Dynamic Tree Block Coordinate Ascent - Semantic Scholar
Dynamic
Tree
Block
Coordinate
Ascent
Daniel Tarlow1, Dhruv Batra2 Pushmeet Kohli3, Vladimir Kolmogorov4 1: University of Toronto 2: TTI Chicago
3: Microso3 Research Cambridge 4: University College London
InternaAonal Conference on Machine Learning (ICML), 2011
MAP in Large Discrete Models • Many important problems can be expressed as a discrete Random Field (MRF, CRF) • MAP inference is a fundamental problem
min E(x) = min
x∈X
x∈X
! i∈V
θi (xi ) +
!
θij (xi , xj )
(i,j)∈E
InpainAng
Stereo
Object Class Labeling
Protein Design / Side Chain PredicAon
Primal and Dual Primal min x
!
Dual θA (xA )
A∈V∪E
≥
!
min θ˜A (xA ) =
A∈V∪E
xA
!
h∗A
A∈V∪E
‐ Dual is a lower bound: less constrained version of primal ‐ is a reparameteriza)on, determined by messages ‐ hA* is height of unary or pairwise potenAal ‐ DefiniAon of reparameterizaAon:
!
A∈V∪E
θA (xA ) =
!
A∈V∪E
θ˜A (xA )
∀{xA }
LP‐based message passing: find reparameterizaAon to maximize dual
Standard Linear Program‐based Message Passing • Max Product Linear Programming (MPLP) – Update edges in fixed order
• SequenAal Tree‐Reweighted Max Product (TRW‐S) – SequenAally iterate over variables in fixed order
• Tree Block Coordinate Ascent (TBCA) [Sontag & Jaakkola, 2009] – Update trees in fixed order
Key: these are all energy oblivious Can we do be^er by being energy aware?
Example TBCA with StaJc Schedule: 630 messages needed
TBCA with Dynamic Schedule: 276 messages needed
Benefit of Energy Awareness StaJc seMngs – Not all graph regions are equally difficult – RepeaAng computaAon on easy parts is wasteful Harder region Easy region
Dynamic seMngs (e.g., learning, search) – Small region of graph changes. – ComputaAon on unchanged part is wasteful Unchanged Changed Image
Previous OpAmum Change Mask
References and Related Work • [Elidan et al., 2006], [Su^on & McCallum, 2007] – Residual Belief PropagaAon. Pass most different messages first.
• [Chandrasekaran et al., 2007] – Works only on conAnuous variables. Very different formulaAon.
• [Batra et al., 2011] – Local Primal Dual Gap for Tightening LP relaxaAons.
• [Kolmogorov, 2006] – Weak Tree Agreement in relaAon to TRW‐S.
• [Sontag et al., 2009] – Tree Block Coordinate Descent.
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
G
x1 R G B
G‐G, B‐B
(can assume "good" has cost 0, otherwise cost 1)
G x2
G‐G, B‐B
G x3
G‐B, B‐B
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
G
x1
G‐G, B‐B "Don't be R or B" "Don't be R or B"
G x2
G‐G, B‐B "Don't be R or B" "Don't be R or B"
G x3
G‐B, B‐B "Don't be R or G" "Don't be R"
R G B HypotheJcal messages that e.g. residual max‐product would send.
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
G
x1
G‐G, B‐B
G x2
G‐G, B‐B
G x3
G‐B, B‐B
B x4
R G B But we don't need to send any messages. We are at the global opJmum. Our scores (see later slides) are 0, so we wouldn't send any messages here.
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B
B x2
G‐G, B‐B
B
G‐B, B‐B
x3
R G B Change unary potenJals (e.g., during learning or search)
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B
B x2
G‐G, B‐B
B
G‐B, B‐B
x3
R G B Locally, best assignment for some variables change.
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B "Don't be R or G" "Don't be R or G"
B x2
G‐G, B‐B "Don't be R or G" "Don't be R or G"
B x3
G‐B, B‐B "Don't be R or G" "Don't be R"
R G B HypotheJcal messages that e.g. residual max‐product would send.
