lti Is convex Perceptron Boosting Max-Margin Conditional - - PowerPoint PPT Presentation

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Softmax-Margin CRFs: Training Log-Linear Models with Cost Functions

Kevin Gimpel and Noah A. Smith

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Risk Perceptron Minimum Error Rate Training Conditional Likelihood Max-Margin MIRA Boosting

Uses a cost function Is convex Based on probabilistic inference

Latent Variable Conditional Likelihood

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Risk Perceptron Minimum Error Rate Training Conditional Likelihood Max-Margin MIRA Boosting

Uses a cost function Is convex

Latent Variable Conditional Likelihood Softmax-Margin

Based on probabilistic inference

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Risk Perceptron Minimum Error Rate Training Conditional Likelihood Max-Margin MIRA Boosting

Uses a cost function Is convex

Latent Variable Conditional Likelihood Softmax-Margin Jensen Risk Bound

Based on probabilistic inference

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Linear Models for Structured Prediction

θ| {θ⊤f }

• ′∈ {θ⊤f ′}

For probabilistic interpretation, exponentiate and normalize:

weights features

• ∈

θ⊤f

input

• utput

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• θ
• −θ⊤f
• Training
• θ
• 

−θ⊤f  

Standard approach is to maximize conditional likelihood: Another approach maximizes margin (Taskar et al., 2003):

• ∈

{ } θ⊤f

• ∈
• θ⊤f

task-specific cost function

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• θ
• −θ⊤f
• Training
• θ
• 

−θ⊤f  

Standard approach is to maximize conditional likelihood: Another approach maximizes margin (Taskar et al., 2003):

• ∈

{ } θ⊤f

• ∈
• θ⊤f

cost-augmented decoding

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• θ
• −θ⊤f
• Training
• θ
• 

−θ⊤f  

Standard approach is to maximize conditional likelihood: Another approach maximizes margin (Taskar et al., 2003):

• θ
• 

−θ⊤f  

Softmax-margin: replace “max” with “softmax”

• ∈

{ } θ⊤f

• ∈
• θ⊤f
• ∈

{ }

θ⊤f

“cost-augmented summing”

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• θ
• −θ⊤f
• Training
• θ
• 

−θ⊤f  

Standard approach is to maximize conditional likelihood: Another approach maximizes margin (Taskar et al., 2003):

• θ
• 

−θ⊤f  

Softmax-margin: replace “max” with “softmax”

• ∈

{ } θ⊤f

• ∈
• θ⊤f
• ∈

{ }

θ⊤f

Sha and Saul (2006), Povey et al. (2008)

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Has a probabilistic interpretation in the minimum

divergence framework (Jelinek, 1997)

Details in technical report

Is a bound on:

Max-margin Conditional likelihood Risk

Properties of Softmax-Margin

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Has a probabilistic interpretation in the minimum

divergence framework (Jelinek, 1997)

Details in technical report

Is a bound on:

Max-margin (because “softmax” bounds “max”)

• Conditional likelihood

Risk

Properties of Softmax-Margin

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Risk?

• θ
• θ|

Risk is the expected value of the cost function

(Smith and Eisner, 2006; Li and Eisner, 2009):

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• 

−θ⊤f

• ∈

{θ⊤f }  

• −θ⊤f
• { }

Conditional likelihood Bound on risk via Jensen’s inequality

Softmax-margin:

Bounding Conditional Likelihood and Risk

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• 

−θ⊤f

• ∈

{θ⊤f }  

• −θ⊤f
• { }

Conditional likelihood Bound on risk via Jensen’s inequality

Softmax-margin:

Bounding Conditional Likelihood and Risk

Softmax-margin is a convex bound on max-margin, conditional likelihood, and risk

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• 

−θ⊤f

• ∈

{θ⊤f }  

• −θ⊤f
• { }

Conditional likelihood

Softmax-margin:

Bounding Conditional Likelihood and Risk

Bound on risk via Jensen’s inequality

Jensen Risk Bound Easier to optimize than risk (cf. Li and Eisner, 2009)

• −θ⊤f

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Implementation

Conditional likelihood → Softmax-margin

If cost function factors the same way as the

features, it’s easy:

Add additional features for the cost function Keep their weights fixed

If not, use a simpler cost function or use

approximate inference

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Experiments

English named-entity recognition (CoNLL 2003) Compared softmax-margin and Jensen risk bound with five

baselines:

Perceptron (Collins, 2002) 1-best MIRA with cost-augmented decoding (Crammer et al., 2006) Max-margin via subgradient descent (Ratliff et al., 2006) Conditional likelihood (Lafferty et al., 2001) Risk (Xiong et al., 2009)

For risk and Jensen risk bound, initialized using output of

conditional likelihood training

Used Hamming cost for cost function

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Results

85.84* Softmax-Margin 85.65* Jensen Risk Bound 85.59* Risk 85.46* Conditional Likelihood 85.28* Max-Margin 85.72* MIRA 83.98* Perceptron Test F1 Method

* Indicates significance (compared with softmax-margin)

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Results

85.84* Softmax-Margin 85.65* Jensen Risk Bound 85.59* Risk 85.46* Conditional Likelihood 85.28* Max-Margin 85.72* MIRA 83.98* Perceptron Test F1 Method

* Indicates significance (compared with softmax-margin) Significant improvement with equal training time and implementation difficulty

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Results

85.84* Softmax-Margin 85.65* Jensen Risk Bound 85.59* Risk 85.46* Conditional Likelihood 85.28* Max-Margin 85.72* MIRA 83.98* Perceptron Test F1 Method

* Indicates significance (compared with softmax-margin) Comparable performance with half the training time

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Risk Conditional Likelihood Max-Margin MIRA

Uses a cost function Is convex

Softmax-Margin Jensen Risk Bound Perceptron

Based on probabilistic inference

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Risk Conditional Likelihood Max-Margin MIRA

Time Performance

Perceptron Softmax-Margin Jensen Risk Bound

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Risk Conditional Likelihood Max-Margin MIRA

Time Performance

Perceptron

(Cost-Augmented) Decoding (Cost-Augmented) Summing Expectations

• f Products

Softmax-Margin Jensen Risk Bound (Cost-Augmented) Decoding (Cost-Augmented) Summing Expectations

• f Products

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See extended technical report for:

Probabilistic interpretation for softmax-margin in

minimum divergence framework (Jelinek, 1997)

Softmax-margin training with hidden variables Additional experiments

Thank you!

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Loss Functions for Binary Classification

• 3
• 2
• 1

1 2 3 4 1 2 3 × ! " − ! # − $%! " − #"! × ≤ # & '$ (! −

−

'$!

− −

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• cost-augmented summing

Softmax-Margin

• cost-augmented summing

Jensen Risk Bound

• expectations of products

Risk

• summing

Conditional Likelihood

• cost-augmented decoding

Max-Margin

• cost-augmented decoding

MIRA

• decoding

Perceptron Prob. Interp. Convex Cost Function Requirements Training Method