Computationally Efficient Decision Making Under Uncertainty in … · 2020-06-18 · 3 Introduction § Information-theoretic decision making – Objective: find action that minimizes

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Computationally Efficient Decision Making Under Uncertainty in High-Dimensional State Spaces

International Conference on Intelligent Robots and Systems (IROS), October 2016

Dmitry Kopitkov and Vadim Indelman

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Introduction

D. Kopitkov, V. Indelman, Computationally Efficient Decision Making Under Uncertainty in High-Dimensional State Spaces

§ Decision making under uncertainty - fundamental problem in autonomous systems and artificial intelligence

§ Examples– Belief space planning in uncertain/unknown environments

(e.g. for autonomous navigation)– Active simultaneous localization and mapping (SLAM) – Informative planning, active sensing– Sensor selection, sensor deployment– Multi-agent informative planning and active SLAM– Graph sparsification for long-term autonomy

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Introduction

§ Information-theoretic decision making– Objective: find action that minimizes an information-theoretic metric

(e.g. entropy, information gain, mutual information)– Problem types: Unfocused (entropy of all variables), and

Focused (entropy only of subset of variables)

§ Decision making over high-dimensional state spaces is expensive!

§ Evaluating action impact typically involves determinant calculation: (smaller for sparse matrices)

O(n3)


⇤ ⌘ ⌃�1 2 Rn⇥nX 2 Rn

§ Existing approaches typically calculate posterior information (covariance) matrix for each candidate action, and then its determinant

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§ High-level overview: – Avoid calculating determinants of large matrices– Re-use of calculations

– Per-action evaluation does not depend on state dimension– Yet - exact and general solution

Key idea

§ Resort to matrix determinant lemma and calculation re-use techniques for information-theoretic decision making


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Problem Formulation§ Given action and new observations , future belief is:


b [Xk+L

] = p (Xk+L

|Z0:k+L

, u0:k+L�1) / p (Xk

|Z0:k, u0:k�1)k+LY

l=k+1

p (xl

|xl�1, ul�1) p (Zl

|Xo

l

)

motion model measurement likelihood

Zk+1:k+La = uk:k+L�1

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Problem Formulation

§ Information-theoretic cost (entropy):


JH (a) = H (p (Xk+L|Z0:k+L, u0:k+L�1))

a? = argmina2A

JH (a) A .= {a1, a2, . . . , aN}§ Decision making:

§ Gaussian distributions:

b [Xk+L

] = p (Xk+L

|Z0:k+L

, u0:k+L�1) / p (Xk

|Z0:k, u0:k�1)k+LY

l=k+1

p (xl

|xl�1, ul�1) p (Zl

|Xo

l

)

⇤k+L = ⇤k +ATA

JH(a) =n

2(1 + ln (2⇡))� 1

2ln |⇤k+L|

Action Jacobian

§ Impact evaluation for a candidate action is in the general case: O(n3)

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§ Use IG instead of entropy:

Information Gain (IG) & Matrix Determinant Lemma

§ Applying matrix determinant lemma:


JIG (a).= H (b [Xk])�H (b [Xk+L])=

1

2ln

��⇤k +ATA��

|⇤k|

JIG (a) =1

2ln

��Im +A · ⌃k ·AT��

⌃k ⌘ ⇤�1k 2 Rn⇥n

A 2 Rm⇥n

m ⌧ nTypically:

Examples: sensor deployment, active SLAM

Cheap, given !⌃k

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§ Jacobian of action represents new terms in the joint pdf:

Sparse Structure of Action Jacobian

§ Illustration example:

Factors Appropriate rows in Action Jacobian


A 2 Rm⇥n a

b [Xk+L

] = p (Xk+L

|Z0:k+L

, u0:k+L�1) / p (Xk

|Z0:k, u0:k�1)k+LY

l=k+1

p (xl

|xl�1, ul�1) p (Zl

|Xo

l

)

A§ Columns of that are not involved in new terms, contain only zeros

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Using Sparsity of Action Jacobian

– is set of variables involved in – is constructed from by removing all zero columns– is prior marginal covariance of


JIG (a) =1

2ln

��Im +A · ⌃k ·AT�� JIG (a) =

1

2ln��Im +AC · ⌃M,C

k ·ATC

��

C

C

AAC

A

⌃M,Ck

§ Only few entries from the prior covariance are actually required!

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§ Key observations:– Given , calculation of action impact depends only on action Jacobian:

– Different candidate actions often share many involved variables

Re-use of Calculation

§ Re-use calculations:– Combine variables involved in all candidate actions into set – Perform one-time calculation of

• Can be calculated efficiently from square root information matrix • Depends on state dimension

– Calculate for each action, using


JIG (a) =1

2ln��Im +AC · ⌃M,C

k ·ATC

��

CAll

⌃M,CAll

k

⌃M,CAll

kJIG(a)

n

⌃M,Ck

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§ Posterior entropy over focused variables

Focused Decision Making

§ Applying IG and matrix determinant lemma of Schur complement:

– columns of related to– is inverse of


JFH (a) = H �

XFk+L

�=

nF

2(1 + ln (2⇡)) +

1

2ln

��⌃M,Fk+L

��

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Application to Sensor Deployment Problems

§ Significant time reduction in Unfocused case


Uncertainty field (dense prior information matrix)

Uncertainty field after sensors’ deployment

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Application to Sensor Deployment Problems

§ Significant time reduction in Focused case


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Application to Measurement Selection (in SLAM Context)


‘Standard’: for each action, calculate posterior sqrt information

matrix via iSAM2, then its determinant

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Conclusions

§ Decision making via matrix determinant lemma and calculation re-use– Exact (no approximations applied)– General (any measurement model)– Per-candidate complexity does not depend on – Unfocused and Focused problem formulations– Applicable to Sensor Deployment, Measurement Selection, Graph

Sparsification, and many more..

§ Multiple extensions to be investigated


Documents

Computationally Efficient Decision Making Under Uncertainty in … · 2020-06-18 · 3 Introduction § Information-theoretic decision making – Objective: find action that minimizes