1. Bayesian Optimization with Experimental Constraints Javad Azimi Advisor: Dr. Xiaoli Fern PhD Proposal Exam April 2012 1

2. Outline Introduction to Bayesian Optimization Completed Works – Constrained Bayesian Optimization – Batch Bayesian Optimization – Scheduling Methods for Bayesian Optimization Future Works – Hybrid Bayesian optimization Timeline 2

3. Bayesian Optimization We have a black box function and we don’t know anything about its distribution We are able to sample the function but it is very expensive We are interested to find the maximizer (minimizer) of the function Assumption: – lipschitz continuity 3 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

4. Big Picture Introduction to BOConstrained BOBatch BOSchedulingFuture Work Current Experiments Posterior Model Select Experiment(s) Run Experiment(s) 4

5. Posterior Model (1): Regression approaches Simulates the unknown function distribution based on the prior – Deterministic (Classical Linear Regression,…) There is a deterministic prediction for each point x in the input space – Stochastic (Bayesian regression, Gaussian Process,…) There is a distribution over the prediction for each point x in the input space. (i.e. Normal distribution) – Example Deterministic : f(x 1 )=y 1, f(x 2 )=y 2 Stochastic: f(x 1 )=N(y 1,0.2) f(x 2 )=N(y 2,5) Introduction to BOConstrained BOBatch BOSchedulingFuture Work 5

6. Posterior Model (2): Gaussian Process Gaussian Process is used to build the posterior model – The prediction output at any point is a normal random variable – Variance is independent from observation y – The mean is a linear combination of observation y Points with high output expectation Points with high output variance 6

7. Selection Criterion Goal: Which point should be selected next to get to the maximizer of the function faster. Maximum Mean (MM) – Selects the points which has the highest output mean – Purely exploitative Maximum Upper bound Interval (MUI) – Select point with highest 95% upper confidence bound – Purely explorative approach Maximum Probability of Improvement (MPI) – It computes the probability that the output is more than (1+m) times of the best current observation, m>0. – Explorative and Exploitative Maximum Expected of Improvement (MEI) – Similar to MPI but parameter free – It simply computes the expected amount of improvement after sampling at any point Introduction to BOConstrained BOBatch BOSchedulingFuture Work MMMUIMPIMEI 7

8. 8 Introduction to BOConstrained BOBatch BOSchedulingFuture Work Motivating Application: Fuel Cell Anode Cathode bacteria Oxidation products (CO 2 ) Fuel (organic matter) e-e- e-e- O2O2 H2OH2O H+H+ This is how an MFC works SEM image of bacteria sp. on Ni nanoparticle enhanced carbon fibers. Nano-structure of anode significantly impact the electricity production. We should optimize anode nano-structure to maximize power by selecting a set of experiment.

9. Other Applications Financial Investment Reinforcement Learning Drug test Destructive tests And … Introduction to BOConstrained BOBatch BOSchedulingFuture Work 9

10. Constrained Bayesian optimization ( AAAI 2010, to be submitted Journal ) Introduction to BOConstrained BOBatch BOSchedulingFuture Work 10

11. Problem Definition(1) Introduction to BOConstrained BOBatch BOSchedulingFuture Work BO assumes that we can ask for specific experiment This is unreasonable assumption in many applications – In Fuel Cell it takes many trials to create a nano- structure with specific requested properties. – Costly to fulfill 11

12. Problem Definition(2) It is less costly to fulfill a request that specifies ranges for the nanostructure properties E.g. run an experiment with Averaged Area in range r1 and Average Circularity in range r2 We will call such requests “constrained experiments” Space of Experiments Average Circularity Averaged Area Constrained Experiment 1 large ranges low cost high uncertainty about which experiment will be run Constrained Experiment 2 small ranges high cost low uncertainty about which experiment will be run Introduction to BOConstrained BOBatch BOSchedulingFuture Work 12

