Geometry of Online Packing Linear Programs

1. Geometry of Online Packing Linear Programs Marco Molinaroand R. Ravi Carnegie Mellon University

2. Packing Integer Programs (PIPs) n • Non-negative c, A, b • Max st • A has entries in [0,1] A x b m ≤

3. Online Packing Integer Programs • Adversary chooses values for c, A, b • …but columns are presented in random order • …when column comes, set variable to 0/1 irrevocably • b and n are known upfront c A A A A A A x 1 b A 0 ≤ n

4. Online Packing Integer Programs • Goal: Find feasible solution that maximizes expected value • -competitive:

5. Previous Results • First online problem: secretary problem [Dynkin 63] • B-secretary problem (m=1, b=B, A is all 1’s) [Kleinberg 05] -competitive for • PIPs (B=min bi) [FHKMS 10, AWY] -competitive for need do not depend on n depends on n

6. Main Question and Result • Q:Do general PIPs become more difficult for larger n? • A:No! Main result Algorithm -competitive when

7. High-level Idea • Online PIP as learning • Improving learning error using tailored covering bounds • Geometry of PIPs that allow good covering bounds • Reduce general PIP to above • For this talk: • Every right-hand side • Show weaker bound

8. Online PIP as Learning • Reduction to learning a classifier[DH 09] Linear classifier: given (dual) vector , 0 0 1 1 0 1 1

9. Online PIP as Learning • Reduction to learning a classifier[DH 09] Linear classifier: given (dual) vector , Claim:If the classification 𝑥(𝑝) given by satisfies 1) 2) then 𝑥(𝑝) is (1−𝜖) optimal. Moreover, such classification always exists. [Feasible] [Packs tightly] If , then

10. Online PIP as Learning • Reduction to learning [DH 09] Linear classifier: given (dual) vector , set Claim:If the classification 𝑥(𝑝) given by satisfies 1) 2) then 𝑥(𝑝) is (1−𝜖) optimal. Moreover, such classification always exists. [Feasible] [Packs tightly] If , then

11. Online PIP as Learning • S fraction of columns • Compute appropriate for sampled IP • Use to classify remaining columns • Solving PIP via learning

12. Online PIP as Learning • S fraction of columns • Compute appropriate for sampled IP • Use to classify remaining columns • Solving PIP via learning • Probability of learning good classifier: • Consider a classifier that overfills some budget: • Can only learn if sample is skewed. Happens with probability at most • At most distinct bad classifiers • Union bounding over all bad classifiers, learn bad classifier with prob. at most • When to get good classifier with high probability

13. Online PIP as Learning • Solving PIP via learning Improve this… • Probability of learning good classification: • Consider a classification that overfills some budget: • Can only learn if sample is skewed. Happens with probability at most • At most distinct bad classifications • Union bounding over all bad classifications, learn desired good classification with prob. at least • When to get good classification with high probability

14. Improved Learning Error • Idea 1: Covering bounds via witnesses (handling multiple bad classifiers at a time) • -witness: is a +-witness of for constraint if • Columns picked by columns picked by • Total occupation of constraint by columns picked by is -witness: similar… Total weight • Lemma: Suppose there is a witness set of size . • Then probability of learning a bad classifier is

15. Geometry of PIPs with Small Witness Set • For some PIPs, size of witness set is at least • Idea 2: Consider PIPs whose columns lie on few () 1-d subspaces

16. Geometry of PIPs with Small Witness Set • For some PIPs, size of witness set is at least • Idea 2: Consider PIPs whose columns lie on few () 1-d subspaces =2 • Lemma: For such PIPs, can find witness set of size

17. Geometry of PIPs with Small Witness Set • Covering bound + witness size: it suffices • Final step: Convert any PIP into one with , loses value Algorithm -competitive when

18. Conclusion • Guarantee for online PIPs independent of number of columns • Asymptotically matches that for single constraint version [Kleinberg 05] • Ideas • Tailored covering bound based on witnesses • Analyze geometry of columns to obtain small witness set Make the learning problem more robust Open problems • Obtain optimal ? Can do if sample columns with replacement [DJSW 11] • Generalize to AdWords-type problem • Better online models: infinite horizon? less randomness?

19. Thank you!