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CS 574: Randomized Algorithms Lecture 14. Introduction to Martingales October 8, 2015 Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms Martingales Introduction For independent r.v.s X i we showed tight concentration of

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CS 574: Randomized Algorithms

Lecture 14. Introduction to Martingales October 8, 2015

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Introduction

For independent r.v.s Xi we showed tight concentration of their sum around the mean.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Introduction

For independent r.v.s Xi we showed tight concentration of their sum around the mean. We can also show similar results for dependent r.v’s.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Introduction

For independent r.v.s Xi we showed tight concentration of their sum around the mean. We can also show similar results for dependent r.v’s. Definition A sequence of r.v.’s X1, X2 · · · is called a discrete time martingale, if E[Xi+1|X0, X1, · · · , Xi] = Xi, for every i = 0, 1, 2, · · · .

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Introduction

For independent r.v.s Xi we showed tight concentration of their sum around the mean. We can also show similar results for dependent r.v’s. Definition A sequence of r.v.’s X1, X2 · · · is called a discrete time martingale, if E[Xi+1|X0, X1, · · · , Xi] = Xi, for every i = 0, 1, 2, · · · . More generally, Xi sequence is a martingale with respect to a sequence Yi if E[Xi+1|Y0, Y1, · · · , Yi] = Xi, for every i = 0, 1, 2, · · · .

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Introduction

For independent r.v.s Xi we showed tight concentration of their sum around the mean. We can also show similar results for dependent r.v’s. Definition A sequence of r.v.’s X1, X2 · · · is called a discrete time martingale, if E[Xi+1|X0, X1, · · · , Xi] = Xi, for every i = 0, 1, 2, · · · . More generally, Xi sequence is a martingale with respect to a sequence Yi if E[Xi+1|Y0, Y1, · · · , Yi] = Xi, for every i = 0, 1, 2, · · · . Equivalently, E[Xi+1 − Xi|Y0, · · · , Yi] = 0 if the set of Y0, · · · , Yi is all the information up to time i. Namely, the difference Xi+1 − Xi is unbiased on the past up to time i.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Introduction

For independent r.v.s Xi we showed tight concentration of their sum around the mean. We can also show similar results for dependent r.v’s. Definition A sequence of r.v.’s X1, X2 · · · is called a discrete time martingale, if E[Xi+1|X0, X1, · · · , Xi] = Xi, for every i = 0, 1, 2, · · · . More generally, Xi sequence is a martingale with respect to a sequence Yi if E[Xi+1|Y0, Y1, · · · , Yi] = Xi, for every i = 0, 1, 2, · · · . Equivalently, E[Xi+1 − Xi|Y0, · · · , Yi] = 0 if the set of Y0, · · · , Yi is all the information up to time i. Namely, the difference Xi+1 − Xi is unbiased on the past up to time i.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Doob Martingales

Classic example of a Gambler whose bank roll is X0. At each time, she chooses to play some game in the casino at some

stakes. If we assume that every game is fair (expected utility
f playing is 0), but games need not be independent and

stakes need not be independent, then the sequence X0, X1, ... is a martingale, where Xi is the amount of money she has at time i.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Doob Martingales

Classic example of a Gambler whose bank roll is X0. At each time, she chooses to play some game in the casino at some

stakes. If we assume that every game is fair (expected utility
f playing is 0), but games need not be independent and

stakes need not be independent, then the sequence X0, X1, ... is a martingale, where Xi is the amount of money she has at time i. Tower rule of conditional expectations E[V |W ] = E[E[V |U, W ]|W ].

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Doob Martingales

Classic example of a Gambler whose bank roll is X0. At each time, she chooses to play some game in the casino at some

stakes. If we assume that every game is fair (expected utility
f playing is 0), but games need not be independent and

stakes need not be independent, then the sequence X0, X1, ... is a martingale, where Xi is the amount of money she has at time i. Tower rule of conditional expectations E[V |W ] = E[E[V |U, W ]|W ]. Define Doob Martingale: Let X0, X1, · · · be a sequence or r.v.s. Let Y be also an r.v. with E[Y ] < ∞. Then Zi = E[Y |X0, X1, · · · , Xi] is a Doob Martingale. Doob martingales try to estimate function Y with finer and finer estimates.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Doob Martingales

Classic example of a Gambler whose bank roll is X0. At each time, she chooses to play some game in the casino at some

stakes. If we assume that every game is fair (expected utility
f playing is 0), but games need not be independent and

stakes need not be independent, then the sequence X0, X1, ... is a martingale, where Xi is the amount of money she has at time i. Tower rule of conditional expectations E[V |W ] = E[E[V |U, W ]|W ]. Define Doob Martingale: Let X0, X1, · · · be a sequence or r.v.s. Let Y be also an r.v. with E[Y ] < ∞. Then Zi = E[Y |X0, X1, · · · , Xi] is a Doob Martingale. Doob martingales try to estimate function Y with finer and finer estimates. Frequently, in application we have Y = f (Z1, ..., Zn). In this case, Z0 = E(Y ) and Zn = E(Y |Z1, ..., Zn) = Y .

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Examples

Fair,independent coin tosses: Martingale with independent differences.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Examples

Fair,independent coin tosses: Martingale with independent differences. Balls in Bins example: How may empty bins are there if I throw m balls in n bins randomly?

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Examples

Fair,independent coin tosses: Martingale with independent differences. Balls in Bins example: How may empty bins are there if I throw m balls in n bins randomly? The vertex/edge exposure martingale for random graphs and chromatic number.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Martingales Examples

Fair,independent coin tosses: Martingale with independent differences. Balls in Bins example: How may empty bins are there if I throw m balls in n bins randomly? The vertex/edge exposure martingale for random graphs and chromatic number.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Azuma Inequality

We say that the martingale {Xi} has L-bounded increments if |Xi+1 − Xi| ≤ L for every i.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Azuma Inequality

We say that the martingale {Xi} has L-bounded increments if |Xi+1 − Xi| ≤ L for every i. Theorem For every L > 0, if {Xi} is a martingale with L-bounded increments, then for every λ > 0 and every n ≥ 0 we have P[Xn ≥ X0 + λ] ≤ e− λ2

2L2n

and P[Xn ≥ X0 − λ] ≤ e− λ2

2L2n Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Class Assignment: Show the special case for independent r.v.s: Corollary If Zi are independent r.v.s taking values in [−L, L], Z = Zi and µ = E(Z), then for every λ > 0 we have P[Z ≥ µ + λ] ≤ e− λ2

2L2n

and P[Z ≥ µ − λ] ≤ e− λ2

2L2n Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Lipschitz condition and Application to Balls in Bins

Function f (z1, z2, ..., zn) is L-Lipschitz is changing any one coordinate changes the value of f by at most c in absolute value.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Lipschitz condition and Application to Balls in Bins

Function f (z1, z2, ..., zn) is L-Lipschitz is changing any one coordinate changes the value of f by at most c in absolute value. If f (Z1, ...Zn) is L-Lipschitz and Zi independent, then the Doob martingale of f with respect to Zi has increments bounded by L.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms

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Lipschitz condition and Application to Balls in Bins

Function f (z1, z2, ..., zn) is L-Lipschitz is changing any one coordinate changes the value of f by at most c in absolute value. If f (Z1, ...Zn) is L-Lipschitz and Zi independent, then the Doob martingale of f with respect to Zi has increments bounded by L. Apply Azuma to balls in bins for concentration of the number

f empty bins.

Lecture 14. Introduction to Martingales CS 574: Randomized Algorithms