Data-Driven Optimization under Distributional Uncertainty · UTC-IASE, Apr 2017 Shuo Han Challenges and My Contribution • Challenge: Finding the worst-case distribution-Infinite-dimensional

Data-Driven Optimization under Distributional Uncertainty

Shuo Han

Postdoctoral ResearcherElectrical and Systems Engineering

University of Pennsylvania

Shuo HanUTC-IASE, Apr 2017

Internet of Things (IoT)

• A network of physical objects- Devices- Vehicles- Buildings

• Allows objects to be - Sensed and controlled- Remotely across the network

• Growing rapidly, by 2020:- 50 billion devices- 6.58 devices per person

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What IoT Brings: More Sensing (and More Data)

• Total size of dataset: 267 GB

• 1.1 billion taxi and Uber trips (2009 - 2015)

• Pick-up and drop-off dates/times, locations, distances...

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[Source: nyc.gov]

TLC: Taxi and Limousine Commission


What IoT Brings: More Control

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Connected and Autonomous VehiclesSmart Home Appliances

Wireless Traffic Light Control Smart Buildings


Smart Cities: IoT + Decision Support

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City Infrastructure

Control & Optimization Algorithm

Sensor Data

Actions


Investment in Smart Cities

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“... an infrastructure to continuously improve the collection, aggregation, and use of data to improve the life of their residents – by harnessing the growing data revolution, low-cost sensors, and research collaborations, and doing so securely to protect safety and privacy.”

The Smart Cities Initiative from the White House (Sep 2015)

TerraSwarm


TerraSwarm: Swarm at The Edge of The Cloud

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“How should we make use of data?”

“How should we send data?”

“How should we collect data?”

TerraSwarm


Research Interests

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Convex Optimization Control Theory Statistics

Theory Applications

Energy

Transportation

Research Topics

Multi-Agent Systems

Stochastic Systems

Network Dynamics


Research Overview

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Data-Driven Optimization

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[ACC13], [SIOPT15] [CDC15], [TASE16], [ICCPS17]

Pricing for Ridesharing

[Allerton14], [TAC16] [ACC17]

Privacy Solutions for Cyber-Physical Systems


Research Overview

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Data-Driven Optimization

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[ACC13], [SIOPT15] [CDC15], [TASE16], [ICCPS17]

Privacy Solutions for Cyber-Physical Systems

[Allerton14], [TAC16] [ACC17]

Pricing for Ridesharing


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Motivation: Wind Energy Integration

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conventional power plant

wind power storage devices

+

Control Action: Allocation of energy storage

How can we make use of the wind power generation data to maximally utilize wind power?

Wind Energy Data


Motivation: On-Demand Ridesharing in Cities

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How can we make use of the trip data to reduce the average wait time for passengers?

- Pick-up and drop-off times- Pick-up/drop-off locations- Travel distances

Cause: Mismatch between supply and demand

Control Action: Redistribution of empty vehicles


Probability distribution that models the stochastic phenomenon

Background: Stochastic Programming

• : Objective function

• : Decision variable- Ridesharing: Redirection of empty vehicles- Wind power integration: Allocation of storage

• : Stochastic phenomenon- Ridesharing: Future passenger demand- Wind power integration: Wind power generation

• : Probability distribution of

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minimizex

E✓⇠d

[f(x, ✓)]

f

x

✓

d ✓


Distribution is Not Always Available

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We often do not have: Instead, we have:

Question: How should these samples be used in a computationally tractable way with performance guarantees?


Using Sampled Data: Previous Methods

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Sample average approximation

minimizex

1

n

nX

i=1

f(x, ✓i

)

• Weak guarantee on performance

Robust optimization

minimize

x

max

✓2⇥f(x, ✓)

✓i

⇥

• Can be extremely conservative

Distributional Information + Uncertainty ?


Using Sampled Data: Distributional Uncertainty

• Distributional uncertainty- An ambiguity set in the space of probability distributions- No assumption on the type (continuous vs discrete,

Gaussian, uniform, ...) of distributions- Contains the true distribution with high probability- Informally: “Uncertainty of uncertainty”

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D

d(true distribution)

minimizex

E✓⇠d

[f(x, ✓)](vs. )minimize

x

max

d2DE✓⇠d

[f(x, ✓)]

• Decision making problem: Distributionally robust optimization

- Strong worst-case guarantees- Subsumes conventional robust optimization


Distributional Uncertainty

• Method 1: Based on certain (pseudo)metric - KL divergence- Wasserstein metric (earth mover’s distance)

• Metric ball centered at the empirical distribution

• The ball contains with high probability

• Advantage: “Nonparametric” characterization

• Disadvantage: Complexity of decision making against grows quickly with the number of samples

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Dd

bd

empirical distribution

d

D

(true distribution)

M

D(✏) =n

d : M(d, bd) ✏o


Distributional Uncertainty (cont’d)

• Method 2: Based on generalized moments (this talk)

• Assume: is easily bounded

• Examples- Moments: , - Tail probability:

• Classical concentration inequalities can be used to compute the probability that contains

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E[✓] = ✓̂ cov[✓] = b⌃

P(✓ � ✓̄) ✏

g

D = {d : E✓⇠d[g(✓)] � 0}

D


D d

P 1

n

nX

i=1

✓i � E✓ + t

! exp(�2nt2)Example (Hoeffding’s inequality):