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B
B x2
G‐G, B‐B
B x3
G‐B, B‐B
B x4
R G B But we don't need to send any messages. We are at the global opJmum. Our scores (see later slides) are 0, so we wouldn't send any messages here.
VisualizaJon of reparameterized energy "Good" local settings: B G‐G, B‐B B x1
G‐G, B‐B
x2
B
G‐B, B‐B
x3
B x4
Possible fix: look at how much sending messages on edge would improve dual. • Would work in above case, but incorrectly ignores e.g. the subgraph below: "Good" local settings: B x1
R G B
B‐B
G, B x2
G‐G
R, G x3
R‐R
R x4
Key Slide
B x1
B‐B
G, B x2
B‐B
R,G
R‐R
x3
Locally, everything looks opAmal
R x4
Key Slide
B x1
B‐B
G, B x2
B‐B
R,G
R‐R
x3
Try assigning a value to each variable
R x4
Key Slide Our main contribuAon
Use primal (and dual) informaJon to choose regions on which to pass messages B
B‐B
G, B
x1
x2
al
Am p o b u S
B‐B
R,G
R‐R
R
x3
x4
Am Subop
Try assigning a value to each variable
al
Our FormulaJon • Measure primal‐dual local agreement at edges and variables – Local Primal Dual Gap (LPDG). – Weak Tree Agreement (WTA).
• Choose forest with maximum disagreement – Kruskal's algorithm, possibly terminated early
• Apply TBCA update on maximal trees
Important! Minimize overhead. Use quanAAes that are already computed during inference, and carefully cache computaAons
Local Primal‐Dual Gap (LPDG) Score • Difference between primal and dual objecAves – Given primal assignment xp and dual variables (messages) defining , primal‐dual gap is Primal‐dual gap
!
θA (xpA )
A∈V∪E
−
!
A∈V∪E
primal
=
! "
A∈V∪E
min θ˜A (xA ) xA
dual
# ! p θ˜A (xA ) − min θ˜A (xA ) = LPDG (A)
Primal cost of node/edge
xA
A∈V∪E
Dual bound at node/edge
e: “local disagreement” measure: eA = LPDG (A)
Shortcoming of LPDG Score: Loose RelaxaJons
LPDG > 0, but dual opAmal
Filled circle means
, black edge means
Weak Tree Agreement (WTA) [Kolmogorov 2006] Reparameterized potenAals are said to saAsfy WTA if there exist non‐empty subsets for each node i Di ⊆ Xi such that θ˜i (xi ) = h∗i ∀xi ∈ Di min θ˜ij (xi , xj ) = h∗ij ∀xi ∈ Di , (i, j) ∈ E xj ∈Dj
Black edge means labels
labels
Filled circle means
At Weak Tree Agreement
Not at Weak Tree Agreement
Weak Tree Agreement (WTA) [Kolmogorov 2006] Reparameterized potenAals are said to saAsfy WTA if there exist non‐empty subsets for each node i Di ⊆ Xi such that θ˜i (xi ) = h∗i ∀xi ∈ Di min θ˜ij (xi , xj ) = h∗ij ∀xi ∈ Di , (i, j) ∈ E xj ∈Dj
Black edge means labels
labels
Filled circle means
At Weak Tree Agreement D1={0} D2={0,2} D2={0,2} D3={0}
Not at Weak Tree Agreement D1={0}
D2={2}
D2={0,2}
D3={0}
Weak Tree Agreement (WTA) [Kolmogorov 2006] Reparameterized potenAals are said to saAsfy WTA if there exist non‐empty subsets for each node i Di ⊆ Xi such that θ˜i (xi ) = h∗i ∀xi ∈ Di min θ˜ij (xi , xj ) = h∗ij ∀xi ∈ Di , (i, j) ∈ E xj ∈Dj
Black edge means labels
labels
Filled circle means
At Weak Tree Agreement
Not at Weak Tree Agreement D1={0}
D2={2}
D2={0,2}
D3={0}
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max(min(a,high), min(b,c)) − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max(min(a,high), min(b,c)) − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max( a, c ) − c = a − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max( a, c ) − c = a − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure: node measure
ei = max θ˜i (x i ) − min θ˜i (x i ) x i ∈Di
€
xi
Single FormulaJon of LPDG and WTA • Set a max history size parameter R. • Store most recent R labelings of variable i in label set Di R=1: LPDG score. R>1: WTA score. Combine scores into undirected edge score:
ProperJes of LPDG/WTA Scores • LPDG measure gives upper bound on possible dual improvement from passing messages on forest • LPDG may overesAmate "usefulness" of an edge e.g., on non‐ Aght relaxaAons.
