Computational Sustainability Andreas Krause Master Class at - PowerPoint PPT Presentation

Submodular Optimization in Computational Sustainability Andreas Krause Master Class at CompSust 2012

Combinatorial optimization in computational sustainability Many applications in computational sustainability require solving large discrete optimization problems: Given finite set V wish to select subset A (subject to some constraints) maximizing utility F(A) These problems are the focus of this tutorial. 2

Wind farm Deployment [Changshui et al, Renewable Energy, 2011] 1 2 3 How should we deploy wind farms to maximize efficiency? 3

Conservation Planning [w Golovin, Converse, Gardner, Morey – AAAI ‘11 Outstanding Paper+ Which patches of land should we recommend?

Robotic monitoring of rivers and lakes [with Singh, Guestrin, Kaiser, Journal of AI Research ’ 09] Need to monitor large spatial phenomena Temperature, nutrient distribution, fluorescence, … NIMS Kaiser et.al. Can only make a limited (UCLA) number of measurements! Actual temperature Predicted temperature Use robotic sensors to cover large areas Depth Location across lake Where should we sense to get most accurate predictions? 5

Monitoring water networks [with Leskovec, Guestrin, VanBriesen, Faloutsos, J Wat Res Mgt ‘ 08] Contamination of drinking water could affect millions of people Contamination Sensors ~ $14K Place sensors to detect contaminations Where should we place sensors to quickly detect contamination? 6

Quantifying utility of sensor placements Model predicts impact of contaminations For each subset A of V compute sensing quality F(A) Model predicts Low impact High impact Contamination location Medium impact S 3 location S 1 S 2 S 3 S 4 S 2 S 1 Sensor reduces S 4 impact through Set V of all early detection! S 1 network junctions Low sensing quality F(A)=0.01 High sensing quality F(A) = 0.9 7

Sensor placement Given : finite set V of locations, sensing quality F Want : such that S 3 S 1 NP-hard! S 5 S 2 S 4 Greedy algorithm: S 6 Start with A = {} For i = 1 to k s* := argmax s F(A U {s}) A := A U {s*} How well can this simple heuristic do? 8

Performance of greedy algorithm Optimal 0.9 0.9 Population protected Population affected Population affected (higher is better) 0.8 0.8 Greedy Small subset of 0.7 0.7 Water networks data 0.6 0.6 0.5 0.5 2 4 6 8 10 2 4 6 8 10 Number of sensors placed Number of sensors placed Number of sensors placed Greedy score empirically close to optimal. Why? 9

Key property 1: Monotonicity Placement A = {S 1 , S 2 } Placement B = {S 1 , S 2 , S 3 , S 4 } S 2 S 2 S 3 S 1 S 1 S 4 F is monotonic: Adding sensors can only help 10

Key property 2: Diminishing returns Placement A = {S 1 , S 2 } Placement B = {S 1 , S 2 , S 3 , S 4 } S 2 S 2 S 3 S 1 S 1 S 4 Adding S ’ Adding S ’ S ’ will help a lot! doesn ’ t help much New New sensor Y ’ sensor S ’ + S ’ Large improvement Submodularity: A B S ’ + Small improvement 11

One reason submodularity is useful Theorem [Nemhauser et al ‘ 78] Suppose F is monotonic and submodular . Then greedy algorithm gives constant factor approximation: ~63% Greedy algorithm gives near-optimal solution! In general, guarantees best possible unless P = NP! 12

Battle of the Water Sensor Networks Competition [with Leskovec, Guestrin, VanBriesen, Faloutsos, J Wat Res Mgt 2008] Real metropolitan area network (12,527 nodes) Water flow simulator provided by EPA 3.6 million contamination events Multiple objectives: Detection time, affected population, … Place sensors that detect well “ on average ” 13

Reward function is submodular Claim: Reward function is monotonic submodular Consider event i : R i (u 2 ) R i (u k ) = benefit from sensor u k in event i R i (A) = max R i (u k ), u k  A u 2  R i is submodular u 1 Overall objective: R i (u 1 ) Event i F(A) =  Prob(i) R i (A) Submodular?? 14

Closedness properties F 1 ,…, F m submodular functions on V and  1 ,…,  m ≥ 0 Then: F(A) =  i  i F i (A) is submodular! Submodularity closed under nonnegative linear combinations! Extremely useful fact!! F  (A) submodular    P(  ) F  (A) submodular! Multicriterion optimization: F 1 ,…, F m submodular,  i > 0   i  i F i (A) submodular 15

Reward function is submodular Claim: Reward function is monotonic submodular Consider event i : R i (u 2 ) R i (u k ) = benefit from sensor u k in event i R i (A) = max R i (u k ), u k  A u 2  R i is submodular u 1 Overall objective: R i (u 1 ) Event i F(A) =  Prob(i) R i (A)  F is submodular!  Can use greedy algorithm to solve ! 16

