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

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

3. GENERALIZED LOAD-BALANCING.2 0.8 0.9.3.4.8.1.7.2.4.2.1.6.5.4.1.3.4.7.2 ++ … 0

4. GENERALIZED LOAD-BALANCING.9.3.4.8.1.7.2.4.2.1.6.5.4.1.3.4.7.2 0 1 0 0.1.7.2.4

5. Captures scheduling on unrelated machines (diagonal matrices) GENERALIZED LOAD-BALANCING 0.2 0.5 0.1 0.7 0.5 0 1 0 0

6. GENERALIZED LOAD-BALANCING Random permutation model + +…

7. GENERALIZED LOAD-BALANCING Random permutation model

8. GENERALIZED LOAD-BALANCING

9. 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

10. ALGORITHM

11. Online linear optimization

12. ONLINE LINEAR OPTIMIZATION

13. ALGORITHM

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

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

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

19. Maximal Bernstein ANALYSIS (2/3): IN EXPECTATION

20. ANALYSIS (3/3): HIGH PROB.

21. ONLINE PACKING/COVERING LP

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

23. 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

24. 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

25. THANK YOU!