multiple instance learning with manifold bagsverma/papers/mil_icml_slides.pdf · take ‐home...
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Multiple Instance Learning with Manifold Bags
Boris Babenko, Nakul Verma, Piotr Dollar, Serge Belongie
ICML 2011
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Supervised Learning
• (example label) pairs provided during training(example, label) pairs provided during training
( +) ( +) ( ) ( )( , +) ( , +) ( , ‐) ( , ‐)
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Multiple Instance Learning (MIL)
• (set of examples label) pairs provided(set of examples, label) pairs provided• MIL lingo: set of examples = bag of instances
d i l b l• Learner does not see instance labels• Bag labeled positive if at least one instance in bag is positive
[Dietterich et al. ‘97]
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MIL Example: Face Detection
{ }+ { … }+Instance: image patch
{ … }+Instance: image patchInstance Label: is face?Bag: whole image
{ }‐
g gBag Label: contains face?
{ … }
[Andrews et al. ’02, Viola et al. ’05, Dollar et al. 08, Galleguillos et al. 08]
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PAC Analysis of MIL
• Bound bag generalization error in terms ofBound bag generalization error in terms of empirical error
• Data model (bottom up)• Data model (bottom up)Draw instances and their labels from fixed distributiondistributionCreate bag from instances, determine its label (max of instance labels)(max of instance labels)Return bag & bag label to learner
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Data Model
Bag 1: positiveBag 1: positive
(instance space)
N ti i t P iti i tNegative instance Positive instance
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Data Model
Bag 2: positiveBag 2: positive
N ti i t P iti i tNegative instance Positive instance
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Data Model
Bag 3: negativeBag 3: negative
N ti i t P iti i tNegative instance Positive instance
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PAC Analysis of MIL
• Blum & Kalai (1998)Blum & Kalai (1998)If: access to noise tolerant instance learner, instances drawn independentlyinstances drawn independentlyThen: bag sample complexity linear in
• Sabato & Tishby (2009)• Sabato & Tishby (2009)If: can minimize empirical error on bagsTh b l l it l ith i iThen: bag sample complexity logarithmic in
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MIL Applications
• Recently MIL has become popular in appliedRecently MIL has become popular in applied areas (vision, audio, etc)
• Disconnect between theory and many of• Disconnect between theory and many of these applications
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MIL Example: Face Detection (Images)
{ }+ { … }+
{ … } Bag: whole imageInstance: image patch+
{ }‐ { … }
[Andrews et al. ’02, Viola et al. ’05, Dollar et al. 08, Galleguillos et al. 08]
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MIL Example: Phoneme Detection (Audio)
Detecting ‘sh’ phonemeg p
+ Bag: audio of wordInstance: audio clip
“machine”
+
‐“learning”
[Mandel et al. ‘08]
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MIL Example: Event Detection (Video)
+ Bag: videoInstance: few frames
+
‐
[Ali et al. ‘08, Buehler et al. ’09, Stikic et al. ‘09]
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Observations for these applications
• Top down process: draw entire bag from a bagTop down process: draw entire bag from a bag distribution, then get instances
• Instances of a bag lie on amanifold• Instances of a bag lie on a manifold
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Manifold Bags
N ti i P iti iNegative region Positive region
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Manifold Bags
• For such problems:For such problems:Existing analysis not appropriate because number of instances is infiniteof instances is infiniteExpect sample complexity to scale with manifoldparameters (curvature, dimension, volume, etc)parameters (curvature, dimension, volume, etc)
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Manifold Bags: Formulation
• Manifold bag drawn from bag distributionManifold bag drawn from bag distribution• Instance hypotheses:
• Corresponding bag hypotheses:
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Typical Route: VC Dimension
• Error Bound:Error Bound:
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Typical Route: VC Dimension
• Error Bound:Error Bound:
generalization error # of training bags
empirical error
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Typical Route: VC Dimension
• Error Bound:Error Bound:
VC Dimension of bag hypothesis class
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Relating to
• We do have a handle onWe do have a handle on• For finite sized bags, Sabato & Tishby:
• Question: can we assume manifold bags are smooth and use a covering argument?
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VC of bag hypotheses is unbounded!
• Let be half spaces (hyperplanes)Let be half spaces (hyperplanes)• For arbitrarily smooth bags can always construct any number of bags s t all possibleconstruct any number of bags s.t. all possiblelabelings achieved byTh b d d!• Thus, unbounded!
