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A temporally abstracted Viterbi algorithm (TAV). Shaunak Chatterjee and Stuart Russell University of California, Berkeley July 17, 2011. Earth’s history – A timescale view. Widely varying timescales are pervasive in data Planning, simulation & state estimation

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a temporally abstracted viterbi algorithm tav

A temporally abstracted Viterbi algorithm (TAV)

ShaunakChatterjeeand Stuart Russell

University of California, Berkeley

July 17, 2011

earth s history a timescale view
Earth’s history – A timescale view
  • Widely varying timescales are pervasive in data
  • Planning, simulation & state estimation
    • More efficient if timescale information is cleverly exploited

4.5Ga

1Ma

10000 yrs

600 yrs

1 yr

2 days

1 min

where is shaunak

Images: berkeley.edu, wikipedia, food.com

Where is Shaunak?

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Berkeley

Berkeley

Philadelphia

Barcelona

Barcelona

Barcelona

Barcelona

Burger

Cheese steak

Paella

Gazpacho

Tapas

Gazpacho

Burger

state time trellis
State time trellis

Berkeley

Philadelphia

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 1967
The Viterbi algorithm – Viterbi, 1967

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Berkeley

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Philadelphia

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Montreal

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Toronto

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Barcelona

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Madrid

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Paris

4

Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 19671
The Viterbi algorithm – Viterbi, 1967

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Berkeley

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Philadelphia

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Montreal

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Toronto

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Barcelona

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Madrid

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Paris

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Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 19672
The Viterbi algorithm – Viterbi, 1967

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Berkeley

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Philadelphia

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Montreal

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Toronto

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Barcelona

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Paris

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Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 19673
The Viterbi algorithm – Viterbi, 1967

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Berkeley

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13

Philadelphia

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Montreal

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Toronto

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Barcelona

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Madrid

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Paris

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Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 19674
The Viterbi algorithm – Viterbi, 1967

1

3

10

13

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Berkeley

2

4

7

13

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Philadelphia

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4

15

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Montreal

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Toronto

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Barcelona

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Madrid

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Paris

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Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 19675
The Viterbi algorithm – Viterbi, 1967

1

3

10

13

14

15

Berkeley

2

4

7

13

15

16

Philadelphia

3

4

15

15

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Montreal

3

4

15

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Toronto

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Barcelona

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18

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Madrid

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Paris

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18

15

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Marseille

t=1

t=2

t=3

t=4

t=5

t=6

the viterbi algorithm viterbi 19676
The Viterbi algorithm – Viterbi, 1967
  • O(N2T) by using dynamic programming
    • NT possible state sequences
  • Used in signal decoding, speech recognition, parsing and many other applications
  • For large N and T, this cost could be quite prohibitive
  • Every possible transition is considered
    • In some cases, many of these transitions are very unlikely to feature in the optimal path
abstraction
Abstraction

Berkeley

U.S.A.

Philly

North America

Montreal

Canada

Toronto

Barcelona

Spain

Madrid

Europe

Paris

France

Marseille

t=1

t=2

t=3

t=4

t=5

t=6

coarse to fine dynamic programming cfdp raphael 2001
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

slide14
CFDP
  • Step 1: Find the most likely sequence in the current state-time trellis
coarse to fine dynamic programming cfdp raphael 20011
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

slide16
CFDP
  • Step 1: Find the most likely sequence in the current state-time trellis
  • Step 2: Refine along the most likely sequence
cfdp refinement
CFDP Refinement
  • Node-based refinement

Spain

Europe

France

Node Refinement

N.America

N.America

coarse to fine dynamic programming cfdp raphael 20012
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

coarse to fine dynamic programming cfdp raphael 20013
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

slide20
CFDP
  • Step 1: Find the most likely sequence in the current state-time trellis
  • Step 2: Refine along the most likely sequence
  • Step 3: Go to step 1 if step 2 performed any refinement; else terminate
coarse to fine dynamic programming cfdp raphael 20014
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

coarse to fine dynamic programming cfdp raphael 20015
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

coarse to fine dynamic programming cfdp raphael 20016
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

coarse to fine dynamic programming cfdp raphael 20017
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

coarse to fine dynamic programming cfdp raphael 20018
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

coarse to fine dynamic programming cfdp raphael 20019
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

cost bounds for abstract links
Cost bounds for abstract links
  • Cost of an abstract link should be a lower bound of the link refinements it encapsulates
  • Standard heuristic admissibility argument  Correctness
coarse to fine dynamic programming cfdp raphael 200110
Coarse-to-fine dynamic programming (CFDP) – Raphael, 2001

