88 research outputs found
The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues
A classic result of Banaszczyk (Random Str. & Algor. 1997) states that given any n vectors in Rm with ℓ2-norm at most 1 and any convex body K in Rm of Gaussian measure at least half, there exists a ±1 combination of these vectors that lies in 5K. Banaszczyk’s proof of this result was non-constructive and it was open how to find such a ±1 combination in polynomial time. In this paper, we give an efficient randomized algorithm to find a ±1 combination of the vectors which lies in cK for some fixed constant c > 0. This leads to new efficient algorithms for several problems in discrepancy theory
An Lp Analog to AAK Theory for p⩾2
AbstractWe develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication
Online Discrepancy Minimization for Stochastic Arrivals
In the stochastic online vector balancing problem, vectors
chosen independently from an arbitrary distribution in
arrive one-by-one and must be immediately given a sign.
The goal is to keep the norm of the discrepancy vector, i.e., the signed
prefix-sum, as small as possible for a given target norm.
We consider some of the most well-known problems in discrepancy theory in the
above online stochastic setting, and give algorithms that match the known
offline bounds up to factors. This substantially
generalizes and improves upon the previous results of Bansal, Jiang, Singla,
and Sinha (STOC' 20). In particular, for the Koml\'{o}s problem where
for each , our algorithm achieves
discrepancy with high probability, improving upon the previous
bound. For Tusn\'{a}dy's problem of minimizing the
discrepancy of axis-aligned boxes, we obtain an bound for
arbitrary distribution over points. Previous techniques only worked for product
distributions and gave a weaker bound. We also consider the
Banaszczyk setting, where given a symmetric convex body with Gaussian
measure at least , our algorithm achieves discrepancy with
respect to the norm given by for input distributions with sub-exponential
tails.
Our key idea is to introduce a potential that also enforces constraints on
how the discrepancy vector evolves, allowing us to maintain certain
anti-concentration properties. For the Banaszczyk setting, we further enhance
this potential by combining it with ideas from generic chaining. Finally, we
also extend these results to the setting of online multi-color discrepancy
- …