LPDG > 0 WTA = 0
• WTA measure addresses overesAmate problem: is zero shortly a3er normal message passing would converge. • Both only change when messages are passed on nearby region of graph.
Experiments Computer Vision: • Stereo • Image SegmentaAon • Dynamic Image SegmentaAon Protein Design: • StaAc problem • CorrelaAon between measure and dual improvement • Dynamic search applicaAon Algorithms • TBCA: StaAc Schedule, LPDG Schedule, WTA Schedule • MPLP [Sontag and Globerson implementaAon] • TRW‐S [Kolmogorov ImplementaAon]
Experiments: Stereo
383x434 pixels, 16 labels. Po^s potenAals.
Experiments: Image SegmentaJon
(
+
+,!"
'(%&
, %*%,
!&!)( !&!) ->?@!AB5C ->?@!D-@ ->?@!?E/:2 -FD!G H@B
!&!'( !&!' ,
!""
#"" -./0,12034
$""
%""
.;2?541/@5
5678,9:;030 6HG!I JCE
%*%)+ %*%) (
!
" # $ %& -(./012345(67230897:0
%! ) '(%&
375x500 pixels, 21 labels. General potenAals based on label co‐occurence.
Experiments: Dynamic Image SegmentaJon
*
!&'#
Sheep
+,!"
, 6>?!@,.A2B 6>?!@,C7/:2B 6DEF!?6F,.A2B 6DEF!?6F,C7/:2B
!&'*
Previous Opt
Modify White Unaries
;357
Daniel Tarlow1, Dhruv Batra2 Pushmeet Kohli3, Vladimir Kolmogorov4 1: University of Toronto 2: TTI Chicago
3: Microso3 Research Cambridge 4: University College London
InternaAonal Conference on Machine Learning (ICML), 2011
MAP in Large Discrete Models • Many important problems can be expressed as a discrete Random Field (MRF, CRF) • MAP inference is a fundamental problem
min E(x) = min
x∈X
x∈X
! i∈V
θi (xi ) +
!
θij (xi , xj )
(i,j)∈E
InpainAng
Stereo
Object Class Labeling
Protein Design / Side Chain PredicAon
Primal and Dual Primal min x
!
Dual θA (xA )
A∈V∪E
≥
!
min θ˜A (xA ) =
A∈V∪E
xA
!
h∗A
A∈V∪E
‐ Dual is a lower bound: less constrained version of primal ‐ is a reparameteriza)on, determined by messages ‐ hA* is height of unary or pairwise potenAal ‐ DefiniAon of reparameterizaAon:
!
A∈V∪E
θA (xA ) =
!
A∈V∪E
θ˜A (xA )
∀{xA }
LP‐based message passing: find reparameterizaAon to maximize dual
Standard Linear Program‐based Message Passing • Max Product Linear Programming (MPLP) – Update edges in fixed order
• SequenAal Tree‐Reweighted Max Product (TRW‐S) – SequenAally iterate over variables in fixed order
• Tree Block Coordinate Ascent (TBCA) [Sontag & Jaakkola, 2009] – Update trees in fixed order
Key: these are all energy oblivious Can we do be^er by being energy aware?
Example TBCA with StaJc Schedule: 630 messages needed
TBCA with Dynamic Schedule: 276 messages needed
Benefit of Energy Awareness StaJc seMngs – Not all graph regions are equally difficult – RepeaAng computaAon on easy parts is wasteful Harder region Easy region
Dynamic seMngs (e.g., learning, search) – Small region of graph changes. – ComputaAon on unchanged part is wasteful Unchanged Changed Image
Previous OpAmum Change Mask
References and Related Work • [Elidan et al., 2006], [Su^on & McCallum, 2007] – Residual Belief PropagaAon. Pass most different messages first.
