Big learning: challenges and opportunities€¦ · Unsupervised learning through matrix...

Big learning:challenges and opportunities

Francis Bach

SIERRA Project-team, INRIA - Ecole Normale Superieure

October 2013

Scientific context

Big data

• Omnipresent digital media

– Multimedia, sensors, indicators, social networks

– All levels: personal, professional, scientific, industrial

– Too large and/or complex for manual processing

– Computational challenges

– Dealing with large databases

– Statistical challenges

– What can be predicted from such databases and how?

– Looking for hidden information

– Opportunities (and threats)

Scientific context

Big data

• Omnipresent digital media

– Multimedia, sensors, indicators, social networks

– All levels: personal, professional, scientific, industrial

– Too large and/or complex for manual processing

• Computational challenges

– Dealing with large databases

• Statistical challenges

– What can be predicted from such databases and how?

– Looking for hidden information

• Opportunities (and threats)

Machine learning for big data

• Large-scale machine learning: large p, large n, large k

– p : dimension of each observation (input)

– n : number of observations

– k : number of tasks (dimension of outputs)

• Examples: computer vision, bioinformatics, etc.

Search engines - advertising

Advertising - recommendation

Object recognition

Learning for bioinformatics - Proteins

• Crucial components of cell life

• Predicting multiple functions and

interactions

• Massive data: up to 1 millions for

humans!

• Complex data

– Amino-acid sequence

– Link with DNA

– Tri-dimensional molecule

Machine learning for big data

• Large-scale machine learning: large p, large n, large k

– p : dimension of each observation (input)

– n : number of observations

– k : number of tasks (dimension of outputs)

• Examples: computer vision, bioinformatics, etc.

• Two main challenges:

1. Computational: ideal running-time complexity = O(pn+ kn)

2. Statistical: meaningful results

Big learning: challenges and opportunities

Outline

• Scientific context

– Big data: need for supervised and unsupervised learning

• Beyond stochastic gradient for supervised learning

– Few passes through the data

– Provable robustness and ease of use

• Matrix factorization for unsupervised learning

– Looking for hidden information through dictionary learning

– Feature learning

Supervised machine learning

• Data: n observations (xi, yi) ∈ X × Y, i = 1, . . . , n

• Prediction as a linear function θ⊤Φ(x) of features Φ(x) ∈ Rp

• (regularized) empirical risk minimization: find θ solution of

minθ∈Rp

1

n

n∑

i=1

ℓ(yi, θ

⊤Φ(xi))

+ µΩ(θ)

convex data fitting term + regularizer

Supervised machine learning

• Data: n observations (xi, yi) ∈ X × Y, i = 1, . . . , n

• Prediction as a linear function θ⊤Φ(x) of features Φ(x) ∈ Rp

• (regularized) empirical risk minimization: find θ solution of

minθ∈Rp

1

n

n∑

i=1

ℓ(yi, θ

⊤Φ(xi))

+ µΩ(θ)

convex data fitting term + regularizer

• Applications to any data-oriented field

– Computer vision, bioinformatics

– Natural language processing, etc.

Supervised machine learning

• Data: n observations (xi, yi) ∈ X × Y, i = 1, . . . , n

• Prediction as a linear function θ⊤Φ(x) of features Φ(x) ∈ Rp

• (regularized) empirical risk minimization: find θ solution of

minθ∈Rp

1

n

n∑

i=1

ℓ(yi, θ

⊤Φ(xi))

+ µΩ(θ)

convex data fitting term + regularizer

• Main practical challenges

– Designing/learning good features Φ(x)

– Efficiently solving the optimization problem

Stochastic vs. deterministic methods

• Minimizing g(θ) =1

n

n∑

i=1

fi(θ) with fi(θ) = ℓ(yi, θ

⊤Φ(xi))+ µΩ(θ)

• Batch gradient descent: θt = θt−1−γtg′(θt−1) = θt−1−

γtn

n∑

i=1

f ′i(θt−1)

– Linear (e.g., exponential) convergence rate in O(e−αt)

– Iteration complexity is linear in n (with line search)

Stochastic vs. deterministic methods

• Minimizing g(θ) =1

n

n∑

i=1

fi(θ) with fi(θ) = ℓ(yi, θ

⊤Φ(xi))+ µΩ(θ)