13. Proposed Approach We introduced two different formulation Non Sequential – Select all experiments at the same time Sequential – Only one constraint experiment is selected at each iteration Two challenges: – How to compute heuristics for constrained experiment? – How to take experimental cost into account?(which has been ignored by most of the approaches in BO) Introduction to BOConstrained BOBatch BOSchedulingFuture Work 13

14. Non-Sequential All experiments must be chosen at the same time Objective function: – A sub set of experiments (with cost B) which jointly have the highest expected maximum is selected, i.e. E[Max(.)] Introduction to BOConstrained BOBatch BOSchedulingFuture Work 14

15. Submodularity It simply means adding an element to the smaller set provides us with more improvement than adding an element to the larger set Example: We show that max (.) is submodular – S 1 ={1, 2, 4}, S 2 ={1, 2, 4, 8}, (S 1 is a subset of S 2 ), g=max(.) and x=6 – g(S 1, x) - g(S 1 )=2, g(S 2,x)-g(S 2 )=0 E[max(.)] over a set of jointly normal random variable is a submodular function Greedy algorithm provides us with a “constant” approximation bound Introduction to BOConstrained BOBatch BOSchedulingFuture Work 15

16. Greedy Algorithm Introduction to BOConstrained BOBatch BOSchedulingFuture Work 16

17. Sequential Policies Having the posterior distribution of p(y|x,D) and p x (.|D) we can calculate the posterior of the output of each constrained experiment which has a closed form solution Therefore we can compute standard BO heuristics for constrained experiments – There are closed form solution for these heuristics 17 Input space Discretization Level Introduction to BOConstrained BOBatch BOSchedulingFuture Work

18. Budgeted Constrained We are limited with Budget B. Unfortunately heuristics will typically select the smallest and most costly constrained experiments which is not a good use of budget How can we consider the cost of each constrained experiment in making the decision? – Cost Normalized Policy (CN) – Constraint Minimum Cost Policy(CMC) 18 -Low uncertainty -High uncertainty -Better heuristic value -Lower heuristic value -Expensive-Cheap Introduction to BOConstrained BOBatch BOSchedulingFuture Work

19. Cost Normalized Policy It selects the constrained experiment achieving the highest expected improvement per unit cost We report this approach for MEI policy only 19 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

20. Constraint Minimum Cost Policy (CMC) Motivation: 1.Approximately maximizes the heuristic value 2.Has expected improvement at least as great as spending the same amount of budget on random experiments Example: Very expensive: 10 random experiments likely to be better Selected Constrained experiment Poor heuristic value: not select due to 1 st condition 20 Cost=4 random Cost=10 randomCost=5 random Introduction to BOConstrained BOBatch BOSchedulingFuture Work

21. Results (1) 21 CMC-MEI Cosines Fuel Cell Real Rosenbrock Introduction to BOConstrained BOBatch BOSchedulingFuture Work

22. Results (2) 22 NS Cosines Fuel Cell Real Rosenbrock Introduction to BOConstrained BOBatch BOSchedulingFuture Work

23. Batch Bayesian Optimization (NIPS 2010) Sometimes it is better to select batch. (Javad Azimi) 23

24. Motivation Traditional BO approach request a single experiment at each iteration This is not time efficient when running an experiment is very time consuming and there is enough facilities to run up to k experiments concurrently We would like to improve performance per unit time by selecting/running k experiments in parallel A good batch approach can speedup the experimental procedure without degrading the performance Introduction to BOConstrained BOBatch BOSchedulingFuture Work 24

25. Main Idea We Use Monte Carlo simulation to select a batch of k experiments that closely match what a good sequential policy selection in k steps Introduction to BOConstrained BOBatch BOSchedulingFuture Work 25 Given a sequential Policy and batch size k x 11 x 12 x 13 x 1k.......... x 21 x 22 x 23 x 2k.......... x 31 x 32 x 33 x 3k.......... x n1 x n2 x n3 x nk.......................... Return B*={x 1,x 2,…,x k }