• [Chandrasekaran et al., 2007] – Works only on conAnuous variables. Very different formulaAon.
• [Batra et al., 2011] – Local Primal Dual Gap for Tightening LP relaxaAons.
• [Kolmogorov, 2006] – Weak Tree Agreement in relaAon to TRW‐S.
• [Sontag et al., 2009] – Tree Block Coordinate Descent.
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
G
x1 R G B
G‐G, B‐B
(can assume "good" has cost 0, otherwise cost 1)
G x2
G‐G, B‐B
G x3
G‐B, B‐B
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
G
x1
G‐G, B‐B "Don't be R or B" "Don't be R or B"
G x2
G‐G, B‐B "Don't be R or B" "Don't be R or B"
G x3
G‐B, B‐B "Don't be R or G" "Don't be R"
R G B HypotheJcal messages that e.g. residual max‐product would send.
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
G
x1
G‐G, B‐B
G x2
G‐G, B‐B
G x3
G‐B, B‐B
B x4
R G B But we don't need to send any messages. We are at the global opJmum. Our scores (see later slides) are 0, so we wouldn't send any messages here.
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B
B x2
G‐G, B‐B
B
G‐B, B‐B
x3
R G B Change unary potenJals (e.g., during learning or search)
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B
B x2
G‐G, B‐B
B
G‐B, B‐B
x3
R G B Locally, best assignment for some variables change.
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B "Don't be R or G" "Don't be R or G"
B x2
G‐G, B‐B "Don't be R or G" "Don't be R or G"
B x3
G‐B, B‐B "Don't be R or G" "Don't be R"
R G B HypotheJcal messages that e.g. residual max‐product would send.
B x4
VisualizaJon of reparameterized energy States for each variable: red (R), green (G), or blue (B) "Good" local settings:
B
x1
G‐G, B‐B
B x2
G‐G, B‐B
B x3
G‐B, B‐B
B x4
R G B But we don't need to send any messages. We are at the global opJmum. Our scores (see later slides) are 0, so we wouldn't send any messages here.
VisualizaJon of reparameterized energy "Good" local settings: B G‐G, B‐B B x1
G‐G, B‐B
x2
B
G‐B, B‐B
x3
B x4
Possible fix: look at how much sending messages on edge would improve dual. • Would work in above case, but incorrectly ignores e.g. the subgraph below: "Good" local settings: B x1
R G B
B‐B
G, B x2
G‐G
R, G x3
R‐R
R x4
Key Slide
B x1
B‐B
G, B x2
B‐B
R,G
R‐R
x3
Locally, everything looks opAmal
R x4
Key Slide
B x1
B‐B
G, B x2
B‐B
R,G
R‐R
x3
Try assigning a value to each variable
R x4
Key Slide Our main contribuAon
Use primal (and dual) informaJon to choose regions on which to pass messages B
B‐B
G, B
x1
x2
al
Am p o b u S
B‐B
R,G
R‐R
R
x3
x4
Am Subop
Try assigning a value to each variable
al
Our FormulaJon • Measure primal‐dual local agreement at edges and variables – Local Primal Dual Gap (LPDG). – Weak Tree Agreement (WTA).
• Choose forest with maximum disagreement – Kruskal's algorithm, possibly terminated early
• Apply TBCA update on maximal trees
Important! Minimize overhead. Use quanAAes that are already computed during inference, and carefully cache computaAons
Local Primal‐Dual Gap (LPDG) Score • Difference between primal and dual objecAves – Given primal assignment xp and dual variables (messages) defining , primal‐dual gap is Primal‐dual gap
!
θA (xpA )
A∈V∪E
−
!