• Batch gradient descent: θt = θt−1−γtg′(θt−1) = θt−1−

γtn

n∑

i=1

f ′i(θt−1)

Stochastic vs. deterministic methods

• Minimizing g(θ) =1

n

n∑

i=1

fi(θ) with fi(θ) = ℓ(yi, θ

⊤Φ(xi))+ µΩ(θ)

• Batch gradient descent: θt = θt−1−γtg′(θt−1) = θt−1−

γtn

n∑

i=1

f ′i(θt−1)

– Linear (e.g., exponential) convergence rate in O(e−αt)

– Iteration complexity is linear in n (with line search)

• Stochastic gradient descent: θt = θt−1 − γtf′i(t)(θt−1)

– Sampling with replacement: i(t) random element of 1, . . . , n

– Convergence rate in O(1/t)

– Iteration complexity is independent of n (step size selection?)

Stochastic vs. deterministic methods

• Minimizing g(θ) =1

n

n∑

i=1

fi(θ) with fi(θ) = ℓ(yi, θ

⊤Φ(xi))+ µΩ(θ)

• Batch gradient descent: θt = θt−1−γtg′(θt−1) = θt−1−

γtn

n∑

i=1

f ′i(θt−1)

• Stochastic gradient descent: θt = θt−1 − γtf′i(t)(θt−1)

Stochastic vs. deterministic methods

• Goal = best of both worlds: Linear rate with O(1) iteration cost

Robustness to step size

time

log(

exce

ss c

ost)

stochastic

deterministic

Stochastic vs. deterministic methods

• Goal = best of both worlds: Linear rate with O(1) iteration cost

Robustness to step size

hybridlog(

exce

ss c

ost)

stochastic

deterministic

time

Stochastic average gradient

(Le Roux, Schmidt, and Bach, 2012)

• Stochastic average gradient (SAG) iteration

– Keep in memory the gradients of all functions fi, i = 1, . . . , n

– Random selection i(t) ∈ 1, . . . , n with replacement

– Iteration: θt = θt−1 −γtn

n∑

i=1

yti with yti =

f ′i(θt−1) if i = i(t)

yt−1i otherwise

Stochastic average gradient

(Le Roux, Schmidt, and Bach, 2012)

• Stochastic average gradient (SAG) iteration

– Keep in memory the gradients of all functions fi, i = 1, . . . , n

– Random selection i(t) ∈ 1, . . . , n with replacement

– Iteration: θt = θt−1 −γtn

n∑

i=1

yti with yti =

f ′i(θt−1) if i = i(t)

yt−1i otherwise

• Stochastic version of incremental average gradient (Blatt et al., 2008)

• Simple implementation

– Extra memory requirement: same size as original data (or less)

– Simple/robust constant step-size

Stochastic average gradient

Convergence analysis

• Assume each fi is L-smooth and g=1

n

n∑

i=1

fi is µ-strongly convex

• Constant step size γt =1

16L. If µ >

2L

n, ∃C ∈ R such that

∀t > 0, E[g(θt)− g(θ∗)

]6 C exp

(

−t

8n

)

• Linear convergence rate with iteration cost independent of n

– After each pass through the data, constant error reduction

– Breaking two lower bounds

Stochastic average gradient

Simulation experiments

• protein dataset (n = 145751, p = 74)

• Dataset split in two (training/testing)

0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

Effective Passes

Ob

jec

tiv

e m

inu

s O

ptim

um

Steepest

AFG

L−BFGS

pegasos

RDA

SAG (2/(L+nµ))

SAG−LS

0 5 10 15 20 25 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

4

Effective Passes

Test

Lo

gis

tic

Lo

ss

Steepest

AFG

L−BFGS

pegasos

RDA

SAG (2/(L+nµ))

SAG−LS

Training cost Testing cost

Stochastic average gradient

Simulation experiments

• covertype dataset (n = 581012, p = 54)

• Dataset split in two (training/testing)

0 5 10 15 20 25 30

10−4

10−2

100

102

Effective Passes

Ob

jec

tiv

e m

inu

s O

ptim

um

Steepest

AFG

L−BFGS

pegasos

RDA

SAG (2/(L+nµ))

SAG−LS

0 5 10 15 20 25 30

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

2

x 105

Effective Passes

Test

Lo

gis

tic

Lo

ss

Steepest

AFG

L−BFGS

pegasos

RDA

SAG (2/(L+nµ))

SAG−LS

Training cost Testing cost

Large-scale supervised learning

Convex optimization

• Simplicity

– Few lines of code

• Robustness

– Step-size

– Adaptivity to problem difficulty

• On-going work

– Single pass through the data (Bach and Moulines, 2013)

– Distributed algorithms

- Convexity as a solution to all problems?

