Efficient Learning of Topic Ranking by Soft-projections onto - - PowerPoint PPT Presentation

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Efficient Learning of Topic Ranking by Soft-projections onto Polyhedra

Yoram Singer

Part of the work was performed at The Hebrew University, Jerusalem, Israel

AIM Workshop on the Mathematics of Ranking, Aug. 19, 2010

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Acknowledgements

• Koby Crammer, Technion

Initial framework & numerous algorithms for online ranking using dual decomposition

• Shai Shalev-Shwartz, Hebrew U.

Specialized efficient ranking algorithm, Regret bound analysis using primal-dual method

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This Workshop (thus far & today)

• Overviews of structure of ranking problems
• Foundations from economics to social choice
• Probabilistic & statistical analysis of orderings
• Models for expressing & generating orderings
• Page rank, graph-based methods, Hodge theory
• Overview of machine learning 4 ranking (MLR)
• Learning theoretic analysis of MLR (to follow)

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This Talk

• Specific ranking problem setting
• Loss minimization framework
• An efficient learning algorithm
• Brief experimental discussion
• High-level overview of formal analysis

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ECONOMICS CORPORATE / INDUSTRIAL REGULATION / POLICY MARKETS LABOUR GOVERNMENT / SOCIAL LEGAL/JUDICIAL REGULATION/POLICY SHARE LISTINGS PERFORMANCE ACCOUNTS/EARNINGS COMMENT / FORECASTS SHARE CAPITAL BONDS / DEBT ISSUES LOANS / CREDITS STRATEGY / PLANS INSOLVENCY / LIQUIDITY

Desired Ordering Document The higher minimum wage signed into law… will be welcome relief for millions of workers … . The 90- cent-an-hour increase … . Relevant topics

MARKETS LABOUR ECONOMICS REGULATION / POLICY CORPORATE / INDUSTRIAL GOVERNMENT / SOCIAL

Multi-label , Bipartite feedback

Topic Ranking

Courtesy of K. Crammer

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• Instances - vectors in

(documents, images, speech signals, …)

• Predefined set of labels

(topics, categories, phonemes)

• Target ranking of labels:

label r preferred over s ( ) iff

• Ranking functions

Topic Ranking: Setting

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Special Cases

• Binary classification:
• Multiclass categorization:
• Multiclass multilabel:

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Preference-Based View

1 5 2 4 3

3 3 2 2 1 1 1 1

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Quality of Predicted Ranking

• Loss for a pair of labels s.t.
• Measures whether we predicted the
• rder of a pair correctly, and with

sufficient confidence

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Quality of Predicted Ranking (cont.)

• Loss of f on a subset E:
• Loss of f on an entire example

predefined subset weights We will assume that each defines a bipartite graph

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Linear Ranking Functions

• Linear predictors
• “Complexity” of a predictor
• Complexity of a function set
• Can be used with Mercer kernels

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Loss Minimization & Regularization

Complexity of Ranker Empirical Risk of Ranker

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Loss  Pair-wise Constraints

• Each pair of (comparable) labels corresponds to

a margin constraint

• Slack variables distinguish between different edge-sets
• Focus on a single instance & a single edge-set
• Make use of the fact that

Loss:

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A Reduced Problem

• Current estimate of ranking functions
• “New” example
• Update ranking function
• Framework for online learning
• Iterative procedure for batch optimization

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Solving the Reduced Problem

• Direct use of Lagrange multipliers & strong duality

leads to variables since each constraint is associated with a variable.

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Reducing the Reduced…

• Introduce new |A|+|B|=k variables

B A

1 2 2 3 1

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Reduced & Compacted Dual

• More compact dual problem:
• Feasible  Feasible
• Feasible  Feasible

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Obtaining (Primal) Solution

• The new set of ranking vectors:
• But, we still need to find

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“Decoupling” the Dual

• Compact Dual
• Suppose we were given
• We can solve two independent problems

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Solving the Decoupled Problems

• We need to solve
• Optimal solution takes the form
• Can be found in linear time using an

improved & generalized (CS’99, SS’06, DSSC’09) projection technique by Bretsekas

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Finding C*

• The sets define |A|+|B|=k “knots”

Value of Dual as function of C*

• Function is piecewise quadratic
• Minimum is unique
• Can be computed in O(k)

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“Coupled” Again

• Locate the global optimum
• Check whether C>C*

Value

• f

Dual

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Recapping

• 1. Focus on a single example
• 2. Find a compact form of the dual
• 3. Decouple the reduced & compact dual
• 4. Compute “knots” for decoupled problems
• 5. Find the optimal value of
• 6. Once C* is known we use soft projection to

find α and β

• 7. Once α and β are known we find w from u

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Back to Multiple Examples

• Cycle through the examples:
• Online mode:
• Visit each example only once
• Can obtain worst case loss bound
• Batch mode:
• Visit each example multiple time an “re-project”
• Can obtain asymptotic convergence
• Generalization bounds
• SOPOPO - SOft Projection Onto POlyhedra

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Empirical Evaluation

(Batch convergence)

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0.5 1 1.5 2 x 10

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0.4 0.5 0.6 0.7 0.8 Perc PA Sop 0.5 1 1.5 2 x 10

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Perc PA Sop

Empirical Evaluation

(Online Error)

Letter (poly ker.) Letter (linear)

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Why Does it “Work” ?

(proof by picture)

Primal Objective Dual Objective Single Iterate If each > then

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Summary

• General framework for online ranking
• Specialized efficient algorithm - SOPOPO
• Ranking: rich structure, interesting, useful, …
• Primal-dual regret bound analysis
• Based on:
• “Efficient Learning of Label Ranking by Soft Projections onto Polyhedra”

Journal Of Machine Learning Research, Vol. 7, 2007

• "A Primal-Dual Perspective of Online Learning Algorithms”

Machine Learning Journal, Vol. 69, 2007

• Code: http://www.cs.huji.ac.il/~shais/code/sopopo.tgz
• See also: http://www.magicbroom.info