Question

6. (a) [Chernoff Bound for a Special Case] Let X1, X2,..., Xn be independently distributed random variables with Pr[X₁ = 1] = Pr[X₁ = -1] = 1/2. Let X = ∑_{i=1}^n X₁. Show that for any a > 0, Pr[X > a] ≤ e^{-a^2/2n}. (b) [Set Balancing] A set S of m objects, each having zero or more of n potential features is specified by an n x m matrix A over {0, 1}, where aij = 1 iff j-th object has the i-th feature. A partition of the set into S1, S2 is given by a vector b∈ {-1,1}m, with 1 (resp. -1) in position j indicating the presence of object j in S1 (resp. S2). If we let c = Ab, then ci denotes the imbalance in feature i. The imbalance of a partition 6 is given by ||A6||∞ = max_{i∈[n]} |ci|. The set balancing problem is to find a partition 6 that minimizes the imbalance ||A6||. Consider the following algorithm: choose entries of 6 uniformly and independently at random from {-1, 1}, completely ignoring A. Show that Pr[||A6||> √4mlnn] ≤ 2/n. Use the bound from the previous part.

          6. (a) [Chernoff Bound for a Special Case] Let X1, X2,..., Xn be independently distributed
random variables with
Pr[X₁ = 1] = Pr[X₁ = -1] = 1/2.
Let X = ∑_{i=1}^n X₁. Show that for any a > 0,
Pr[X > a] ≤ e^{-a^2/2n}.
(b) [Set Balancing] A set S of m objects, each having zero or more of n potential features
is specified by an n x m matrix A over {0, 1}, where aij = 1 iff j-th object has the i-th
feature. A partition of the set into S1, S2 is given by a vector b∈ {-1,1}m, with 1 (resp.
-1) in position j indicating the presence of object j in S1 (resp. S2). If we let c = Ab,
then ci denotes the imbalance in feature i. The imbalance of a partition 6 is given by
||A6||∞ = max_{i∈[n]} |ci|. The set balancing problem is to find a partition 6 that minimizes
the imbalance ||A6||. Consider the following algorithm: choose entries of 6 uniformly and
independently at random from {-1, 1}, completely ignoring A. Show that
Pr[||A6||> √4mlnn] ≤ 2/n.
Use the bound from the previous part.

6 a chernoff bound for a special case let x1 x2 xn be independently distributed random variables with prx1 1 prx1 1 12 let x i1n x1 show that for any a 0 prx a e a22n b set balancing a set 85294

Added by Felix J.

Question

Please give Ace some feedback