A∈V∪E
primal
=
! "
A∈V∪E
min θ˜A (xA ) xA
dual
# ! p θ˜A (xA ) − min θ˜A (xA ) = LPDG (A)
Primal cost of node/edge
xA
A∈V∪E
Dual bound at node/edge
e: “local disagreement” measure: eA = LPDG (A)
Shortcoming of LPDG Score: Loose RelaxaJons
LPDG > 0, but dual opAmal
Filled circle means
, black edge means
Weak Tree Agreement (WTA) [Kolmogorov 2006] Reparameterized potenAals are said to saAsfy WTA if there exist non‐empty subsets for each node i Di ⊆ Xi such that θ˜i (xi ) = h∗i ∀xi ∈ Di min θ˜ij (xi , xj ) = h∗ij ∀xi ∈ Di , (i, j) ∈ E xj ∈Dj
Black edge means labels
labels
Filled circle means
At Weak Tree Agreement
Not at Weak Tree Agreement
Weak Tree Agreement (WTA) [Kolmogorov 2006] Reparameterized potenAals are said to saAsfy WTA if there exist non‐empty subsets for each node i Di ⊆ Xi such that θ˜i (xi ) = h∗i ∀xi ∈ Di min θ˜ij (xi , xj ) = h∗ij ∀xi ∈ Di , (i, j) ∈ E xj ∈Dj
Black edge means labels
labels
Filled circle means
At Weak Tree Agreement D1={0} D2={0,2} D2={0,2} D3={0}
Not at Weak Tree Agreement D1={0}
D2={2}
D2={0,2}
D3={0}
Weak Tree Agreement (WTA) [Kolmogorov 2006] Reparameterized potenAals are said to saAsfy WTA if there exist non‐empty subsets for each node i Di ⊆ Xi such that θ˜i (xi ) = h∗i ∀xi ∈ Di min θ˜ij (xi , xj ) = h∗ij ∀xi ∈ Di , (i, j) ∈ E xj ∈Dj
Black edge means labels
labels
Filled circle means
At Weak Tree Agreement
Not at Weak Tree Agreement D1={0}
D2={2}
D2={0,2}
D3={0}
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max(min(a,high), min(b,c)) − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max(min(a,high), min(b,c)) − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max( a, c ) − c = a − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure
D2={0,2} D3={0,2}
Costs: solid – low do^ed – medium else – high
e23 = max( a, c ) − c = a − c Filled circle means
, black edge means
WTA Score e: “local disagreement” measure: node measure
ei = max θ˜i (x i ) − min θ˜i (x i ) x i ∈Di
€
xi
Single FormulaJon of LPDG and WTA • Set a max history size parameter R. • Store most recent R labelings of variable i in label set Di R=1: LPDG score. R>1: WTA score. Combine scores into undirected edge score:
ProperJes of LPDG/WTA Scores • LPDG measure gives upper bound on possible dual improvement from passing messages on forest • LPDG may overesAmate "usefulness" of an edge e.g., on non‐ Aght relaxaAons.
LPDG > 0 WTA = 0
• WTA measure addresses overesAmate problem: is zero shortly a3er normal message passing would converge. • Both only change when messages are passed on nearby region of graph.
Experiments Computer Vision: • Stereo • Image SegmentaAon • Dynamic Image SegmentaAon Protein Design: • StaAc problem • CorrelaAon between measure and dual improvement • Dynamic search applicaAon Algorithms • TBCA: StaAc Schedule, LPDG Schedule, WTA Schedule • MPLP [Sontag and Globerson implementaAon] • TRW‐S [Kolmogorov ImplementaAon]
Experiments: Stereo
383x434 pixels, 16 labels. Po^s potenAals.
Experiments: Image SegmentaJon
(
+
+,!"
'(%&
, %*%,
!&!)( !&!) ->?@!AB5C ->?@!D-@ ->?@!?E/:2 -FD!G H@B
!&!'( !&!' ,
!""
#"" -./0,12034
$""
%""
.;2?541/@5
5678,9:;030 6HG!I JCE
%*%)+ %*%) (
!
" # $ %& -(./012345(67230897:0
%! ) '(%&
375x500 pixels, 21 labels. General potenAals based on label co‐occurence.
Experiments: Dynamic Image SegmentaJon
*
!&'#
Sheep
+,!"
, 6>?!@,.A2B 6>?!@,C7/:2B 6DEF!?6F,.A2B 6DEF!?6F,C7/:2B
!&'*
Previous Opt
Modify White Unaries
;357
