Non-constant p(·)

• Allow p(x) to vary with x (but bounded away from 1/2 outside any neighborhood of X ∗ ).

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

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References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Non-constant p(·)

• Allow p(x) to vary with x (but bounded away from 1/2 outside any neighborhood of X ∗ ). • The probabilistic bisection algorithm can be used as a heuristic.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

42/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Non-constant p(·)

• Allow p(x) to vary with x (but bounded away from 1/2 outside any neighborhood of X ∗ ). • The probabilistic bisection algorithm can be used as a heuristic. Theorem (Consistency) Xn → X ∗ almost surely as n → ∞.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

42/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

Non-constant p(·)

• Allow p(x) to vary with x (but bounded away from 1/2 outside any neighborhood of X ∗ ). • The probabilistic bisection algorithm can be used as a heuristic. Theorem (Consistency) Xn → X ∗ almost surely as n → ∞. • Pn (·) fails to converge to a point mass in general!

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

42/26

References

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Non-constant p(·): Rate of Convergence

Theorem Let ε > 0. Define A = [X ∗ − ε, X ∗ + ε] and τ (A) := inf{n ≥ 0 : Xn ∈ A}. Then E[τ (A)] ≤

− ln(2ε) , r

where r is some positive constant.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

43/26

Stochastic Root-Finding

The Probabilistic Bisection Algorithm

Algorithm Analysis

Future Research and Conclusions

References

Non-constant p(·): Rate of Convergence

Theorem Let ε > 0. Define A = [X ∗ − ε, X ∗ + ε] and τ (A) := inf{n ≥ 0 : Xn ∈ A}. Then E[τ (A)] ≤

− ln(2ε) , r

where r is some positive constant. Future Research: • Measuring at the median is not necessarily optimal for reducing expected posterior entropy. • Reaching some neighborhood A = [X ∗ − ε, X ∗ + ε] might be sufficient.

Rolf Waeber

A Bayesian Approach to Stochastic Root-Finding

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