4 research outputs found
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
Online discrepancy minimization for stochastic arrivals
In the stochastic online vector balancing problem, vectors v1, v2,..., vT chosen independently from an arbitrary distribution in Rn 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 polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlós problem where kvtk2 ≤ 1 for each t, our algorithm achieves Oe(1) discrepancy with high probability, improving upon the previous Oe(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(logd+4 T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log2d+1 T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves Oe(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. 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. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. 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 multicolor discrepancy
Online discrepancy minimization for stochastic arrivals
In the stochastic online vector balancing problem, vectors v1,v2,…,vT chosen independently from an arbitrary distribution in Rn 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 polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlos problem where ∥v_t∥_2≤1 for each t, our algorithm achieves ˜O(1) discrepancy with high probability, improving upon the previous ˜O(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(log^{d+4}T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log^{2d+1}T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves \tilde{O}(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails.
Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. 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. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. 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.</p