Experimental constraints on fragmentation functions for strange hadrons at RHIC
Bayesian Optimization with Experimental Constraints
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Bayesian Optimization with
Experimental Constraints
Javad AzimiAdvisor: Dr. Xiaoli Fern
PhD Proposal ExamApril 2012
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Outline• Introduction to Bayesian Optimization
• Completed Works– Constrained Bayesian Optimization– Batch Bayesian Optimization– Scheduling Methods for Bayesian Optimization
• Future Works– Hybrid Bayesian optimization
• Timeline
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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
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Big Picture
Introduction to BO Constrained BO Batch BO Scheduling Future Work
Current Experiments Posterior Model Select Experiment(s)
Run Experiment(s)
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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(x1)=y1, f(x2)=y2
• Stochastic: f(x1)=N(y1,0.2) f(x2)=N(y2,5)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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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
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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 BO Constrained BO Batch BO Scheduling Future Work
MM MUI MPI MEI
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Introduction to BO Constrained BO Batch BO Scheduling Future Work
Motivating Application: Fuel Cell
Anode Cathode
bact
eria
Oxidation products (CO2)
Fuel (organic matter)
e-
e-
O2
H2OH+
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.
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Other Applications
• Financial Investment• Reinforcement Learning• Drug test• Destructive tests• And …
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Constrained Bayesian optimization(AAAI 2010, to be submitted Journal)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Problem Definition(1)
Introduction to BO Constrained BO Batch BO Scheduling Future 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
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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
Ave
rage
d A
rea
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 BO Constrained BO Batch BO Scheduling Future Work
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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 BO Constrained BO Batch BO Scheduling Future Work
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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 BO Constrained BO Batch BO Scheduling Future Work
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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– S1={1, 2, 4}, S2={1, 2, 4, 8}, (S1 is a subset of S2), g=max(.) and x=6– g(S1, x) - g(S1)=2, g(S2,x)-g(S2)=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 BO Constrained BO Batch BO Scheduling Future Work
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Greedy Algorithm
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Sequential Policies
• Having the posterior distribution of p(y|x,D) and px(.|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
Input spaceDiscretization
Level
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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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)
-Low uncertainty
-High uncertainty
-Better heuristic value
-Lower heuristic value
-Expensive -Cheap
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Cost Normalized Policy
• It selects the constrained experiment achieving the highest expected improvement per unit cost
• We report this approach for MEI policy only
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Constraint Minimum Cost Policy (CMC)• Motivation:
1. Approximately maximizes the heuristic value2. 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 1st
condition20
Cost=4 random Cost=10 random Cost=5 random
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Results (1)
CMC-MEICosines
Fuel CellReal
Rosenbrock
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Results (2)
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NSCosines
Fuel CellReal
Rosenbrock
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Batch Bayesian Optimization(NIPS 2010)
Sometimes it is better to select batch.(Javad Azimi)
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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 BO Constrained BO Batch BO Scheduling Future Work
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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 BO Constrained BO Batch BO Scheduling Future Work
Given a sequential Policy and batch size k
x11
x12
x13
x1k
.....
x21
x22
x23
x2k
.....
x31
x32
x33
x3k
.....
xn1
xn2
xn3
xnk
...... . . . . . . .
Return B*={x1,x2,…,xk}
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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
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Supermodularity
• Example: Min(.) is a supermodual function– B1={1, 2, 4}, B2={1, 2, 4, -2}, f=min(.) and x = 0 – f(B1) -f(B1, x)=1, f(B2)-f(B2, 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 BO Constrained BO Batch BO Scheduling Future Work
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Algorithm
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Results (5)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
Greedy
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Scheduling Methods for Bayesian Optimization (NIPS 2011(spotlight))
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Extended BO Model
Introduction to BO Constrained BO Batch BO Scheduling Future Work31
Problem:Schedule when to start new experiments and which ones to start
Stochastic Experiment Durations
Lab 1
Lab 2
Lab 3
Lab l
x1
x2
x3
x4 xn-1
x5 x8
xnx7
x6
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
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Challenges
Introduction to BO Constrained BO Batch BO Scheduling Future Work32
Objective 2: maximize info. used in selecting each experiments(favors minimizing concurrency)
x1 x2 ⋯ xn
We present online and offline approaches that effectively trade off these two conflicting objectives
Lab 4
x1
x2
x3
x4
x5
x6
x4
Lab 1
Lab 2
Lab 3
x7
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Objective Function• Cumulative prior experiments (CPE) of E is measured as follows:
• Example: Suppose n1=1, n2=5, n3=5, n4=2, Then CPE=(1*0)+(5*1)+(5*6)+(2*11)=57
• We found a non trivial correlation between CPE and regret
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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
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Staged Schedules• There are N stage and each stage is represent as <ni,di> such that
– CPE is calculated as: – We call an schedule uniform if |ni-nj|<2
Introduction to BO Constrained BO Batch BO Scheduling Future Work
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
x13
x14
d1 d2d3 d4
n1=4 n2=3 n4=3 n3=4
h
• Goal: finding a p-safe uniform schedule with maximum number of stages.
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Staged Schedules: Schedule
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Independent Lab (IL)• Assigns mi 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
x11Lab1
Lab2
Lab3
Lab4
x12 x13 x14
x21 x22 x23 x24
x31 x32 x33
x41 x42 x43
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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
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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
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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 BO Constrained BO Batch BO Scheduling Future Work
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Experimental Results
Introduction to BO Constrained BO Batch BO Scheduling Future Work
Setting: h=4,5,6; pd=Truncated normal distribution, n=20 and L=10
Best CPE in each setting
Best Performance
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Future Work
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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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 BO Constrained BO Batch BO Scheduling Future Work
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Batch Performance (Azimi et.al NIPS 2010)
k=5
k=10
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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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)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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First Idea(NIPS Workshop 2011)
• Based on a given prior (blue circles) and an objective function (MEI), x1 is selected
• To select the next experiment, x2 , we need, y1=f(x1) which is not available
• The statistics of the samples inside the red circle are expected to change after observing at actual y1
• We set y1 =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 x1
x1
x2
x3
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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?
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TimeLine
• Spring 2012: Finishing the Hybrid batch approach
• Summer 2012: Finding a job and final defend (hopefully )
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Publications
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And
• I would like to thank Dr. Xaioli Fern and Dr. Alan Fern
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51Introduction to BO Constrained BO Batch BO Scheduling Future Work
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Results (1)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
Random
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Results (2)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
Sequential
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Results (3)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
EMAX
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Results (4)
Introduction to BO Constrained BO Batch BO Scheduling Future Work
K-means
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Constrained BO: Results
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Random
Cosines
Fuel CellReal
Rosenbrock
CMC-MUI
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Constrained BO: Results
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CN-MEI
Cosines
Fuel CellReal
Rosenbrock
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Constrained BO: Results
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CMC-MPI(0.2)
Cosines
Fuel CellReal
Rosenbrock
Introduction to BO Constrained BO Batch BO Scheduling Future Work
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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
),( ts
),,( ts
)(stC
Introduction to BO Constrained BO Batch BO Scheduling Future Work