Non-myopic Informative Path Planning in Spatio-Temporal

Models

Alexandra Meliou

Andreas Krause

Carlos Guestrin

Joe Hellerstein

Collection Tours

Approximate Queries Approximate representation of the world:

Discrete locations Lossy communication Noisy measurements

Applications do not expect accurate values (tolerance to noise)

Monitored phenomena usually demonstrate strong correlationsCorrelation makes approximation cheap

Example: Return the temperature at all locations ±1C, with 95% confidence

Optimizing Information

: sensing nodes on path

Approximate answers

Search for most informative paths

Continuous Queries

Repeated at periodic intervals Finite horizon

Example: Return the temperature at all locations ±1C, with 95% confidence,

every 10 minutes for the next 5 hours.

Myopic vs Nonmyopic tradeoff

Myopic approach: repeat optimization for every timestep

Timestep 1Timestep 2

Myopic vs Nonmyopic tradeoff

Nonmyopic approach: optimize for all timesteps

Timestep 1Timestep 2 No work! Extra node

Quantify Informativeness

Entropy [Shewry & Wynn ‘87]

Mutual Information [Caselton & Zidek ‘84]

Reduction of predictive variance [Chaloner & Verdinelli ‘95]

Measuring Information

1

4

3

5

2

Observing 1 gives information on 3 and 4

Observing 2 gives information on 3 and 5

After observing 2, observing 3 becomes less useful

Diminishing Returns

Submodular Functions

)()()()( BFXBFAFXAF −∪≥−∪BA⊆

BA

X

X

+

+

More reward

Less reward

Entropy, mutual information and reduction of predictive variance are all submodular.

Non-myopic Spatio-Temporal Path Planning (NSTP)

Given: A collection of submodular functions ft

• ft only depends on data collected at times 1..t

A set of accuracy constraints kt

Find: A collection of paths Pt with

( )( ) ttt

T

ttP

kPfts

PCP

≥

= ∑=

:1

1

*

..

minarg

Minimize cost

Subject to reward constraints

Planning for multiple timesteps

Harder than planning for one

First idea : Solve an equivalent single step problem

instead!

obviously

Nonmyopic Planning Graph

t=1 t=2 t=3

A solution path on the NPG = collection of paths for multiple timesteps

Solve the single step problem

NP hard No good known approximation guarantees

Dual: Submodular Orienteering Problem

€

P* = argmaxP f P( )

s.t. C P( ) ≤ B

€

P* = argminP C P( )

s.t. f P( ) ≥ K

dual: primal:Maximize reward

Subject to budget constraints

Minimize cost

Subject to reward constraints

Good News

The dual algorithm [Chekuri & Pal ’05] provides an O(logn) factor approximation

€

f ( ˆ P ) ≥f (OPT)

logn

(where n is the size of the network)

Covering Algorithm

Transform a dual blackbox solution to a primal solution

€

P* = argmaxP f P( )

s.t. C P( ) ≤ B

€

P* = argminP C P( )

s.t. f P( ) ≥ K

dual:

primal:

Reward required to “cover”

(with α approximation factor)

Call with BOPT

Return solution with reward ≥K/α

CoveringAlgorithm

Transform a dual blackbox solution to a primal solution

Reward required to “cover”

• Call SOP for increasing budgets

• Guaranteed to cover K/α reward when called for BOPT

• Update chosen set and repeat for uncovered reward

• Terminate when ε portion left

Guaranteed to use at most budget

€

2logε

log 1−1

α

⎛

⎝ ⎜

⎞

⎠ ⎟BOPT

• Call for budget 1 : insufficient reward• Call for budget 2• Call for budget BOPT: reward sufficient!

uncovered reward

Bad News

On the unrolled graph the Chekuri-Pal guarantee becomes O(log(nT))

The running time on the unrolled graph is O((BnT)log(nT))

Addressing Computation Complexity

DP Algorithm Algorithm details in proceedings Bug in proof of guarantees. Not fixed (yet)

New algorithm: Nonmyopic Greedy Details on my webpage… Guaranteed to provide O(logn) approximation

Better than the previous O(log(nT))

Approach

Replace expensive blackbox, with cheaper blackbox

Covering transformation

Chekuri-Pal

SOP on NPG

Blackbox for dual

Nonmyopic greedy

algorithm

Blackbox for dual

More efficient:Nonmyopic greedy calls the dual on the smaller network graph instead of the unrolled

graph

Nonmyopic Greedy

Time

Bud

get

dual(b,Gt)

R = 2C = 1

R = 1C = 1

R = 1C = 1

R = 3C = 2

R = 5C = 4

R = 4C = 2

R = 3C = 2

R = 6C = 4

R = 3C = 2

R = 5C = 3

R = 4C = 3

R = 5C = 4

budget

P1

Cost = 2

Time = 2

Best greedy choice condition on A1

A2

R = 2C = 1

R = 1C = 1

R = 1C = 1

R = 2C = 2

X

R = 1C = 2

R = 1C = 2

XX X XX

P2

Cost = 1

Time = 1

R = 0C = 1

R = 0C = 1

R = 1C = 1

X X XP3

Cost = 1

Time = 3

Best ratio R/C1. Condition on picked data2. Recompute matrix

A1

Return best of A1, A2

dual(budget=4,time=1)

dual(budget=1,time=3)

For border cases were A1 is bad, A2 is guaranteed to be good

Nonmyopic Greedy Guarantees

Nonmyopic greedy Chekuri-Pal on NPG

O(B2T(nB)logn) O((nBT)log(nT))

runn

ing

tim

eap

prox

imat

ion

€

f (P) ≥1− e−1

2log nf (OPT)

€

f (P) ≥1

log(nT)f (OPT)

Myopic and Nonmyopic evaluation

Varying Constraints

Setup: 46 nodes on the Intel Berkeley Lab deployment 7 days of data (5 for learning, 2 for testing)

Cost and Runtime

Varying Horizon

Effect of greedy parameters

Varying budget levels

Conclusions Transform any blackbox solution to

nonmyopic

Obtain primal from dual

Nonmyopic greedy provides significant runtime improvements and better theoretical guarantees