Near-optimal Observation Selection using Submodular Functions

1. Near-optimal Observation Selection using Submodular Functions Andreas Krause, Carlos Guestrin Carnegie Mellon University AAAI, 2007 Presented by Haojun Chen

2. Introduction • Observation selection with constraint • Sensor placements with budget constraints (Krause et al. 2005a) • Multi-robot informative path planning (Singh et al. 2007) • Sensor placements for maximizing information at minimum communication cost (Krause et al. 2006) • … • Problems: NP-hard • Heuristic approaches applied but no performance guarantees

3. s s Key Property: Diminishing Returns A={s1,s2} s1 s2 Definition (Nemhauser et al. 1978) A real value set function F on V is called submodular if for all s1 s2 s3 s4 B = {s1,s2,s3,s4} Slides from http://www.select.cs.cmu.edu/tutorials/icml08submodularity.html

4. Maximization of Submodular Functions • Optimization problem • Still NP-hard in general, but can get approximation guarantees • Approximation guarantees for three different constraints are reviewed in this paper for some nonnegative budget B and ,where each has a fixed positive cost

5. Cardinality and Budget Constraints • Unit cost case: • Approximation guarantees: Greedy algorithm: Start with For i = 1 to k

6. 1 2 2 1.5 relay node 1 1 1 1 2 No communicationpossible! C(A) = 1 relay node C(A) = 3.5 F(A) = 3.5 2 2 Sensor Placements with Communication Constraints • Simple heuristic: Greedily optimize submodular utility function F(A) • Then add nodes to minimize communication cost C(A) 2 2 relay node F(A) = 0.2 Most informative F(A) = 4 F(A) = 4 C(A) = 3 C(A) = 10 C(A) = 10 2 Secondmost informative relay node 2 2 efficientcommunication! Not veryinformative Communication cost = Expected # of trials (learned using Gaussian Processes) Very informative, High communicationcost! Want to find optimal tradeoff between information and communication cost Slides from http://www.select.cs.cmu.edu/tutorials/icml08submodularity.html

7. 2 1 3 2 3 1 1 2 pSPIEL Algorithm • Padded Sensor Placements at Informative and cost-Effective Locations (pSPIEL): • Decompose sensing region into small, well-separated clusters • Solve cardinality constrained problem per cluster (greedy) • Combine solutions using k-minimum spanning tree (k-MST) algorithm 1 2 C1 C2 C4 C3 Slides from http://www.select.cs.cmu.edu/tutorials/icml08submodularity.html

8. Guarantees and Performance for pSPIEL • Approximation guarantee • Performance

9. Conclusions • Many natural observation selection objectives: submodular • Key algorithmic problem: Constrained maximization of submodular functions • Efficient approximation algorithms with provable quality guarantees developed by exploiting submodularity

10. Reference • Chekuri, C., and Pal, M. 2005. A recursive greedy algorithm for walks in directed graphs. In FOCS, 245–253 • Krause, A., and Guestrin, C. 2005a. A note on the budgeted maximization of submodular functions. Technical report, CMUCALD-05-103 • Krause, A.; Guestrin, C.; Gupta, A.; and Kleinberg, J. 2006. Near-optimal sensor placements: Maximizing information while minimizing communication cost. In IPSN. • Nemhauser, G.; Wolsey, L.; and Fisher, M. 1978. An analysis of the approximations for maximizing submodular set functions.Mathematical Programming 14:265–294. • Singh, A.; Krause, A.; Guestrin, C.; Kaiser, W.; and Batalin, M.2007. Efficient planning of informative paths for multiple robots.In IJCAI.