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


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)


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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


MM MUI MPI MEI

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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)


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Problem Definition(1)


• 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


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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)


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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(.)]


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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


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Greedy Algorithm


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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


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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


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


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Results (1)

CMC-MEICosines

Fuel CellReal

Rosenbrock


Results (2)

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NSCosines

Fuel CellReal

Rosenbrock


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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


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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


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


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Algorithm


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Results (5)


Greedy

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

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

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


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


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Experimental Results


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


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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


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Batch Performance (Azimi et.al NIPS 2010)

k=5

k=10


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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)


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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


51Introduction to BO Constrained BO Batch BO Scheduling Future Work

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Results (1)


Random

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Results (2)


Sequential

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Results (3)


EMAX

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Results (4)


K-means

Constrained BO: Results

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Random

Cosines

Fuel CellReal

Rosenbrock

CMC-MUI



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CN-MEI

Cosines

Fuel CellReal

Rosenbrock



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CMC-MPI(0.2)

Cosines

Fuel CellReal

Rosenbrock


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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


Bayesian Optimization with Experimental Constraints

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