Chernoff Bounds Life in Los Angeles ORBAN STRES RATE LOW MED Herman Chernoff
Chernoff Bounds Herman Chernoff
Concentration Flip a coin for many times: flips -1 flips-5 flips-50 flips -500 fips=500000 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.4 0.4 04 0.4 0.4 0.2 0.2 0.2 0.2 02 0 1 1 1 0 0
Concentration Flip a coin for many times:
Chernoff bound: For independent trials X1,X2,...,XnE(0,1} Let X=>2 Xi,and u=E[X]. For any 6 >0, Pr[X≥(1+6)W≤ PrX≤I-oW a
Chernof bound: Pr[X ⇥ (1+)µ] ⇥ e (1+)(1+) µ Pr[X ⇥ (1)µ] ⇥ ⇥ e (1)(1) µ For any > 0, For independent trials X1,X2,...,Xn 2 {0, 1}. Let X = Pn i=1 Xi , and µ = E[X]
Chernoff Bounds 0.8 0.6 Pr[X≥1+6川≤ 0.4 0.2 0 0.5 1 1.5 6
Chernoff Bounds Pr[ X ⇥ (1 + ) µ ] ⇥ e (1 + )(1 + ) µ
A Generalization of Markov's Inequality Theorem: For any X,for h:XR,for any t>0, E[h(X)] Pr[h(X)≥t1≤ t
A Generalization of Markov’s Inequality Theorem: For any X, for h : X ⇥ R+, for any t > 0, Pr[h(X) ⇥ t] E[h(X)] t
Moment Generating Functions Definition(moment generating functions): The moment generating function of X is M(A)=EcAX. Taylor's expansion: A 三营刘
Moment Generating Functions Definition (moment generating functions): The moment generating function of X is M() = E eX ⇥ . E ⌅ eX ⇧ = E ⇤ k=0 k k! Xk ⇥ = ⇤ k=0 k k! E ⌅ Xk ⇧ Taylor’s expansion:
independent X1,X2,.,Xn∈{0，l} X=Xi E[X]= 2=1 Pr[X≥(1+)4≤？ for入>0 s preix≥en+Ase Markov eA(1+8)u ●--直 Independence!
X = X n i=1 Xi E[X] = µ independent X1, X2,...,Xn 2 {0, 1} Pr[X (1 + )µ] ? X (1+)µ Pr e e E e⇥X ⇥ e⇥(1+)µ Markov for > 0 = E ⇥ en i=1 Xi ⇤ = E ⇤ n i=1 eXi ⇥ = n i=1 E ⇥ eXi ⇤ Independence!
independent X1,X2,..,Xm∈{0，l} X=Xi EX= 2=1 Pr[X≥(1+0)4 for入>0 prle1X≥e1+4=gLeD e1(1+o)μ ●=e eP:(e!-1)=e(e-Du ≤】 i= i=1 pi PrXi 1 =∑ i=1 =pe1+(1-p)e0 ≤ePi(e-l)
X = X n i=1 Xi E[X] = µ independent X1, X2,...,Xn 2 {0, 1} Pr[X (1 + )µ] X (1+)µ Pr e e E e⇥X ⇥ e⇥(1+)µ for > 0 = n i=1 E ⇥ eXi ⇤ pi = Pr[Xi = 1] µ = X n i=1 pi = pi · e·1 +(1 pi)· e·0 ⇥ epi(e1) n i=1 epi(e1) e(e1)µ =
independent X1,.X2,.,Xn∈{0，l} X=Xi E[X]= i=1 Pr[X≥(1+o)W for入>0 ●≤ee-1r
X = X n i=1 Xi E[X] = µ independent X1, X2,...,Xn 2 {0, 1} Pr[X (1 + )µ] X (1+)µ Pr e e E e⇥X ⇥ e⇥(1+)µ for > 0 e(e1)µ ∑ √ e(e∏°1) e∏(1+±) !µ
independent X1,X2,.,Xn∈{0，l} m X=∑X: E[X]= i=1 入>0 1.3 1.2 1.1 0.9 0.8 0.7 0. 0 0. 1.5 whenλ=ln(1+6)
X = X n i=1 Xi E[X] = µ independent X1, X2,...,Xn 2 {0, 1} Pr[X (1 + )µ] X (1+)µ Pr e e for > 0 ∑ √ e(e∏°1) e∏(1+±) !µ when ⇥ = ln(1+)