Berkeley

Philly

Montreal

Toronto

Barcelona

Madrid

Paris

Marseille

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6

analyzing cfdp
Analyzing CFDP
  • Great when large portions of the state-time trellis are very unlikely
    • Leading to fewer refinements
  • An appropriate abstraction hierarchy is required
an actual state trajectory
An actual state trajectory

Europe trip

Sardinia

Venice

Milan

Interlaken

India trip

Los Angeles road trip

Yosemite road trip

San Francisco

Berkeley

Stanford

Jan

May

Sep

Dec

persistence a k a timescales
Persistence a.k.a. Timescales

Europe trip

Sardinia

Venice

Milan

Interlaken

India trip

Los Angeles road trip

Yosemite road trip

San Francisco

Berkeley

Stanford

Jan

May

Sep

Dec

a set of really good paths
A set of really good paths
  • Set 1: All paths within California for the entire month of April
  • Set 2: All paths that visit California and at least one other state in April
  • Cost(PathsApril-in-California) < Cost(PathsApril-in-1+-states)
  • |PathsApril-in-California | << | PathsApril-in-1+-states|
  • An abstraction scheme which can distinguish between these two sets of paths!
temporally abstract link
Temporally abstract link
  • Each link encapsulates a set of paths at the specified abstraction level over a temporal interval [T1,T2]
  • Just specifying start and end points is pointless!

Europe

Europe

N.America

N.America

T2

T1

links
Links

Europe

Europe

Direct links

Paths that stay within N. America for the entire interval [T1,T2]

N.America

N.America

T2

T1

links1
Links

Europe

Europe

Cross links

Paths that start in Europe at T1 and end in N. America at T2

N.America

N.America

T2

T1

links2
Links

Europe

Europe

Re-entry links

Paths that start and end in N. America at T1 and T2 respectively, but leave N.Americaat least once in that interval

N.America

N.America

T2

T1

link refinement
Link Refinement
  • No longer refining nodes!
  • Two types of refinement
    • Direct links undergo spatial refinement
    • Cross and re-entry links undergo temporal refinement
slide38

Europe

Europe

Europe

Europe

Spatial Refinement

N.America

N.America

N.America

N.America

T1

T2

T1

T2

U.S.A.

U.S.A.

Canada

Canada

slide39

Europe

Europe

Europe

Europe

Europe

Temporal Refinement

N.America

N.America

N.America

N.America

N.America

T2

T1

T2

T1

T’

tav algorithm
TAV algorithm
  • Identical to CFDP in structure
  • Step 1: Find the most likely sequence in the current state-time trellis
  • Step 2: Refine along the most likely sequence
    • Link refinement instead of node refinement
  • Step 3: Go to step 1 if step 2 performed any refinement; else terminate
tav example
TAV example

Europe

Europe

N.America

N.America

0

T

design choices
Design choices
  • Temporal refinement
    • One link or all links
    • Splitting point
  • Computing cost bounds
    • Bound on paths within California in April
    • Tradeoff between precision and computation cost
  • Abstraction hierarchy
    • Deepvs shallow
simulation setup
Simulation setup
  • TAV works well when the system has a wide range of timescales
  • We set up a DBN with n binary variables
    • Similar to the continent, country, city example but with more levels
  • The ith variable had a timescale of (1/ε)i
results
Results
  • TAV outperforms CFDP and Viterbi for various values of T, N and ε
  • 1/ε is the timescale gap between hierarchy levels
comments
Comments
  • Much faster than CFDP and Viterbi in a system with multiple timescales
  • The speedup is a function of the range of timescales
  • Not suited for applications without timescales (persistence). In the worst case, TAV is much slower than CFDP and Viterbi
conclusion
Conclusion
  • Efficient inference algorithms can be designed for systems with a wide range of timescales
    • Conventional algorithms often cannot exploit this extra structure
  • TAV benefits significantly from considering locally constrained trajectories
    • Using such constrained local search to build a global solution was something that previous DP formulations did not do
thank you
On a lighter note:

Counter for grad students being currently tracked by TAV:

0

THANK YOU!