Robust High Performance Optimization for Clustering, Multi Dimensional Scaling and Mixture Models

1

Robust High Performance

Optimization for Clustering,

Multi-Dimensional Scaling

and Mixture Models

CGB Indiana University Lunchtime Tal January 22 2008

Geoffrey Fox discussing work with:

Seung-Hee Bae, Neil Devadasan, Rajarshi Guha, Marlon Pierce, Xiaohong Qiu, David Wild, Huapeng Yuan

Community Grids Laboratory, Research Computing UITS, School of informatics and POLIS Center Indiana University

George Chrysanthakopoulos, Henrik Frystyk Nielsen Microsoft Research, Redmond WA

http://www.infomall.org/multicore

Abstract

n We first review the pros and cons of various approaches to non linear

optimization in the presence of local minima, ill conditioned matrices and ambiguous choice of appropriate number of degrees of freedom (over and under fitting).

n We define constraints on approaches from need to run well in parallel on

systems of multicore CPU's.

n We present a uniform approach to data clustering and Gaussian mixture model

ling that uses deterministic (not Monte Carlo) annealing to mitigate the local minima problem and naturally relates the appropriate number of parameters (clusters or mixture components) to the scale at which problem is examined.

n New clusters (mixtures) are introduced at phase transitions as the annealing

temperature is lowered and second derivative matrix becomes singular.

n We contrast three ways of visualizing this structure in low (2) dimensions with

Principal Component Analysis PCA, Generative Topographic Mapping GTM

and Multi-Dimensional Scaling MDS using annealing to regularize GTM and MDS.

n Currently we have implemented in preliminary fashion deterministic annealing

clustering and GTM in a fashion that runs well on multicore systems.

n We have applied these techniques to Geographical Information Systems

(clustering demographic data in 2D) and Cheminformatics in 1052 and lower dimensions.

n We would like to understand other applications that can constrain and test

Too much Computing?

n Historically both grids and parallel computing have tried to

increase computing capabilities by

• Optimizing performance of codes at cost of re-usability • Exploiting all possible CPU’s such as Graphics

co-processors and “idle cycles” (across administrative domains)

• Linking central computers together such as NSF/DoE/DoD

supercomputer networks without clear user requirements

n Next Crisis in technology area will be the opposite problem –

commodity chips will be 32-128way parallel in 5 years time and we currently have no idea how to use them on commodity systems – especially on clients

• Only 2 releases of standard software (e.g. Office) in this

time span so need solutions that can be implemented in next 3-5 years

n Intel RMS analysis: Gaming and Generalized decision

Too much Data to the Rescue?

n Multicore servers have clear “universal parallelism” as many

users can access and use machines simultaneously

n Maybe also need application parallelism (e.g. datamining) as

needed on client machines

n Over next years, we will be submerged of course in data

deluge

• Scientific observations for e-Science • Local (video, environmental) sensors

• Data fetched from Internet defining users interests

n Maybe data-mining of this “too much data” will use up the

“too much computing” both for science and commodity PC’s

• PC will use this data(-mining) to be intelligent user

assistant?

30 Clusters

Renters

Asian

Hispanic

Total

30 Clusters

Clustering algorithm annealing by decreasing distance scale and gradually finds more clusters as resolution improved

Parallel Multicor

Deterministic Annealing Clustering

10000/(Grain Size n = points per core) Overhead = Constant1 + Constant2/n

Constant1 = 0.05 to 0.1 (Windows) due to threa

runtime fluctuations

10 Clusters

20 Clusters Parallel Overhea

on 8 Threads on Intel 8 core

Speedup = 8/(1+Overhead)

Basic Nonlinear Optimization I

subject to well known problems including diverging or converging but to non optimal solution (local minima)

n If we have n parameters _i and need to minimize

F = 2 _{or – ln L}

n Then naively one expands

n F = F(0) + _k=1n _k F(0) /_i +  _k,l=1n _k_l 2F(0)

