Deep Inverse Optimization Yingcong Tan 1 , Andrew Delong 2 , Daria - - PowerPoint PPT Presentation

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Deep Inverse Optimization Yingcong Tan 1 , Andrew Delong 2 , Daria - - PowerPoint PPT Presentation

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CPAIOR 2019 Deep Inverse Optimization Yingcong Tan 1 , Andrew Delong 2 , Daria Terekhov 1 1 Department of Mechanical, Industrial and Aerospace Engineering Concordia University 2 Department of Computer Science and Software Engineering Concordia

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CPAIOR 2019

Deep Inverse Optimization

Yingcong Tan 1, Andrew Delong 2, Daria Terekhov 1

1Department of Mechanical, Industrial and Aerospace Engineering

Concordia University

2Department of Computer Science and Software Engineering

Concordia University

Thursday, June 6th, 2019

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Agenda

i. Motivation ii. Methodology iii. Experiments iv. Summary

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 2/35

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What is Inverse Optimization (IO)?

Forward Optimization Problem c minx c′x s.t. Ax ≤ b

Target

c

Target

minc Target, x s.t. x ∈ arg minx{c′x|Ax ≤ b}

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 3/35

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What is Inverse Optimization (IO)?

Forward Optimization Problem c x∗ minx c′x s.t. Ax ≤ b c

Target

minc Target, x s.t. x ∈ arg minx{c′x|Ax ≤ b}

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 4/35

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What is Inverse Optimization (IO)?

Forward Optimization Problem Inverse Optimization Problem c x∗ minx c′x s.t. Ax ≤ b c

Target

minc Target, x s.t. x ∈ arg minx{c′x|Ax ≤ b}

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 5/35

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What is Inverse Optimization (IO)?

Forward Optimization Problem Inverse Optimization Problem c x∗ minx c′x s.t. Ax ≤ b c

Target

x minc Target, x s.t. x ∈ arg minx{c′x|Ax ≤ b}

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 6/35

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Motivation

Routing Problem (i.e., Least Cost Path) Objective Learn the arc cost Production Planning Problem Objective Estimate backorder cost Customer Behavior Objective Estimate customer utility function

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 7/35

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Contribution

Existing algorithms

Highlights Optimization formulations based on optimality conditions Guarantee optimal solution Limitation Algorithms are tailored to solve special cases of IO problems

*Chan et al. [2–4], Troutt et al.[6, 7], Aswani et al. [1], Saez-Gallego and Morales [5]

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 8/35

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Contribution

Existing algorithms

Highlights Optimization formulations based on optimality conditions Guarantee optimal solution Limitation Algorithms are tailored to solve special cases of IO problems

*Chan et al. [2–4], Troutt et al.[6, 7], Aswani et al. [1], Saez-Gallego and Morales [5]

Deep Inverse Optimization

Highlights First deep-learning based approach Learn parameters through backpropogation Generally applicable to different IO problems Limitation Doesn’t guarantee optimal solution

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 9/35

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Agenda

i. Motivation ii. Methodology iii. Experiments iv. Summary

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 10/35

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Methodology

(IO): Find a Cost Vector Consistent With Target

Target

c

Discrepancy ∆c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 11/35

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Methodology

Solve FOP using Interior-Point Method (IPM)

Target

c x∗

Discrepancy ∆c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 12/35

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Methodology

Observe Discrepancy and Compute Gradients

Target

c x∗

Discrepancy ∆c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 13/35

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Methodology

Observe Discrepancy and Compute Gradients

Target

c x∗

Discrepancy ∆c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 14/35

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Methodology

Termination

Target

c x∗

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 15/35

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Methodology

Deep Inverse Optimization Unroll the IPM Unroll a Deep RNN

x(0) = features x(1) = RNN(x(0), weights) x(2) = RNN(x(1), weights) x(n) = RNN(x(n−1), weights)

minweights Loss(Target, x(n)) Dynamic Num. of Steps Backpropogation

∂ Loss ∂ weights

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 16/35

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Methodology

Deep Inverse Optimization Unroll the IPM

c, A, b = DefineLP(features, weights) x(0) = FindFeasible(c, A, b) x(1) = Newton(x(0), c, A, b) x(2) = Newton(x(1), c, A, b) x(n) = Newton(x(n−1), c, A, b)