Challenges and My Contribution

• Challenge: Finding the worst-case distribution- Infinite-dimensional optimization problem- Not numerically tractable

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minimize

x

max

d2DE✓⇠d

[f(x, ✓)]

• My contribution- Formulate equivalent convex optimization problem (under certain conditions)- Tractable numerical solutions- Conditions apply to many resource allocation and scheduling problems

• Previous work on special instances- [Scarf, 1958]: Analytical solution for a special case- [Bertsimas, Popescu, 2005]: Optimal probability inequalities- [Vandenberghe, Boyd, Comanor, 2007]: Optimal Chebyshev bounds- [Delage, Ye, 2010]: Piecewise affine functions


g(✓)

f(✓)

Main Result: Equivalent Convex Optimization Problem

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Theorem: There exists an equivalent convex optimization problem for computing the worst-case distribution if

f

concave

• The objective is piecewise concave

f(✓) = max

kf (k)

(✓) f (k)

g

convex

• The constraint is piecewise convex

g(✓) = minl

g(l)(✓) g(l)

Shuo Han, Molei Tao, Ufuk Topcu, Houman Owhadi, Richard M. Murray, “Convex optimal uncertainty quantification,” SIAM Journal on Optimization, 25(3), 1368–1387, 2015.


concave piecewise affine 0-1 indicator

resource allocation/schedulingresource allocation/scheduling failure rate

Piecewise Concave Functions

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0

1I(✓ � a)max

k2K{aT

k ✓ + bk}


Piecewise Convex Functions

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linear convex 0-1 indicator

mean covariance & higher moments tail probability

0

1I(✓ � a)


The Convex Optimization Problem

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maximize

{pkl,�kl}k,l

X

k,l

pklf(k)

(�kl/pkl)

subject to

X

k,l

pkl = 1

pkl � 0, 8k, lX

k,l

pklg(l)

(�kl/pkl) 0

K · L

f(✓) = max

k2{1,2,··· ,K}f (k)

(✓)

g(✓) = minl2{1,2,··· ,L}

g(l)(✓)

For:

• The worst case is always attained by a discrete distribution

• Total number of Dirac masses in the distribution:


Storage Allocation for Power Grid

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xi

power flowfij

storage

✓iwind power(stochastic)

min.

x

max

d

E✓⇠d

min

f

Wind Energy Wasted(x, ✓, f)

�Storage Allocation Problem

optimal power flow

✓piecewise concave in

| {z }


Numerical Example: IEEE 14-Bus Test Case

• Network with 5 generators

• Time: one day, 3-hour interval

• Mean and covariance obtained from real wind generation data

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0 5 10 150

10

20

30

40

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Total storage

Expe

cted

cos

t

The Influence of Information Constraints

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Exact distribution

Support Information Only

Support + Mean + Covariance

Shuo Han, Ufuk Topcu, Molei Tao, Houman Owhadi, Richard M. Murray, “Convex optimal uncertainty quantification: Algorithms and a case study in energy storage placement for power grids study,” American Control Conference, 2013.


On-Demand Ridesharing

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Dispatch Center

Vehicle

Customer

Predicted Demand

Dispatch Command

min.X1:T

TX

t=1

[JD(Xt) + JE(Xt, rt)]

Distribution of Customer Demand

Vehicle Flows Wait TimeCost of Rebalancing

Demand


Robust vs. Non-Robust

• Robust optimization against demand uncertainty

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Fei Miao, Shuo Han, Shan Lin, George J. Pappas, “Taxi dispatch under model uncertainties,” IEEE Conference on Decision and Control, 2015.

12 16 20 24 28 32 36 40 44 48 520

10

20

30

40

Cost rangeNum

ber o

f exp

erim

ents

Costs distribution of dispatch solutions

non−robust solutionsrobust solutions

Robust solution: 35.5% reduction

min.

X1:T

max

r1:T2�

TX

t=1


100

200

300

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Total

requ

ests

numb

er

Boxplot of total requests number during one hour

Region ID

NYC Dataset: 4 years, 100 GB


Conventional vs. Distributionally Robust

• Distributionally robust formulation

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 117

8

9

10x 104 Average cost comparison of DRO and RO

ϵ

Aver

age

cost

SOCBoxNRDRO

min.

X1:T

max

d2DEr⇠d

(TX

t=1


)

Fei Miao, Shuo Han, Abdeltawab Hendawi, et al., “Data-driven distributionally robust vehicle balancing with dynamic region partition,” ACM/IEEE Intl. Conf. on Cyber-Physical Systems, 2017.

(non-robust)(distr. robust)

(robust #1)(robust #2)

confidence levelNote: Confidence level not optimized for distr. robust opt.

Confidence level: Probability the true parameter/distribution lies outside ambiguity set


Future Directions

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• Distributed computation• Approximate algorithms

• Markov properties• Prior knowledge

• When to discard old data• When to re-learn

Large Datasets Structured Models Online Optimization


Summary

• Distributional uncertainty: A new approach to data-driven optimization

• A rigorous way to make use of sampled data- Probabilistic guarantees- Worst-case analysis/design: Often required for engineering applications

• Computationally efficient- Convex formulation available for a large class of problems - Examples: Resource allocation and scheduling

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Documents

Data-Driven Optimization under Distributional Uncertainty · UTC-IASE, Apr 2017 Shuo Han Challenges and My Contribution • Challenge: Finding the worst-case distribution-Infinite-dimensional