A temporally abstracted Viterbi algorithm (TAV)

A temporally abstracted Viterbi algorithm (TAV)Shaunak Chatterjee and Stuart RussellUniversity of California, Berkeley

July 17, 2011

Earths history A timescale viewWidely varying timescales are pervasive in dataPlanning, simulation & state estimationMore efficient if timescale information is cleverly exploited

4.5Ga1Ma10000 yrs600 yrs1 yr2 days1 min

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State time trellist=1t=2t=3t=4t=5BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseillet=6

The Viterbi algorithm Viterbi, 196712334444BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseillet=1t=2t=3t=4t=5t=6

1233444434446688t=1t=2t=3t=4t=5t=6

BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseilleThe Viterbi algorithm Viterbi, 19671233444434446681071515171815188t=1t=2t=3t=4t=5t=6

BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseilleThe Viterbi algorithm Viterbi, 19671233444434446681071515171815188t=1t=2t=3t=4t=5t=6

BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseille1313151510111315The Viterbi algorithm Viterbi, 19671233444434446681071515171815188t=1t=2t=3t=4t=5t=6

BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseille131315151011131514151617111214151516171812131516The Viterbi algorithm Viterbi, 19671233444434446681071515171815188t=1t=2t=3t=4t=5t=6

BerkeleyPhiladelphiaMontrealTorontoBarcelonaMadridParisMarseille13131515101113151415161711121415151617181213151610The Viterbi algorithm Viterbi, 1967The Viterbi algorithm Viterbi, 1967O(N2T) by using dynamic programmingNT 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 consideredIn some cases, many of these transitions are very unlikely to feature in the optimal path

AbstractionU.S.A.CanadaSpainFranceEuropeNorth AmericaBerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseillet=1t=2t=3t=4t=5t=6

Abstraction treeAbstractness EuropeSpainBarcelonaFranceMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCFDPStep 1: Find the most likely sequence in the current state-time trellis

Coarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCFDPStep 1: Find the most likely sequence in the current state-time trellis

Step 2: Refine along the most likely sequence

CFDP RefinementNode-based refinementN.AmericaEuropeN.AmericaSpainFranceNode RefinementCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCFDPStep 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, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleCost bounds for abstract linksCost of an abstract link should be a lower bound of the link refinements it encapsulates

Standard heuristic admissibility argument CorrectnessCoarse-to-fine dynamic programming (CFDP) Raphael, 2001t = 1t = 2t = 3t = 4t = 5t = 6BerkeleyPhillyMontrealTorontoBarcelonaMadridParisMarseilleAnalyzing CFDPGreat when large portions of the state-time trellis are very unlikelyLeading to fewer refinementsAn appropriate abstraction hierarchy is required

Analyzing CFDPGreat when large portions of the state-time trellis are very unlikelyLeading to fewer refinementsAn appropriate abstraction hierarchy is required

Computation complexity Best case O(B2T (log N)3)B is the branching factor of the hierarchyWorst case O(N4T2)

An actual state trajectoryJanDecBerkeley

San FranciscoStanfordSardiniaVeniceMilanInterlakenLos Angeles road tripYosemite road tripIndia tripEurope tripMaySepPersistence a.k.a. TimescalesJanDecBerkeley

San FranciscoStanfordSardiniaVeniceMilanInterlakenLos Angeles road tripYosemite road tripIndia tripEurope tripMaySepPersistence and timescalesPeople stay within the same metropolitan area for weeksChange countries in monthsContinent changes are more rareWe do not need to consider transitions from the Bay Area to other continents and countries at every time step

A set of really good pathsSet 1: All paths within California for the entire month of AprilSet 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 |