Nearoptimal Observation Selection using Submodular Functions Andreas Krause

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

Introduction • Observation selection with constraint – Sensor placements with budget constraints (Krause et al. 2005 a) – 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

Key Property: Diminishing Returns s 1 A={s 1, s 2} s 2 s s 1 s 3 s 2 s 4 s B = {s 1, s 2, s 3, s 4} Definition (Nemhauser et al. 1978) A real value set function F on V is called submodular if for all Slides from http: //www. select. cs. cmu. edu/tutorials/icml 08 submodularity. html

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

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

Sensor Placements with Communication Constraints • Simple heuristic: Greedily optimize submodular utility function F(A) • Then add nodes to minimize communication cost C(A) 1 Most informative 11 2 1. 5 F(A) = 0. 2 relay No communication 1 2 node possible! 1 C(A) 1 =3 2 = C(A) relay node F(A)2= 3. 5 F(A) = 4 C(A) = 10 Second C(A) 2= 3. 5 most informative efficient Communication cost = Expected # of trials (learned Gaussian Processes) Want to findusing optimal tradeoff between information and communication! very Very. Not informative, Highinformative communication cost! cost Slides from http: //www. select. cs. cmu. edu/tutorials/icml 08 submodularity. html

p. SPIEL Algorithm • Padded Sensor Placements at Informative and cost -Effective Locations (p. SPIEL): 1 2 2 1 C 1 1 3 C 2 3 C 4 2 1 2 C 3 • 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 Slides from http: //www. select. cs. cmu. edu/tutorials/icml 08 submodularity. html

Guarantees and Performance for p. SPIEL • Approximation guarantee • Performance

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

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. 2005 a. A note on the budgeted maximization of submodular functions. Technical report, CMUCALD 05 -103 • Krause, A. ; Guestrin, C. ; Gupta, A. ; and Kleinberg, J. 2006. Nearoptimal 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.