Dynamic Num. of Steps minweights Loss(Target, x(n)) Unroll a Deep RNN

x(0) = features x(1) = RNN(x(0), weights) x(2) = RNN(x(1), weights) x(n) = RNN(x(n−1), weights)

minweights Loss(Target, x(n))

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 17/35

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Agenda

i. Motivation ii. Methodology iii. Experiments iv. Summary

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 18/35

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Experiments on Three Learning Tasks

Task 1 Single-point non-parametric LP Goal Learn cost vector – Closed-form solution proposed by Chan et al. [2, 4]

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 19/35

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Experiments on Three Learning Tasks

Task 1 Single-point non-parametric LP Goal Learn cost vector – Closed-form solution proposed by Chan et al. [2, 4] Task 2 Single-point non-parametric LP Goal Learn cost vector and constraints jointly – Maximum likelihood estimation approach proposed by Troutt et al. [6]

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 20/35

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Experiments on Three Learning Tasks

Task 1 Single-point non-parametric LP Goal Learn cost vector – Closed-form solution proposed by Chan et al. [2, 4] Task 2 Single-point non-parametric LP Goal Learn cost vector and constraints jointly – Maximum likelihood estimation apprach proposed by Troutt et al. [6] Task 3 Multi-point parametric LP, i.e., c, A, b = f (features, weights) Goal Learn weights – Not addressed in literature

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 21/35

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Experiment on Task 1

Goal Learn cost vector consistent with a single observed target X ∗ Target c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 22/35

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Experiment on Task 1

Goal Learn cost vector consistent with a single observed target X ∗ Target X ∗ c c Test on 300 random LP instances

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 23/35

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Experiment on Task 1

N = 10 Variables M = 20 Constraints Before Learning After Learning

X ∗ Target X ∗ c c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 24/35

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Squared Error (Learned vs Target)

N = 2 , M = 4 N = 2 , M = 8 N = 2 , M = 1 6 N = 1 , M = 2 N = 1 , M = 3 6 N = 1 , M = 8 10

11

10

9

10

7

10

5

10

3

10

1

101

loss (squared error) Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 25/35

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Experiment on Task 2

Goal Learn cost vector and constraints consistent with a single

  • bserved target

X Target c

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 26/35

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Experiment on Task 2

Goal Learn cost vector and constraints consistent with a single

  • bserved target

X Target X c c Test on 300 random LP instances

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 27/35

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Squared Error (Learned vs Target)

N = 2 , M = 4 N = 2 , M = 8 N = 2 , M = 1 6 N = 1 , M = 2 N = 1 , M = 3 6 N = 1 , M = 8 10

11

10

9

10

7

10

5

10

3

10

1

101

loss (squared error) Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 28/35

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Experiment on Task 3

Goal Learn weights such that decisions are consistent with observed targets across multiple conditions

TARGET DISCREPANCY

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 29/35

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Experiment on Task 3

Goal Learn weights such that decisions are consistent with observed targets across multiple conditions

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 30/35

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Squared Error (Learned vs Target)

TARGET Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 31/35

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Squared Error (Learned vs Target)

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 32/35

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Agenda

i. Motivation ii. Methodology iii. Experiments iv. Summary

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 33/35

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Summary

General-purpose framework for solving IO problems

Solves parametric or non-parametric problems Learns all parameters individually or jointly Easily extends to non-linear problems

Deep-Inv-Opt package is now available on https://github.com/tankconcordia/deep inv opt

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 34/35

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Thank you !

Yingcong Tan, Andrew Delong, Daria Terekhov Deep Inverse Optimization 35/35