A ccelerated, P arallel and PROX imal coordinate descent

Accelerated, Parallel and PROXimal coordinate descent

Moscow February 2014

A P PROXPeter Richtárik

(Joint work with Olivier Fercoq - arXiv:1312.5799)

Optimization Problem

Problem

Convex (smooth or nonsmooth)

Convex (smooth or nonsmooth)- separable- allow

Loss Regularizer

Regularizer: examples

No regularizer Weighted L1 norm

Weighted L2 normBox constraints

e.g., SVM dual

e.g., LASSO

Loss: examples

Quadratic loss

L-infinity

L1 regression

Exponential loss

Logistic loss

Square hinge loss

BKBG’11RT’11bTBRS’13RT ’13a

FR’13

RANDOMIZED COORDINATE DESCENT

IN 2D

Find the minimizer of

2D OptimizationContours of a function

Goal:

a2 =b2

Randomized Coordinate Descent in 2D

a2 =b2

N

S

EW

Randomized Coordinate Descent in 2D

a2 =b2

1

N

S

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Randomized Coordinate Descent in 2D

a2 =b2

1

N

S

EW

2

Randomized Coordinate Descent in 2D

a2 =b2

1

23 N

S

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Randomized Coordinate Descent in 2D

a2 =b2

1

23

4N

S

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Randomized Coordinate Descent in 2D

a2 =b2

1

23

4N

S

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5

Randomized Coordinate Descent in 2D

a2 =b2

1

23

45

6

N

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Randomized Coordinate Descent in 2D

a2 =b2

1

23

45

N

S

EW

67SOLVED!

CONTRIBUTIONS

Variants of Randomized Coordinate Descent Methods

• Block– can operate on “blocks” of coordinates – as opposed to just on individual coordinates

• General – applies to “general” (=smooth convex) functions – as opposed to special ones such as quadratics

• Proximal– admits a “nonsmooth regularizer” that is kept intact in solving subproblems – regularizer not smoothed, nor approximated

• Parallel – operates on multiple blocks / coordinates in parallel– as opposed to just 1 block / coordinate at a time

• Accelerated– achieves O(1/k^2) convergence rate for convex functions– as opposed to O(1/k)

• Efficient– complexity of 1 iteration is O(1) per processor on sparse problems – as opposed to O(# coordinates) : avoids adding two full vectors

Brief History of Randomized Coordinate Descent Methods

+ new long stepsizes

APPROX

“PROXIMAL”“PAR

ALLE

L”

“ACCELERATED”

A P PROX

PCDM (R. & Takáč, 2012) = APPROX if we force

APPROX: Smooth Case

Want this to be as large as possible

Update for coordinate i

Partial derivative of f

CONVERGENCE RATE

Convergence Rate

average # coordinates updated / iteration

# coordinates# iterations

implies

Theorem [FR’13b]

Key assumption

Special Case: Fully Parallel Variant

all coordinates are updated in each iteration

# normalized weights (summing to n)

# iterations

implies

Special Case: Effect of New Stepsizes

Average degree of separability

“Average” of the Lipschitz constants

With the new stepsizes (will mention later!), we have:

“EFFICIENCY” OF

APPROX

Cost of 1 Iteration of APPROX

Assume N = n (all blocks are of size 1)and that

Sparse matrixThen the average cost of 1 iteration of APPROX is

Scalar function: derivative = O(1)

arithmetic ops

= average # nonzeros in a column of A

Bottleneck: Computation of Partial Derivatives

maintained

PRELIMINARYEXPERIMENTS

L1 Regularized L1 Regression

Dorothea dataset:

Gradient Method

Nesterov’s Accelerated Gradient Method

SPCDM

APPROX

L1 Regularized L1 Regression

L1 Regularized Least Squares (LASSO)

KDDB dataset:

PCDM

APPROX

Training Linear SVMs

Malicious URL dataset:

Choice of Stepsizes:

How (not) to ParallelizeCoordinate Descent

Convergence of Randomized Coordinate Descent

Strongly convex F(Simple Mehod)

Smooth or ‘simple’ nonsmooth F(Accelerated Method)

‘Difficult’ nonsmooth F(Accelerated Method)

or smooth F(Simple method)

‘Difficult’ nonsmooth F(Simple Method)

Focus on n

(big data = big n)

Parallelization Dream

Depends on to what extent we can add up individual updates, which depends on the properties of F and the

way coordinates are chosen at each iteration

Serial Parallel

What do we actually get?WANT

“Naive” parallelization

Do the same thing as before, but

for MORE or ALL coordinates &

ADD UP the updates

Failure of naive parallelization

1a

1b

0

Failure of naive parallelization

1

1a

1b

0

Failure of naive parallelization

1

2a

2b

Failure of naive parallelization

1

2a

2b

2

Failure of naive parallelization

2

OOPS!

1

1a

1b

0

Idea: averaging updates may help

SOLVED!

Averaging can be too conservative

1a

1b

0

12a

2b

2

and so on...

Averaging may be too conservative 2

WANT

BAD!!!But we wanted:

What to do?

