Martingales and the Method of Bounded Differences Advanced - PowerPoint PPT Presentation

Fundamental Facts about Conditional Expectation Example : 𝑍 : height of the chosen human being 𝑌 : country of origin of the chosen human being 𝑎 : gender of the chosen human being 𝔽 𝑍 = 𝔽 𝔽 𝑍 | 𝑌 average height of all human beings = weighted average of the country-by-country average heights 𝔽 𝑍 | 𝑎 = 𝔽 𝔽 𝑍 | 𝑌, 𝑎 | 𝑎 average height of all male/female human beings = weighted average of the country-by-country average male/female heights 𝔽 𝔽 𝑔 𝑌 𝑕 𝑌, 𝑍 | 𝑌 = 𝔽 𝑔 𝑌 𝔽 𝑕 𝑌, 𝑍 | 𝑌

Fundamental Facts about Conditional Expectation Example : 𝑍 : height of the chosen human being 𝑌 : country of origin of the chosen human being 𝑎 : gender of the chosen human being 𝔽 𝑍 = 𝔽 𝔽 𝑍 | 𝑌 average height of all human beings = weighted average of the country-by-country average heights 𝔽 𝑍 | 𝑎 = 𝔽 𝔽 𝑍 | 𝑌, 𝑎 | 𝑎 average height of all male/female human beings = weighted average of the country-by-country average male/female heights 𝔽 𝔽 𝑔 𝑌 𝑕 𝑌, 𝑍 | 𝑌 = 𝔽 𝑔 𝑌 𝔽 𝑕 𝑌, 𝑍 | 𝑌 once 𝑌 is fixed to some 𝑦 , 𝔽 𝑔 𝑌 𝑕 𝑌, 𝑍 | 𝑌 = 𝑦 = 𝑔(𝑦)𝔽 𝑕 𝑌, 𝑍 | 𝑌 = 𝑦

Fundamental Facts about Conditional Expectation (Cont.) 𝔽 𝑍 = 𝔽 𝔽 𝑍 | 𝑌 𝔽 𝑍 | 𝑎 = 𝔽 𝔽 𝑍 | 𝑌, 𝑎 | 𝑎 𝔽 𝔽 𝑔 𝑌 𝑕 𝑌, 𝑍 | 𝑌 = 𝔽 𝑔 𝑌 𝔽 𝑕 𝑌, 𝑍 | 𝑌

Fundamental Facts about Conditional Expectation (Cont.) 𝔽 𝑍 = 𝔽 𝔽 𝑍 | 𝑌 𝔽 𝑍 | 𝑎 = 𝔽 𝔽 𝑍 | 𝑌, 𝑎 | 𝑎 𝔽 𝔽 𝑔 𝑌 𝑕 𝑌, 𝑍 | 𝑌 = 𝔽 𝑔 𝑌 𝔽 𝑕 𝑌, 𝑍 | 𝑌 Generalization to the multivariate case: 𝔽 𝑍 = 𝔽 𝔽 𝑍 | Ԧ 𝑌 𝔽 𝑍 | Ԧ 𝑎 = 𝔽 𝔽 𝑍 | Ԧ 𝑌, Ԧ 𝑎 | Ԧ 𝑎 𝔽 𝔽 𝑔 Ԧ 𝑌 𝑕 Ԧ 𝑌, 𝑍 | Ԧ = 𝔽 𝑔 Ԧ 𝑌 𝔽 𝑕 Ԧ 𝑌, 𝑍 | Ԧ 𝑌 𝑌

Martingales originally refers to a betting strategy: “double your bet after every loss”

Martingales originally refers to a betting strategy: “double your bet after every loss” 𝑜−1 2 𝑗 = 1 when you get a win after 𝑜 losses: 2 𝑜 − σ 𝑗=0

Martingales originally refers to a betting strategy: “double your bet after every loss” 𝑜−1 2 𝑗 = 1 when you get a win after 𝑜 losses: 2 𝑜 − σ 𝑗=0 consider a fair game, with any betting strategy

Martingales originally refers to a betting strategy: “double your bet after every loss” 𝑜−1 2 𝑗 = 1 when you get a win after 𝑜 losses: 2 𝑜 − σ 𝑗=0 consider a fair game, with any betting strategy let 𝑌 𝑗 be our wealth after 𝑗 rounds

Martingales originally refers to a betting strategy: “double your bet after every loss” 𝑜−1 2 𝑗 = 1 when you get a win after 𝑜 losses: 2 𝑜 − σ 𝑗=0 consider a fair game, with any betting strategy let 𝑌 𝑗 be our wealth after 𝑗 rounds 𝔽 𝑌 𝑗+1 | 𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑗 =

Martingales originally refers to a betting strategy: “double your bet after every loss” 𝑜−1 2 𝑗 = 1 when you get a win after 𝑜 losses: 2 𝑜 − σ 𝑗=0 consider a fair game, with any betting strategy let 𝑌 𝑗 be our wealth after 𝑗 rounds 𝔽 𝑌 𝑗+1 | 𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑗 = 𝑌 𝑗

Martingales originally refers to a betting strategy: “double your bet after every loss” 𝑜−1 2 𝑗 = 1 when you get a win after 𝑜 losses: 2 𝑜 − σ 𝑗=0 consider a fair game, with any betting strategy let 𝑌 𝑗 be our wealth after 𝑗 rounds 𝔽 𝑌 𝑗+1 | 𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑗 = 𝑌 𝑗 since the game is fair , conditioned on past history, we expect no change to current value after one round

Martingales

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Coin Flipping toss a fair coin for many times measure the differences between # of heads and # of tails

Example: Random Walk a dot starting from the origin in each step, move equiprobably to one of four neighbors

Example: Random Walk a dot starting from the origin in each step, move equiprobably to one of four neighbors after 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance)

Example: Random Walk a dot starting from the origin in each step, move equiprobably to one of four neighbors after 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance)

Example: Random Walk a dot starting from the origin in each step, move equiprobably to one of four neighbors after 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance)

Example: Random Walk a dot starting from the origin in each step, move equiprobably to one of four neighbors after 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance) How far the dot is away from the origin after 𝑜 steps?

