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 { X i } is a martingale with | X i − X i − 1 | ≤ c i , then for every λ > 0 and every n ≥ 0 we have λ 2 2 � ci 2 − P [ X n ≥ X 0 + λ ] ≤ e and λ 2 2 � ci 2 − P [ X n ≥ X 0 − λ ] ≤ e Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Azuma’s Inequality and Proof Theorem For every L > 0 , if { X i } is a martingale with | X i − X i − 1 | ≤ c i , then for every λ > 0 and every n ≥ 0 we have λ 2 2 � ci 2 − P [ X n ≥ X 0 + λ ] ≤ e and λ 2 2 � ci 2 − P [ X n ≥ X 0 − λ ] ≤ e 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 { X i } is a martingale with | X i − X i − 1 | ≤ c i , then for every λ > 0 and every n ≥ 0 we have λ 2 2 � ci 2 − P [ X n ≥ X 0 + λ ] ≤ e and λ 2 2 � ci 2 − P [ X n ≥ X 0 − λ ] ≤ e 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 [ e tY ] ≤ e t 2 / 2 . Lecture 15. Martingales and Applications CS 574: Randomized Algorithms

Azuma’s Inequality and Proof Theorem For every L > 0 , if { X i } is a martingale with | X i − X i − 1 | ≤ c i , then for every λ > 0 and every n ≥ 0 we have λ 2 2 � ci 2 − P [ X n ≥ X 0 + λ ] ≤ e and λ 2 2 � ci 2 − P [ X n ≥ X 0 − λ ] ≤ e 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 [ e tY ] ≤ e t 2 / 2 . Can also show something similar for | X i − X i − 1 | ∈ [ a i , b i ]. 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 G n , 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 G n , p around its mean. Theorem (Shamir and Spencer) Let χ ( G ) be the chromatic number of G ∈ G n , p . Pr [ χ ( G ) − E [ χ ( G )] ≥ λ ] ≤ 2 exp ( − λ 2 2 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 n E [ χ ( G )] ≥ 2 log p / (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 n E [ χ ( G )] ≥ 2 log p / (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 n E [ χ ( G )] ≥ 2 log p / (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 n E [ χ ( G )] ≥ 2 log p / (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 n E [ χ ( G )] ≥ 2 log p / (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