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Example (3 bags)
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Example (3 bags)
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Example (3 bags)
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Example (3 bags)
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Example (3 bags)
Want labeling (101)Want labeling (101)
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Example (3 bags)
+–
Achieves labeling (101)Achieves labeling (101)
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Example (3 bags)
(011)(100)
(101)
(110)
(010)
(001)(110)
(111) ( )(111) (000)
All possible labelingsAll possible labelings
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Issue
• Bag hypothesis class too powerfulBag hypothesis class too powerfulFor positive bag, need to only classify 1 instance as positiveas positiveInfinitely many instances ‐> too much flexibility for bag hypothesisbag hypothesis
• Would like to ensure a non‐negligible portion of positive bags is labeled positiveof positive bags is labeled positive
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Solution
• Switch to real‐valued hypothesis classSwitch to real valued hypothesis class
corresponding bag hypothesis also real valuedcorresponding bag hypothesis also real‐valuedbinary label via thresholdingt l b l till bitrue labels still binary
• Require that is (lipschitz) smooth• Incorporate a notion of margin
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Example (3 bags)
small margin
+–
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Fat‐shattering Dimension
• = “Fat‐shattering” dimension of real‐= Fat shattering dimension of realvalued hypothesis class
Analogous to VC dimension
[Anthony & Bartlett ‘99]
Analogous to VC dimension
• Relates generalization error to empirical error at marginat margin
i.e. not only does binary label have to be correct, margin has be tomargin has be to
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Fat‐shattering of Manifold Bags
• Error Bound:Error Bound:
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Fat‐shattering of Manifold Bags
• Error Bound:Error Bound:
generalization error # of training bags
empirical error at margin
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Fat‐shattering of Manifold Bags
• Error Bound:Error Bound:
fat shattering of bag hypothesis class
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Fat‐shattering of Manifold Bags
• Bound in terms ofBound in terms ofUse covering arguments – approximate manifold with finite number of pointswith finite number of pointsAnalogous to Sabato & Tishby’s analysis of finite size bagssize bags
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Error Bound
• With high probability:With high probability:
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Error Bound
• With high probability:With high probability:
generalization error complexity term
empirical error at margin
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Error Bound
• With high probability:With high probability:
fat shattering of instance hypothesis class
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Error Bound
• With high probability:With high probability:
number of training bags
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Error Bound
• With high probability:With high probability:
manifold dimension
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Error Bound
• With high probability:With high probability:
manifold volume
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Error Bound
• With high probability:With high probability:
term depends (inversely) on smoothness of manifolds &smoothness of instance hypothesis class
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Error Bound
• With high probability:With high probability:
• Obvious strategy for learner:Obvious strategy for learner:Minimize empirical error & maximize marginThis is what most MIL algorithms already doThis is what most MIL algorithms already do
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Learning from Queried Instances
• Previous result assumes learner has accessPrevious result assumes learner has access entire manifold bag
• In practice learner will only access smallIn practice learner will only access small number of instances ( )
{ … }• Not enough instances ‐> might not draw a pos. instance from pos baginstance from pos. bag
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Learning from Queried Instances
• BoundBound
holds with failure probability increased by if
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Take‐home Message
• Increasing reduces complexity termIncreasing reduces complexity term• Increasing reduces failure probability
Seems to contradict previous results (smaller bagSeems to contradict previous results (smaller bag size is better)Important difference between and ! If is too small we may only get negative instances from a positive bag
• Increasing requires extra labels, increasing does not
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Iterative Querying Heuristic (IQH)
• Problem: want many instances/bag, but haveProblem: want many instances/bag, but have computational limits
• Heuristic solution:Heuristic solution:Grab small number of instances/bag, run standard MIL algorithmQuery more instances from each bag, only keep the ones that get high score from current classifier
• At each iteration, train with small # of instances
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Experiments
• Synthetic Data (will skip in interest of time)Synthetic Data (will skip in interest of time)• Real Data
INRIA H d (i )INRIA Heads (images)TIMIT Phonemes (audio)
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INRIA Heads
pad=16 pad=32
[Dalal et al. ‘05]
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TIMIT Phonemes
+“machine”
+
‐“learning”
[Garofolo et al., ‘93]
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Padding (volume)
INRIA Heads TIMIT Phonemes
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Number of Instances ( )
INRIA Heads TIMIT Phonemes
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Number of Iterations (heuristic)
INRIA Heads TIMIT Phonemes
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Conclusion
• For many MIL problems bags modeled betterFor many MIL problems, bags modeled better as manifolds
• PAC Bounds depend onmanifold properties• PAC Bounds depend on manifold properties• Need many instances per manifold bag• Iterative approach works well in practice, while keeping comp. requirements low
• Further algorithmic development taking advantage of manifold would be interestingg g
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Thanks
• Happy to take questions!Happy to take questions!
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Why not learn directly over bags?
• Some MIL approaches do thisSome MIL approaches do thisWang & Zucker ‘00, Gartner et al. ‘02
• In practice instance classifier is desirable• In practice, instance classifier is desirable• Consider image application (face detection)
Face can be anywhere in imageNeed features that are extremely robust
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Why not instance error?
• Consider this example:Consider this example:
• In practice instance error tends to be low (if bag error is low)
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Doesn’t VC have lower bound?
• Subtle issue with FAT boundsSubtle issue with FAT boundsIf the distribution is terrible, will be high
• Consider SVMs with RBF kernel• Consider SVMs with RBF kernelVC dimension of linear separator is n+1
(FAT dimension only depends on margin (Bartlett & Shawe‐Taylor, 02)
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Aren’t there finite number of image patches?
• We aremodeling the data as a manifoldWe are modeling the data as a manifold• In practice, everything gets discretized
l b f i ( i• Actual number of instances (e.g. image patches with any scale/orientation) may be h i i b d ill ihuge – existing bounds still not appropriate