Averaging:

Summation:

Update to coordinate i

i-th unit coordinate vector

Figure out when one can safely use:

ESO:Expected SeparableOverapproximation

5 Models for f Admitting Small

1

2

3

Smooth partially separable f [RT’11b ]

Nonsmooth max-type f [FR’13]

f with ‘bounded Hessian’ [BKBG’11, RT’13a ]

5 Partially separable f with block smooth components [FR’13b]

5 Models for f Admitting Small

4 Partially separable f with smooth components [NC’13]

Randomized Parallel Coordinate Descent Method

Random set of coordinates (sampling)

Current iterate New iterate i-th unit coordinate vector

Update to i-th coordinate

ESO: Expected Separable Overapproximation

Definition [RT’11b]

1. Separable in h2. Can minimize in parallel3. Can compute updates for only

Shorthand:

Minimize in h

PART II.ADDITIONAL TOPICS

Partial Separability and

Doubly Uniform Samplings

Serial uniform samplingProbability law:

-nice samplingProbability law:

Good for shared memory systems

Doubly uniform sampling

Probability law:

Can model unreliable processors / machines

ESO for partially separable functionsand doubly uniform samplings

Theorem [RT’11b]

1 Smooth partially separable f [RT’11b ]

PCDM: Theoretical Speedup

Much of Big Data is here!

degree of partial separability

# coordinates

# coordinate updates / iter

WEAK OR NO SPEEDUP: Non-separable (dense) problems

LINEAR OR GOOD SPEEDUP: Nearly separable (sparse) problems

n = 1000(# coordinates)

Theory

Practice

n = 1000(# coordinates)

PCDM: Experiment with a 1 billion-by-2 billion

LASSO problem

Optimization with Big Data

* in a billion dimensional space on a foggy day

Extreme* Mountain Climbing=

Coordinate Updates

Iterations

Wall Time

Distributed-Memory Coordinate Descent

Distributed -nice sampling

Probability law:

Machine 2Machine 1 Machine 3

Good for a distributed version of coordinate descent

ESO: Distributed setting

Theorem [RT’13b]

3 f with ‘bounded Hessian’ [BKBG’11, RT’13a ]

spectral norm of the data

Bad partitioning at most doubles # of iterations

spectral norm of the partitioning

Theorem [RT’13b]

# nodes

# iterations = implies

# updates/node

LASSO with a 3TB data matrix

128 Cray XE6 nodes with 4 MPI processes (c = 512) Each node: 2 x 16-cores with 32GB RAM

= # coordinates

• Shai Shalev-Shwartz and Ambuj Tewari, Stochastic methods for L1-regularized loss minimization. JMLR 2011.

• Yurii Nesterov, Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM Journal on Optimization, 22(2):341-362, 2012.

• [RT’11b] P.R. and Martin Takáč, Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function. Mathematical Prog., 2012.

• Rachael Tappenden, P.R. and Jacek Gondzio, Inexact coordinate descent: complexity and preconditioning, arXiv: 1304.5530, 2013.

• Ion Necoara, Yurii Nesterov, and Francois Glineur. Efficiency of randomized coordinate descent methods on optimization problems with linearly coupled constraints. Technical report, Politehnica University of Bucharest, 2012.

• Zhaosong Lu and Lin Xiao. On the complexity analysis of randomized block-coordinate descent methods. Technical report, Microsoft Research, 2013.

References: serial coordinate descent

• [BKBG’11] Joseph Bradley, Aapo Kyrola, Danny Bickson and Carlos Guestrin, Parallel Coordinate Descent for L1-Regularized Loss Minimization. ICML 2011

• [RT’12] P.R. and Martin Takáč, Parallel coordinate descen methods for big data optimization. arXiv:1212.0873, 2012

• Martin Takáč, Avleen Bijral, P.R., and Nathan Srebro. Mini-batch primal and dual methods for SVMs. ICML 2013

• [FR’13a] Olivier Fercoq and P.R., Smooth minimization of nonsmooth functions with parallel coordinate descent methods. arXiv:1309.5885, 2013

• [RT’13a] P.R. and Martin Takáč, Distributed coordinate descent method for big data learning. arXiv:1310.2059, 2013

• [RT’13b] P.R. and Martin Takáč, On optimal probabilities in stochastic coordinate descent methods. arXiv:1310.3438, 2013

References: parallel coordinate descent

Good entry point to the topic (4p paper)

• P.R. and Martin Takáč, Efficient serial and parallel coordinate descent methods for huge-scale truss topology design. Operations Research Proceedings 2012.

• Rachael Tappenden, P.R. and Burak Buke, Separable approximations and decomposition methods for the augmented Lagrangian. arXiv:1308.6774, 2013.

• Indranil Palit and Chandan K. Reddy. Scalable and parallel boosting with MapReduce. IEEE Transactions on Knowledge and Data Engineering, 24(10):1904-1916, 2012.

• [FR’13b] Olivier Fercoq and P.R., Accelerated, Parallel and Proximal coordinate descent. arXiv:1312.5799, 2013

References: parallel coordinate descent