26. Objective Function(1) Simulated Matching: – Having n different trajectories with length k from a given sequential policy – We want to select a batch of k experiments that best matches the behavior of the sequential policy This objective can be viewed as minimizing an upper bound on the expected performance difference between the sequential policy and the selected batch. This objective is similar to weighted k-medoid Introduction to BOConstrained BOBatch BOSchedulingFuture Work 26

27. Supermodularity Example: Min(.) is a supermodual function – B 1 ={1, 2, 4}, B 2 ={1, 2, 4, -2}, f=min(.) and x = 0 – f(B 1 ) -f(B 1, x)=1, f(B 2 )-f(B 2, x)=0 Quiz: What is the difference between submodular and supermodular function? – If the inequality is changed then we have submodular function The proposed objective function is a supermodular function The greedy algorithm provides us with an approximation bound Introduction to BOConstrained BOBatch BOSchedulingFuture Work 27

28. Algorithm Introduction to BOConstrained BOBatch BOSchedulingFuture Work 28

29. Results (5) Introduction to BOConstrained BOBatch BOSchedulingFuture Work Greedy 29

30. Scheduling Methods for Bayesian Optimization (NIPS 2011 (spotlight) ) 30

31. Extended BO Model Introduction to BOConstrained BOBatch BOSchedulingFuture Work 31 Problem: Schedule when to start new experiments and which ones to start Stochastic Experiment Durations Lab 1 Lab 2 Lab 3 Lab l x1x1 x2x2 x3x3 x4x4 x n-1 x5x5 x8x8 xnxn x7x7 x6x6 Time Horizon h We consider the following: Concurrent experiments (up to l exp. at any time) Stochastic exp. durations (known distribution p) Experiment budget (total of n experiments) Experimental time horizon h

32. Challenges Introduction to BOConstrained BOBatch BOSchedulingFuture Work 32 Objective 2 Objective 2: maximize info. used in selecting each experiments (favors minimizing concurrency) x1x1 x2x2 xnxn We present online and offline approaches that effectively trade off these two conflicting objectives Lab 4 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x4x4 Lab 1 Lab 2 Lab 3 x7x7

33. Objective Function Cumulative prior experiments (CPE) of E is measured as follows: Example: Suppose n 1 =1, n 2 =5, n 3 =5, n 4 =2, Then CPE=(1*0)+(5*1)+(5*6)+(2*11)=57 We found a non trivial correlation between CPE and regret Introduction to BOConstrained BOBatch BOSchedulingFuture Work 33

34. Offline Scheduling Assign start times to all n experiments before the experimental process begins The experiment selection is done online Two class of schedules are presented – Staged Schedules – Independent Labs Introduction to BOConstrained BOBatch BOSchedulingFuture Work 34

35. Staged Schedules There are N stage and each stage is represent as such that – CPE is calculated as: – We call an schedule uniform if | n i -n j |<2 Introduction to BOConstrained BOBatch BOSchedulingFuture Work x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x9x9 x 10 x 11 x 12 x 13 x 14 d1d1 d2d2 d3d3 d4d4 n 1 =4n 2 =3n 4 =3n 3 =4 h Goal: finding a p-safe uniform schedule with maximum number of stages. 35

36. Staged Schedules: Schedule Introduction to BOConstrained BOBatch BOSchedulingFuture Work 36

37. Independent Lab (IL) Assigns m i experiment to each lab i such that Experiments are distributed uniformly within the labs Start times of different labs are decoupled The experiments in each lab have equal duration to maximize the finishing probability within horizon h Mainly designed for policy switching schedule h x 11 Lab1 Lab2 Lab3 Lab4 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 41 x 42 x 43 Introduction to BOConstrained BOBatch BOSchedulingFuture Work 37

38. Online Schedules p -safe guarantee is fairly pessimistic and we can decrease the parallelization degree in practice Selects the start time of experiments online rather than offline More flexible than offline schedule Introduction to BOConstrained BOBatch BOSchedulingFuture Work 38

