CS 574: Randomized Algorithms Lecture 15. Martingales and - - PowerPoint PPT Presentation

CS 574: Randomized Algorithms

Lecture 15. Martingales and Applications October 13, 2015

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Azuma’s Inequality and Proof

Theorem For every L > 0, if {Xi} is a martingale with |Xi − Xi−1| ≤ ci, then for every λ > 0 and every n ≥ 0 we have P[Xn ≥ X0 + λ] ≤ e

−

λ2 2 ci 2

and P[Xn ≥ X0 − λ] ≤ e

−

λ2 2 ci 2 Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Azuma’s Inequality and Proof

Theorem For every L > 0, if {Xi} is a martingale with |Xi − Xi−1| ≤ ci, then for every λ > 0 and every n ≥ 0 we have P[Xn ≥ X0 + λ] ≤ e

−

λ2 2 ci 2

and P[Xn ≥ X0 − λ] ≤ e

−

λ2 2 ci 2

We will see the proof next

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Azuma’s Inequality and Proof

Theorem For every L > 0, if {Xi} is a martingale with |Xi − Xi−1| ≤ ci, then for every λ > 0 and every n ≥ 0 we have P[Xn ≥ X0 + λ] ≤ e

−

λ2 2 ci 2

and P[Xn ≥ X0 − λ] ≤ e

−

λ2 2 ci 2

We will see the proof next Lemma Let Y be a random variable such that Y ∈ [−1, +1] and E[Y ] = 0. Then for any t ≥ 0, we have E[etY ] ≤ et2/2.

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Azuma’s Inequality and Proof

Theorem For every L > 0, if {Xi} is a martingale with |Xi − Xi−1| ≤ ci, then for every λ > 0 and every n ≥ 0 we have P[Xn ≥ X0 + λ] ≤ e

−

λ2 2 ci 2

and P[Xn ≥ X0 − λ] ≤ e

−

λ2 2 ci 2

We will see the proof next Lemma Let Y be a random variable such that Y ∈ [−1, +1] and E[Y ] = 0. Then for any t ≥ 0, we have E[etY ] ≤ et2/2. Can also show something similar for |Xi − Xi−1| ∈ [ai, bi].

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Concentration of the Chromatic Number of Random Graphs

We will use the vertex-exposure martingale and Azuma’s inequality to show sharp concentration of the chromatic number of Gn,p around its mean.

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Concentration of the Chromatic Number of Random Graphs

We will use the vertex-exposure martingale and Azuma’s inequality to show sharp concentration of the chromatic number of Gn,p around its mean. Theorem (Shamir and Spencer) Let χ(G) be the chromatic number of G ∈ Gn,p. Pr[χ(G) − E[χ(G)] ≥ λ] ≤ 2exp(−λ2 2n)

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

A Tighter Bound on Chromatic Number

Note that we did not know E[χ(G)]. We will see next that E[χ(G)] ≥

n 2 logp/(1−p) n.

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

A Tighter Bound on Chromatic Number

Note that we did not know E[χ(G)]. We will see next that E[χ(G)] ≥

n 2 logp/(1−p) n.

This implies that the expectation is tightly concentrated around it mean, since E[χ(G)] >> √n.

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

A Tighter Bound on Chromatic Number

Note that we did not know E[χ(G)]. We will see next that E[χ(G)] ≥

n 2 logp/(1−p) n.

This implies that the expectation is tightly concentrated around it mean, since E[χ(G)] >> √n. Easy lower-bound follows from the largest clique size of random graph.

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

A Tighter Bound on Chromatic Number

Note that we did not know E[χ(G)]. We will see next that E[χ(G)] ≥

n 2 logp/(1−p) n.

This implies that the expectation is tightly concentrated around it mean, since E[χ(G)] >> √n. Easy lower-bound follows from the largest clique size of random graph. Upper-bound much harder.

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

A Tighter Bound on Chromatic Number

Note that we did not know E[χ(G)]. We will see next that E[χ(G)] ≥

n 2 logp/(1−p) n.

This implies that the expectation is tightly concentrated around it mean, since E[χ(G)] >> √n. Easy lower-bound follows from the largest clique size of random graph. Upper-bound much harder. We also show a tighter concentration: Chromatic number is concentrated in 4 values w.h.p!

Lecture 15. Martingales and Applications CS 574: Randomized Algorithms