Learning to Predict Combinatorial Structures Shankar Vembu Joint work with Thomas G

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Learning to Predict Combinatorial
StructuresShankar VembuJoint work with Thomas
Gärtner
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Learning Structured Prediction Models

• Exact vs. Approximate inference

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Learning Structured Prediction Models

• Is it possible to learn structured prediction
  models without using any inference algorithm?

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Learning Structured Prediction Models

• Is it possible to learn structured prediction
  models without using any inference algorithm?
• Yes For combinatorial structures

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Combinatorial Structures

• Partially ordered sets
• Permutations (label ranking)
• Directed cycles
• Graphs
• Multiclass, multilabel, ordinal, hierarchical
  classification

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Outline

• Structured prediction and limitations of existing
  models
• Training combinatorial structures
• Constructing combinatorial structures
• Application settings

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Structured Prediction

• Input and output spaces
• Training data
• Joint scoring function
• Prediction

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Discriminative Structured Prediction
exponential number of constraints ?
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Assumptions

• Decoding (polynomial time)
• given h, x find
• Separation (polynomial time)
• given h, x, y find any
  or prove that none exists
• Optimality (in NP)
• given h, x, y decide

decoding is the strongest assumption and
optimality is the weakest
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Optimality vs. Non-optimality

• optimality
• given h, x, y decide
• non-optimality
• given h, x, y decide

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Dicycles - Optimality vs. Non-optimality

• optimality - is there no longer cycle?
• given h, x, y decide
• non-optimality - is there any longer cycle?
• given h, x, y decide

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Dicycles - Optimality vs. Non-optimality

• optimality - is there no longer cycle?
• given h, x, y decide
• non-optimality - is there any longer cycle?
• given h, x, y decide

Proposition The non-optimality problem for
directed cycles is NP-complete
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Optimality vs. Non-optimality
Optimality
we are interested mostly in problems where
non-optimality is NP-complete
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Training Combinatorial Structures
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Loss Functions

• AUC-loss

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Loss Functions

• AUC-loss
• Exponential loss

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Loss Functions

• AUC-loss
• Exponential loss
• 2nd-order Taylor expansion at 0

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Regularised Risk Minimisation

• Assumption 1 Tensor product of Hilbert spaces
• representer theorem
• Assumption 2
  has bases

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Regularised Risk Minimisation
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Optimisation with Finite Output Embedding
Let Using the canonical orthonormal bases of
, optimise
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Recipe for Training Combinatorial Structures
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Recipe for Training Combinatorial Structures

• Finite dimensional output embedding

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Recipe for Training Combinatorial Structures

• Finite dimensional output embedding

• Polynomial-time computation of

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Recipe for Training Combinatorial Structures

• Finite dimensional output embedding

• Polynomial-time computation of

• Polynomially-sized unconstrained quadratic program

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Constructing Combinatorial Structures
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Approximation Algorithms

• Decoding
• given h, x find
• Approximate decoding
• given h, x find any

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Approximation Measure

• Maximisation problem, Approximation factor 0.65

c(s)
c(s_min)
c(s_max)
0
1
0.65
0.5
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z-approximation

z-approximation is better suited when negative
solutions are possible. It is invariant to (i)
constant offsets, (ii) changing from max to
min, and (iii) using the complement of binary
variables
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Decoding Sibling Systems

• Consider a set system
• with a sibling function and
• an output map such that

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Decoding Sibling Systems

• Consider a set system
• with a sibling function and
• an output map such that

Theorem There is a 1/2-factor z-approximation
algorithm for decoding sibling systems
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Decoding Independence Systems

• Consider a set system
• with

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Decoding Independence Systems

• Consider a set system
• with
• Theorem There is a
  factor z-approximation algorithm for decoding
  independence systems

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Application settings
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Multiclass
Example
Decoding is trivial
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Multilabel
Decoding
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Dicycles

• Digraphs
• (-1,0,1) adjacency matrix

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Dicycles
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and others

• Ordinal regression
• Hierarchical classification
• Partially ordered sets
• Permutations
• Graphs

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Experiments
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Multilabel Classification

• Yeast dataset 1500 training, 917 test, 14
  labels
• Comparison with multi-label SVM

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Hierarchical Classification

• WIPO-alpha data set 1352 training, 358 test
• Number of nodes -188, max. depth - 3

LOSS SVM H-SVM H-RLS H-M3 Hamming H-M3 Tree CSOP
0-1 87.2 76.2 72.1 70.9 65 51.1
Hamming 1.84 1.74 1.69 1.67 1.73 1.84
Tree 0.053 0.051 0.05 0.05 0.048 0.046
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Dicycle Policy Estimation

• Artificial setting
• Predicting the cyclic tour of different people
• Hidden policy for each person s/he takes the
  route that
• maximises reward
• Goal is to estimate the hidden policy

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Dicycle Policy Estimation

• Comparision with SVM-Struct using approximate
  inference

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Ongoing Work

• Probabilistic models
• Negative log likelihood as loss
  function
• Sampling techniques for combinatorial structures
• Markov chain Monte Carlo methods
• Provable guarantees for mixing time

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• THANKS!