BWSN Competition results 13 participants Performance measured in 30 different criteria G: Genetic algorithm D: Domain knowledge E: “ Exact ” method (MIP) H: Other heuristic 30 25 E H G Higher is better 20 Total Score G E 15 H G G 10 H D D 5 G 0 24% better performance than runner-up!  17

What was the trick? Simulated all on 2 weeks / 40 processors 3.6M contaminations 152 GB data on disk , 16 GB in main memory (compressed)  Very accurate computation of F(A) Very slow evaluation of F(A)  30 hours/20 sensors Lower is better Running time (minutes) 300 6 weeks for all Exhaustive search (All subsets) 30 settings  200 Naive greedy 100 0 1 2 3 4 5 6 7 8 9 10 Number of sensors selected 18

“ Lazy ” greedy algorithm [Minoux ’ 78] Lazy greedy algorithm: Benefit  (s | A)  First iteration as usual a a a  Keep an ordered list of marginal benefits  i from previous iteration b b d  Re-evaluate  i only for top b c c element  If  d d e i stays on top, use it, otherwise re-sort c e e Note: Very easy to compute online bounds, use in other algo’s , etc. [Leskovec, Krause et al. ’ 07] 19

Result of lazy evaluation Simulated all on 2 weeks / 40 processors 3.6M contaminations 152 GB data on disk , 16 GB in main memory (compressed)  Very accurate computation of F(A) Very slow evaluation of F(A)  30 hours/20 sensors Lower is better Running time (minutes) 300 6 weeks for all Exhaustive search (All subsets) 30 settings  200 Naive greedy Submodularity 100 Fast greedy to the rescue: 0 Using “ lazy evaluations ” : 1 2 3 4 5 6 7 8 9 10 1 hour/20 sensors Number of sensors selected Done after 2 days!  20

Example: Windfarm deployment [Changshui et al, Renewable Energy, 2011] Wind 1 2 3 Contribution of 2 reduced due to wake effects Total power F(A) is monotonic submodular!  21

Example: Windfarm deployment [Changshui et al, Renewable Energy, 2011] Greedy LazyGreedy Genetic Algo Power (kW) 79,585 79,585 78,850 Runtime 2.5 min 10 sec 1.6 hours 22

Other interesting directions Many sensing problems involve maximization of monotonic submodular functions Can use greedy algorithm to get near-optimal solutions! Lazy evaluations provide dramatic speedup How can we handle more complex settings: Complex constraints / complex cost functions? Sequential decisions? 23

Non-constant cost functions For each s V, let c(s)>0 be its cost (e.g., conservation cost, hardware cost, …) Cost of a set C(A) =  s  A c(s) Want to solve A* = argmax F(A) s.t. C(A) ≤ B Cost-benefit greedy algorithm: Start with A := {} ; While there is an s  V \ A s.t. C(A U {s}) ≤ B A := A U {s*} 24

Performance of cost-benefit greedy Set A F(A) C(A) Want 2   {a} max A F(A) s.t. C(A) ≤ 1 {b} 1 1 Cost-benefit greedy picks a. Then cannot afford b!  Cost-benefit greedy performs arbitrarily badly! 25

Cost-benefit optimization [Wolsey ‘82 , Sviridenko ‘04 , Krause et al ‘05+ Theorem [Krause and Guestrin ‘ 05] A CB : cost-benefit greedy solution and A UC : unit-cost greedy solution (i.e., ignore costs) Then max { F(A CB ), F(A UC ) } ≥ (1-1 /√e) OPT Can still compute online bounds and ~39% speed up using lazy evaluations Note : Can also get [Sviridenko ’ 04] (1-1/e) approximation in time O(n 4 ) (1-1/e) approximation for multiple linear constraints [Kulik ‘09+ 0.38/k approximation for k matroid and m linear constraints [Chekuri et al ‘11+ 26

Application: Conservation Planning [w Golovin, Converse, Gardner, Morey – AAAI ‘11 Outstanding Paper+ How should we select land for conservation to protect rare & endangered species? Case Study: Planned Reserve in Washington State Mazama pocket gopher streaked horned lark Taylor’s checkerspot

Problem Ingredients Land parcel details Geography: Roads, Rivers, etc Model of Species’ Population Dynamics Reproduction, Colonization, Predation, Disease, Famine, Harsh Weather, …

Population Dynamics � � Modeled using a Our Choices Dynamic Bayesian Network Protected Parcels R Time t Z ( i ) Z ( i ) ' & 1 ,t + 1 1 ,t & Environmental $% � ! "# Conditions � Z ( i ) Z ( i ) (Markovian) 2 ,t 2 ,t + 1 . . ( & . . . . & $% Z ( i ) Z ( i ) 5 ,t + 1 ! "# 5 ,t . . . . . . η η t + 1 t Time t+1