- Need good features Φ(x) for linear predictions θ⊤Φ(x) !

Large-scale supervised learning

Convex optimization

• Simplicity

– Few lines of code

• Robustness

– Step-size

– Adaptivity to problem difficulty

• On-going work

– Single pass through the data (Bach and Moulines, 2013)

– Distributed algorithms

• Convexity as a solution to all problems?

– Need good features Φ(x) for linear predictions θ⊤Φ(x) !

Unsupervised learning through matrix factorization

• Given data matrix X = (x⊤1 , . . . ,x

⊤n ) ∈ R

n×p

– Principal component analysis: xi ≈ Dαi

– K-means: xi ≈ dk ⇒ X = DA

Learning dictionaries for uncovering hidden structure

• Fact: many natural signals may be approximately represented as a

superposition of few atoms from a dictionary D = (d1, . . . ,dk)

– Decomposition x =k∑

i=1

αidi = Dα with α ∈ R

k sparse

– Natural signals (sounds, images) (Olshausen and Field, 1997)

- Decoding problem: given a dictionary D, finding α through

regularized convex optimization minα∈Rk ‖x−Dα‖22 + λ‖α‖1

Learning dictionaries for uncovering hidden structure

• Fact: many natural signals may be approximately represented as a

superposition of few atoms from a dictionary D = (d1, . . . ,dk)

– Decomposition x =k∑

i=1

αidi = Dα with α ∈ R

k sparse

– Natural signals (sounds, images) (Olshausen and Field, 1997)

• Decoding problem: given a dictionary D, finding α through

regularized convex optimization minα∈Rk ‖x−Dα‖22 + λ‖α‖1

1

2w

w 1

2w

w

Learning dictionaries for uncovering hidden structure

• Fact: many natural signals may be approximately represented as a

superposition of few atoms from a dictionary D = (d1, . . . ,dk)

– Decomposition x =

k∑

i=1

αidi = Dα with α ∈ R

k sparse

– Natural signals (sounds, images) (Olshausen and Field, 1997)

• Decoding problem: given a dictionary D, finding α through

regularized convex optimization minα∈Rk ‖x−Dα‖22 + λ‖α‖1

• Dictionary learning problem: given n signals x1, . . . ,xn,

– Estimate both dictionary D and codes α1, . . . ,αn

minD

n∑

j=1

minαj∈Rp

∥∥xj −Dαj

∥∥2

2+ λ‖αj‖1

Challenges of dictionary learning

minD

n∑

j=1

minαj∈Rp

∥∥xj −Dαj

∥∥2

2+ λ‖αj‖1

• Algorithmic challenges

– Large number of signals ⇒ online learning (Mairal et al., 2009a)

• Theoretical challenges

– Identifiabiliy/robustness (Jenatton et al., 2012)

• Domain-specific challenges

– Going beyond plain sparsity ⇒ structured sparsity

(Jenatton, Mairal, Obozinski, and Bach, 2011)

Dictionary learning for image denoising

x︸︷︷︸measurements

= y︸︷︷︸

original image

+ ε︸︷︷︸noise

Dictionary learning for image denoising

• Solving the denoising problem (Elad and Aharon, 2006)

– Extract all overlapping 8× 8 patches xi ∈ R64

– Form the matrix X = (x⊤1 , . . . ,x

⊤n ) ∈ R

n×64

– Solve a matrix factorization problem:

minD,A

||X−DA||2F = minD,A

n∑

i=1

||xi −Dαi||22

where A is sparse, and D is the dictionary

– Each patch is decomposed into xi = Dαi

– Average all Dαi to reconstruct a full-sized image

• The number of patches n is large (= number of pixels)

• Online learning (Mairal, Bach, Ponce, and Sapiro, 2009a)

Denoising result

(Mairal, Bach, Ponce, Sapiro, and Zisserman, 2009b)

Denoising result

(Mairal, Bach, Ponce, Sapiro, and Zisserman, 2009b)

Inpainting a 12-Mpixel photograph

Inpainting a 12-Mpixel photograph

Inpainting a 12-Mpixel photograph

Inpainting a 12-Mpixel photograph

Why structured sparsity?

