TY - JOUR

T1 - Competitive online algorithms for resource allocation over the positive semidefinite cone

AU - Eghbali, Reza

AU - Saunderson, James

AU - Fazel, Maryam

PY - 2018/7/1

Y1 - 2018/7/1

N2 - We consider a new and general online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially. We analyze two greedy primal-dual algorithms and provide bounds on their competitive ratios. Our analysis relies on a smooth surrogate of the objective function that needs to satisfy a new diminishing returns (PSD-DR) property (that its gradient is order-reversing with respect to the PSD cone). Using the representation for monotone maps on the PSD cone given by Löwner’s theorem, we obtain a convex parametrization of the family of functions satisfying PSD-DR. We then formulate a convex optimization problem to directly optimize our competitive ratio bound over this set. This design problem can be solved offline before the data start arriving. The online algorithm that uses the designed smoothing is tailored to the given cost function, and enjoys a competitive ratio at least as good as our optimized bound. We provide examples of computing the smooth surrogate for D-optimal and A-optimal experiment design, and demonstrate the performance of the custom-designed algorithm.

AB - We consider a new and general online resource allocation problem, where the goal is to maximize a function of a positive semidefinite (PSD) matrix with a scalar budget constraint. The problem data arrives online, and the algorithm needs to make an irrevocable decision at each step. Of particular interest are classic experiment design problems in the online setting, with the algorithm deciding whether to allocate budget to each experiment as new experiments become available sequentially. We analyze two greedy primal-dual algorithms and provide bounds on their competitive ratios. Our analysis relies on a smooth surrogate of the objective function that needs to satisfy a new diminishing returns (PSD-DR) property (that its gradient is order-reversing with respect to the PSD cone). Using the representation for monotone maps on the PSD cone given by Löwner’s theorem, we obtain a convex parametrization of the family of functions satisfying PSD-DR. We then formulate a convex optimization problem to directly optimize our competitive ratio bound over this set. This design problem can be solved offline before the data start arriving. The online algorithm that uses the designed smoothing is tailored to the given cost function, and enjoys a competitive ratio at least as good as our optimized bound. We provide examples of computing the smooth surrogate for D-optimal and A-optimal experiment design, and demonstrate the performance of the custom-designed algorithm.

KW - Competitive ratio

KW - Löwner’s theorem

KW - Online algorithms

KW - Positive semidefinite cone

UR - http://www.scopus.com/inward/record.url?scp=85048374911&partnerID=8YFLogxK

U2 - 10.1007/s10107-018-1305-1

DO - 10.1007/s10107-018-1305-1

M3 - Article

AN - SCOPUS:85048374911

VL - 170

SP - 267

EP - 292

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1

ER -