39. Baseline online Algorithms Online Fastest Completion policy (OnFCP) – Finish all of the n experiments as quickly as possible – Keeps all l labs busy as long as there are experiments left to run – Achieves the lowest possible CPE Online Minimum Eager Lab Policy (OnMEL) – OnFCP does not attempt to use the full time horizon – use only k labs, where k is the minimum number of labs required to finish n experiments with probability p Introduction to BOConstrained BOBatch BOSchedulingFuture Work 39

40. Policy Switching (PS) PS decides about the number of new experiments at each decision step Assume a set of policies or a policy generator is given The goal is defining a new policy which performs as well as or better than the best given policy at any state s The i-th policy waits to finish i experiments and then call offIL algorithm to reschedule The policy which achieves the maximum CPE is returned The CPE of the switching policy will not be much worse than the best of the policies produced by our generator Introduction to BOConstrained BOBatch BOSchedulingFuture Work 40

41. Experimental Results Introduction to BOConstrained BOBatch BOSchedulingFuture Work Setting: h=4,5,6; p d =Truncated normal distribution, n=20 and L=10 Best CPE in each setting Best Performance 41

42. Future Work 42 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

43. Traditional Approaches Sequential: – Only one experiment is selected at each iteration – Pros: Performance is optimized – Cons: Can be very costly when running one experiment takes long time Batch: – k>1 experiments are selected at each iteration – Pros: k times speed-up comparing to sequential approaches – Cons: Can not performs as well as sequential algorithms Introduction to BOConstrained BOBatch BOSchedulingFuture Work 43

44. Batch Performance (Azimi et.al NIPS 2010) 44 k=5 k=10 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

45. Hybrid Batch Sometimes, the selected points by a given sequential policy at a few consequent steps are independent from each other Size of the batch can change at each time step (Hybrid batch size) 45 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

46. First Idea (NIPS Workshop 2011) 46 Based on a given prior (blue circles) and an objective function (MEI), x 1 is selected To select the next experiment, x 2, we need, y 1 =f( x 1 ) which is not available The statistics of the samples inside the red circle are expected to change after observing at actual y 1 We set y 1 =M and then EI of the next step is upper bounded If the next selected experiment is outside of the red circle, we claim it is independent from x 1 x1x1 x2x2 x3x3 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

47. Next Very pessimistic to set Y=M and then the speedup is small Can we select the next point based on any estimation without degrading the performance? What is the distance of selected experiments in batch and the actual selected experiments by sequential policy? 47 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

48. TimeLine Spring 2012: Finishing the Hybrid batch approach Summer 2012: Finding a job and final defend (hopefully ) Introduction to BOConstrained BOBatch BOSchedulingFuture Work 48

49. Publications 49

50. And I would like to thank Dr. Xaioli Fern and Dr. Alan Fern 50 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

51. 51 Introduction to BOConstrained BOBatch BOSchedulingFuture Work

52. Results (1) Introduction to BOConstrained BOBatch BOSchedulingFuture Work Random 52

53. Results (2) Introduction to BOConstrained BOBatch BOSchedulingFuture Work Sequential 53

54. Results (3) Introduction to BOConstrained BOBatch BOSchedulingFuture Work EMAX 54

55. Results (4) Introduction to BOConstrained BOBatch BOSchedulingFuture Work K-means 55

56. Constrained BO: Results 56 Random Cosines Fuel Cell Real Rosenbrock CMC-MUI Introduction to BOConstrained BOBatch BOSchedulingFuture Work

57. Constrained BO: Results 57 CN-MEI Cosines Fuel Cell Real Rosenbrock Introduction to BOConstrained BOBatch BOSchedulingFuture Work

58. Constrained BO: Results 58 CMC-MPI(0.2) Cosines Fuel Cell Real Rosenbrock Introduction to BOConstrained BOBatch BOSchedulingFuture Work

59. PS Performance Bound is our policy generator at each time step t and state s State s is the current running experiments with their starting time and completed experiments. denotes is the policy switching result where is the base policy selected in the last step The decision by is returned by N independent simulations. is the CPE of policy with error Introduction to BOConstrained BOBatch BOSchedulingFuture Work 59