Azuma’s Inequality

Azuma’s Inequality 𝑌 0 , 𝑌 1 , ⋯ are not necessarily independent

Azuma’s Inequality in Action After 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance) How large is 𝑌 𝑜 ?

Azuma’s Inequality in Action After 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance) How large is 𝑌 𝑜 ?

Azuma’s Inequality in Action After 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance) How large is 𝑌 𝑜 ? We know 𝑌 0 = 0 , and 𝑌 𝑙 − 𝑌 𝑙−1 ≤ 1

Azuma’s Inequality in Action After 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance) How large is 𝑌 𝑜 ? We know 𝑌 0 = 0 , and 𝑌 𝑙 − 𝑌 𝑙−1 ≤ 1

Azuma’s Inequality in Action After 𝑗 steps, use 𝑌 𝑗 to denote # of hops to origin (Manhattan distance) How large is 𝑌 𝑜 ? We know 𝑌 0 = 0 , and 𝑌 𝑙 − 𝑌 𝑙−1 ≤ 1 Within Ο( 𝑜 log 𝑜) w.h.p.

Azuma’s Inequality

Azuma’s Inequality For a sequence of r.v., if in each step: * on average make no change to current value ( martingale ) * no big jump ( bounded difference )

Azuma’s Inequality For a sequence of r.v., if in each step: * on average make no change to current value ( martingale ) * no big jump ( bounded difference ) Then final value does not deviate a lot from the initial value.

Proving Azuma’s Inequality

Proving Azuma’s Inequality Use similar strategy as in proving Chernoff bounds:

Proving Azuma’s Inequality Use similar strategy as in proving Chernoff bounds: (a ) Apply generalized Markov’s inequality to MGF

Proving Azuma’s Inequality Use similar strategy as in proving Chernoff bounds: (a ) Apply generalized Markov’s inequality to MGF (b) * Bound the value of MGF (use Hoeffding’s lemma)

Proving Azuma’s Inequality Use similar strategy as in proving Chernoff bounds: (a ) Apply generalized Markov’s inequality to MGF (b) * Bound the value of MGF (use Hoeffding’s lemma) (c) Optimize the value of MGF

Proving Azuma’s Inequality

Proving Azuma’s Inequality

Proving Azuma’s Inequality

Proving Azuma’s Inequality

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0 𝔽 𝑍 = 𝔽 𝔽 𝑍 | 𝑌

for 𝜇 > 0 𝔽 𝔽 𝑔 𝑌 𝑕 𝑌, 𝑍 |𝑌 = 𝔽 𝑔 𝑌 𝔽 𝑕 𝑌, 𝑍 |𝑌

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0

for 𝜇 > 0 𝑢 minimized when 𝜇 = 𝑜 2 σ 𝑙=1 𝑑 𝑙

Proving Azuma’s Inequality

Proving Azuma’s Inequality ???

Proving Azuma’s Inequality ???

Generalized Martingales

Generalized Martingales betting on a fair game 𝑌 𝑗 : gain/loss of the i th bet 𝑗 : wealth after the i th bet 𝑍

Generalized Martingales betting on a fair game 𝑌 𝑗 : gain/loss of the i th bet 𝑗 : wealth after the i th bet ← martingale (since game is fair) 𝑍

Generalized Azuma’s Inequality

Azuma’s Inequality martingale 𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑜 martingale 𝑌 0 , 𝑌 1 , ⋯ 𝔽 𝑌 𝑗 𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑗−1 ) = 𝑌 𝑗−1 with 𝑌 𝑙 − 𝑌 𝑙−1 ≤ 𝑑 𝑙 , generalization then ℙ 𝑌 𝑜 − 𝑌 0 ≥ 𝑢 ≤ ⋯ martingale 𝑍 0 , 𝑍 1 , ⋯ w.r.t. 𝑌 0 , 𝑌 1 , ⋯ generalization 𝑍 𝑗 = 𝑔(𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑗 ) 𝔽 𝑍 𝑌 0 , 𝑌 1 , ⋯ , 𝑌 𝑗−1 ) = 𝑍 Generalized Azuma’s Inequality 𝑗 𝑗−1 martingale 𝑍 0 , 𝑍 1 , ⋯ w.r.t. 𝑌 0 , 𝑌 1 , ⋯ with 𝑍 𝑙 − 𝑍 𝑙−1 ≤ 𝑑 𝑙 , then ℙ 𝑍 𝑜 − 𝑍 0 ≥ 𝑢 ≤ ⋯

Doob Sequence

Doob Sequence 𝑔( ) , , ,

Doob Sequence 𝑔( ) , , , average over no information 𝔽 𝑔