How the Experts Algorithm Can Help Solve LPs Online Marco Molinaro TU Delft Anupam Gupta Carnegie Mellon University.

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Presentation transcript:

How the Experts Algorithm Can Help Solve LPs Online Marco Molinaro TU Delft Anupam Gupta Carnegie Mellon University

Applications: (optimal) gen load-balancing, packing/covering LPs Primal-dual algo for online random order problems using black-box online learning to compute duals

GENERALIZED LOAD-BALANCING … 0

GENERALIZED LOAD-BALANCING

Captures scheduling on unrelated machines (diagonal matrices) GENERALIZED LOAD-BALANCING

GENERALIZED LOAD-BALANCING Random permutation model + +…

GENERALIZED LOAD-BALANCING Random permutation model

GENERALIZED LOAD-BALANCING

Primal-dual, using black-box online linear optimization for dual Abstracts exponential update of Devanur et al., explains why works Abstraction allow us handle dependencies in random permutation GENERALIZED LOAD-BALANCING

ALGORITHM

Online linear optimization

ONLINE LINEAR OPTIMIZATION

ALGORITHM

ANALYSIS (1/3) (dual) guarantee of online lin optimization (primal) greedy wrt duals

ANALYSIS (2/3): IN EXPECTATION Uses a maximal Bernstein inequality to take care of all time steps in iid

ANALYSIS (2/3): IN EXPECTATION Uses a maximal Bernstein inequality to take care of all time steps in iid

Maximal Bernstein ANALYSIS (2/3): IN EXPECTATION

ANALYSIS (3/3): HIGH PROB.

ONLINE PACKING/COVERING LP

Optimal guarantee for packing (indep Kesselheim et al. 14, Devanur-Agrawal 15) First general result for packing/covering (but requires technical assumption)

Idea: reduce online LP to gen load-balancing Elements – Handle slightly negative loads in gen load balancing (well-bounded instances) – Simple reduction to gen load balancing assuming knows OPT – Estimate OPT: pick out very valuable items, sampling + chernoff on rest Cannot “scale down” solution to get feasibility – Crucially used in Kesselheim et al. 14, Devanur-Agrawal 15… ONLINE PACKING/COVERING LP

Solving random order problems using duals from black-box online linear optimization Clean abstraction, allows to handle dependencies in random perm. – Separates “optimization” and “probability” parts Applications – Generalized load-balancing – (optimal) guarantees for packing/covering LPs Open questions 1.Seems very flexible. Apply techniques to other problems? 2.More general, realistic models 3.Remove technical assumption in packing/covering, or prove LB (minimax?) CONCLUSION

THANK YOU!