TY - JOUR
T1 - Probability bounds for polynomial functions in random variables
AU - He, Simai
AU - Jiang, Bo
AU - Li, Zhening
AU - Zhang, Shuzhong
PY - 2014/8
Y1 - 2014/8
N2 - Random sampling is a simple but powerful method in statistics and in the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ℝn. To do so, one may select a simpler (even finite) subset S0 ⊆S, randomly take some samples over S0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is a d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξi (i D 11 21 ⋯ , n) are i.i.d. Bernoulli random variables taking 1 or -1 with equal probability, then Prob{f (ξ1, ξ2, ⋯ , ξn) ≥ τn-d/2||F||1} ≥ θ, where τ1 θ > 0 are two universal constants and ||·||1 denotes the summation of the absolute values of all its entries. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.
AB - Random sampling is a simple but powerful method in statistics and in the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ℝn. To do so, one may select a simpler (even finite) subset S0 ⊆S, randomly take some samples over S0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f , S and S0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is a d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξi (i D 11 21 ⋯ , n) are i.i.d. Bernoulli random variables taking 1 or -1 with equal probability, then Prob{f (ξ1, ξ2, ⋯ , ξn) ≥ τn-d/2||F||1} ≥ θ, where τ1 θ > 0 are two universal constants and ||·||1 denotes the summation of the absolute values of all its entries. Several new inequalities concerning probabilities of the above nature are presented in this paper. Moreover, we show that the bounds are tight in most cases. Applications of our results in optimization are discussed as well.
KW - Approximation algorithm
KW - Polynomial function
KW - Polynomial optimization
KW - Probability bound
KW - Random sampling
KW - Tensor form
UR - http://www.scopus.com/inward/record.url?scp=84906655770&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84906655770&partnerID=8YFLogxK
U2 - 10.1287/moor.2013.0637
DO - 10.1287/moor.2013.0637
M3 - Article
AN - SCOPUS:84906655770
SN - 0364-765X
VL - 39
SP - 889
EP - 907
JO - Mathematics of Operations Research
JF - Mathematics of Operations Research
IS - 3
ER -