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Differentiable Sparse Coding

David Bradley and J. Andrew BagnellNIPS 2008

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Joint Optimization

100,000 ft View• Complex Systems

VoxelFeatures

VoxelClassifier

2-DPlanner

Cameras

Ladar

Voxel Grid

ColumnCost

Y(Path to goal)

Initialize with “cheap” data

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10,000 ft view

• Sparse Coding = Generative Model

• Semi-supervised learning• KL-Divergence Regularization• Implicit Differentiation

OptimizationUnlabeled Data Latent Variable

Classifierx w

Loss Gradient

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Sparse Coding

Understand X

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As a combination of factors

B

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Sparse coding uses optimization

Projection (feed-forward):

)( Bxfw T

)( wBfx

Some vector

Want to useto classify x

Reconstructionloss function

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Sparse vectors: Only a few significant elements

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Example: X=Handwritten Digits

Basis vectors colored by w

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Input Basis

Projection

Optimization vs. Projection

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Input Basis

KL-regularized Optimization

Optimization vs. Projection

Outputs are sparse for each example

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Generative Model

Latent variables are Independent

PriorLikelihood

Examples are Independent

i

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PriorLikelihood

Sparse Approximation

MAP Estimate

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Sparse Approximation

Distance between reconstruction and input

Distance between weight vector and prior mean

Regularization Constant

pwwBfxxP

||Prior)(||Loss)(log

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Example: Squared Loss + L1

• Convex + sparse (widely studied in engineering)• Sparse coding solves for B as well (non-convex for now…)• Shown to generate useful features on diverse problems

Tropp, Signal Processing, 2006Donoho and Elad, Proceedings of the National Academy of Sciences, 2002Raina, Battle, Lee, Packer, Ng, ICML, 2007

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L1 Sparse Coding

Shown to generate useful features on diverse problems

Optimize B over all examples

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Differentiable Sparse Coding

Bradley & Bagnell, “Differentiable Sparse Coding”, NIPS 2008

Y

X

LearningModule (θ)

Loss Function

OptimizationModule (B)

UnlabeledData

Reconstruction Loss

LabeledData

X

W)(BWf

Sparse Coding

Raina, Battle, Lee, Packer, Ng, ICML, 2007

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L1 Regularization is Not Differentiable

Bradley & Bagnell, “Differentiable Sparse Coding”, NIPS 2008

Y

X

LearningModule (θ)

Loss Function

OptimizationModule (B)

UnlabeledData

Reconstruction Loss

LabeledData

X

W)(BWf

Sparse Coding

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Why is this unsatisfying?

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Problem #1:Instability

L1 Map Estimates are discontinuous

Outputs are not differentiable

Instead use KL-divergenceProven to compete with L1 in online learning

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Problem #2: No closed-form Equation

At the MAP estimate:

pwwBfxww

||Prior)(||Lossminargˆ

0||ˆPrior)ˆ(||Loss pwwBfx ww

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Solution: Implicit DifferentiationDifferentiate both sides with respect to an element of B:

Since is a function of B: Solve for this

0||ˆPrior)ˆ(||Lossij

pwwBfxB ww

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KL-Divergence

Example: Squared Loss, KL prior

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Handwritten Digit Recognition

50,000 digit training set10,000 digit validation set10,000 digit test set

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Handwritten Digit Recognition

Unsupervised Sparse Coding

L2 loss and L1 prior

Training Set

Step #1:

Raina, Battle, Lee, Packer, Ng, ICML, 2007

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Handwritten Digit Recognition

Sparse Approximation

Step #2:Maxent

ClassifierLoss

Function

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Handwritten Digit Recognition

Supervised Sparse Coding

Step #3:Maxent

ClassifierLoss

Function

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KL Maintains Sparsity

Weight concentratedin few elements

Log scale

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KL adds Stability

X

W (KL) W (backprop)

W (L1)

Duda, Hart, Stork. Pattern classification. 2001

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Performance vs. Prior

0123456789

1,000 10,000 50,000

L1KLKL + Backprop

Number of Training Examples

Mis

clas

sific

atio

n E

rror

(%)

Bette

r

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Classifier Comparison

0%

1%

2%

3%

4%

5%

6%

7%

8%

Maxent 2-layer NN SVM (Linear) SVM (RBF)

Mis

clas

sific

ation

Erro

r

PCAL1KLKL+backprop

Bette

r

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Comparison to other algorithms

0.00%0.50%1.00%1.50%2.00%2.50%3.00%3.50%4.00%

50,000

Mis

clas

sific

ation

Erro

r

L1KLKL+backpropSVM2-layer NN

Bette

r

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Transfer to English Characters

8

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24,000 character training set12,000 character validation set12,000 character test set

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Transfer to English Characters

Sparse Approximation

Step #1:

DigitsBasis

MaxentClassifier

Loss Function

Raina, Battle, Lee, Packer, Ng, ICML, 2007

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Transfer to English CharactersStep #2:

MaxentClassifier

Loss Function

Supervised Sparse Coding

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Transfer to English Characters

40%

50%

60%

70%

80%

90%

100%

500 5000 20000

Clas

sific

ation

Acc

urac

y

Training Set Size

RawPCAL1KLKL+backprop

Bette

r

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Text ApplicationX

Unsupervised Sparse Coding

KL loss + KL prior

5,000 movie reviews

10 point sentiment scale1=hated it, 10=loved it

Step #1:

Pang, Lee, Proceeding of the ACL, 2005

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Text ApplicationX

Sparse Approximation

5-fold Cross Validation

10 point sentiment scale1=hated it, 10=loved it

LinearRegression

L2 LossStep #2:

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Text ApplicationX

10 point sentiment scale1=hated it, 10=loved it

5-fold Cross Validation

Step #3:

LinearRegression

L2 Loss

Supervised Sparse Coding

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Movie Review Sentiment

0.00

0.10

0.20

0.30

0.40

0.50

0.60

Pred

ictiv

e R

2 LDAKLsLDAKL+backpropBe

tter

Unsupervisedbasis

Supervisedbasis

State of the artgraphical model

Blei, McAuliffe, NIPS, 2007

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RGB Camera

NIR Camera

Ladar

Future Work

Sparse Coding

EngineeredFeatures Labeled

Training Data

Example Paths

VoxelClassifier

MMPCamera

Laser

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Future Work: Convex Sparse Coding

• Sparse approximation is convex• Sparse coding is not because fixed-size basis is a

non-convex constraint• Sparse coding ↔ sparse approximation on

infinitely large basis + non-convex rank constraint– Relax to a convex L1 rank constraint

• Use boosting for sparse approximation directly on infinitely large basis

Bengio, Le Roux, Vincent, Dellalleau, Marcotte, NIPS, 2005Zhao, Yu. Feature Selection for Data Mining, 2005Riffkin, Lippert. Journal of Machine Learning Research, 2007

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Questions?