• Interpretability

– Structured dictionary elements (Jenatton et al., 2009b)

– Dictionary elements “organized” in a tree or a grid (Kavukcuoglu

et al., 2009; Jenatton et al., 2010; Mairal et al., 2010)

Structured sparse PCA (Jenatton et al., 2009b)

raw data sparse PCA

• Unstructed sparse PCA ⇒ many zeros do not lead to better

interpretability

Structured sparse PCA (Jenatton et al., 2009b)

raw data sparse PCA

• Unstructed sparse PCA ⇒ many zeros do not lead to better

interpretability

Structured sparse PCA (Jenatton et al., 2009b)

raw data Structured sparse PCA

• Enforce selection of convex nonzero patterns ⇒ robustness to

occlusion in face identification

Structured sparse PCA (Jenatton et al., 2009b)

raw data Structured sparse PCA

• Enforce selection of convex nonzero patterns ⇒ robustness to

occlusion in face identification

Modelling of text corpora - Dictionary tree

Why structured sparsity?

• Interpretability

– Structured dictionary elements (Jenatton et al., 2009b)

– Dictionary elements “organized” in a tree or a grid (Kavukcuoglu

et al., 2009; Jenatton et al., 2010; Mairal et al., 2010)

Why structured sparsity?

• Interpretability

– Structured dictionary elements (Jenatton et al., 2009b)

– Dictionary elements “organized” in a tree or a grid (Kavukcuoglu

et al., 2009; Jenatton et al., 2010; Mairal et al., 2010)

• Stability and identifiability

• Prediction or estimation performance

– When prior knowledge matches data (Haupt and Nowak, 2006;

Baraniuk et al., 2008; Jenatton et al., 2009a; Huang et al., 2009)

• How?

– Design of sparsity-inducing norms

Structured sparsity

• Sparsity-inducing behavior from “corners” of constraint sets

Structured dictionary learning- Efficient optimization

minA∈R

k×n

D∈Rp×k

n∑

i=1

‖xi −Dαi‖22 + λψ(αi) s.t. ∀j, ‖dj‖2 ≤ 1.

• Minimization with respect to αi : regularized least-squares

– Many algorithms dedicated to the ℓ1-norm ψ(α) = ‖α‖1

• Proximal methods : first-order methods with optimal convergence

rate (Nesterov, 2007; Beck and Teboulle, 2009)

– Requires solving many times minα∈Rk

12‖y −α‖22 + λψ(α)

• Efficient algorithms for structured sparse problems

– Bach, Jenatton, Mairal, and Obozinski (2011)

– Code available: http://www.di.ens.fr/willow/SPAMS/

Extensions - Digital Zooming

Digital Zooming (Couzinie-Devy et al., 2011)

Extensions - Task-driven dictionaries

inverse half-toning (Mairal et al., 2011)

Extensions - Task-driven dictionaries

inverse half-toning (Mairal et al., 2011)

Big learning: challenges and opportunities

Conclusion

• Scientific context

– Big data: need for supervised and unsupervised learning

• Beyond stochastic gradient for supervised learning

– Few passes through the data

– Provable robustness and ease of use

• Matrix factorization for unsupervised learning

– Looking for hidden information through dictionary learning

– Feature learning

References

F. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence

rate o(1/n). Technical Report 00831977, HAL, 2013.

F. Bach, R. Jenatton, J. Mairal, and G. Obozinski. Optimization with sparsity-inducing penalties.

Foundations and Trends R© in Machine Learning, 4(1):1–106, 2011.

R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde. Model-based compressive sensing. Technical

report, arXiv:0808.3572, 2008.

A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems.

SIAM Journal on Imaging Sciences, 2(1):183–202, 2009.

D. Blatt, A.O. Hero, and H. Gauchman. A convergent incremental gradient method with a constant

step size. 18(1):29–51, 2008.

M. Elad and M. Aharon. Image denoising via sparse and redundant representations over learned

dictionaries. IEEE Transactions on Image Processing, 15(12):3736–3745, 2006.

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