/kl

• with = -0

n As long as number of observables N is large in su

2 _{= }

i=1Nqi2() or – ln L = i=1Nln qi() n One can approximate

2_F(0_{) /kl  i=1}N _{(1/ qi)qi(}0_{) /kqi(}0_{) /l}

dropping term proportional to 2_qi(0_{) /k l}

n This shows that second derivative matrix is positive (semi-)

Basic Nonlinear Optimization II

n This formula can rarely be used as in practice matrix A is

poorly conditioned corresponding to linear combination of 

being (ill/un) determined

n This easy to understand by transforming basis to those of

Basic Nonlinear Optimization III

n In my experience, the lowest eigenvalues _k of A are zero to

rounding error, so corresponding shifts k are unreliable and

incrementing  = _0 _{+  gives a nonsense point and F rapidly}

diverges as one iterates

n There are several approaches and they actually work and with

modification, Newton’s method is very reliable and one can expect to get good results

• Unless n is very large (tens of thousands?) when it can be

computationally unattractive

…….

Reliable Newton’s Method I

n

WNPF

(

W

ell

N

igh

P

erfect Fit – code lost): Here one

diagonalizes A and modifies equation



=

b

/

, by

zeroing shifts



if



<



where is



is

dynamically adjusted and I typically started with



 

with initially



~ 0.1

•  adjusted up or down depending on success of fit • Only shift in well determined directions but do many

directions at once

n

Converges in 10-20 iterations

See my first paper

Reliable Newton’s Method II

n

Manxcat.f

(Code still exists): This can be too expensive

so when I needed millions of such optimizations I used

Marquardt’s method

which corresponds to

• _k  _k + Q accomplished by

• Replacing A  Q  Identity matrix + A

• so removes effect of 1/_k for small _kwithout expense of

determining eigenvalues/vectors

• Has nonzero but small shifts in ill determined directions • Q is dynamically changed in a complicated fashion on a

logarithmic scale

n

Similar performance to WNPF but less flexible

Gradient Descent

n Many optimizations do not use second order Taylor expansion

but rather look at a shift

n =  G(0) so estimate of minimum i

=0_{+  G(}0₎

n Where  may be varied and G has a more or less fancy way of determining

it

n However usually G is related to steepest descent direction n i.e. G = - F which is direction F decreases fastest along

n One can often find a rigorous bound so that one can show F definitely

decreases

n However essentially always a local minima as only explore directions of

largest eigenvalues (still could be a good minima)

n Note unlikely to be a minimum in “other directions” so really a “partial” not

a “local” minima

n Convergence tends to be slow as error lower order in F than for Newton’s

method but derivatives do not need to be calculated

n Computation  n N for n parameters and N data points

Simulated Annealing

n

Here we consider

function F() to be minimized as an

Energy E()

and randomly choose points in a

neighborhood of



n

One accepts a new value of



with a probability P

depending on temperature T and energies E(



) and

E(



₎

n

Typical choice is P=1 if

E() < E(

)

, and

P = exp([E(

_{) − E()]/}

_T

₎

_if

_{E() > E(}

₎

n

Temperatures are decreased gradually

n

This method avoids local minima due to allowing

upward fluctuations at high temperature but execution

time is huge although each iteration has time

CICC Chemical Informatics and Cyberinfrastructure Collaboratory Web Service Infrastructure

Portal Services

RSS Feeds User Profiles

Collaboration as in Sakai

Core Grid Services

Service Registry

Job Submission and Management

Local Clusters

IU Big Red, TeraGrid, Open Science Grid

Varuna.net

Quantum Chemistry OSCAR Document Analysis

InChI Generation/Search

Computational Chemistry (Gamess, Jaguar etc.)

Deterministic Annealing for Data Mining

n We are looking at deterministic annealing algorithms because although

heuristic

• They have clear scalable parallelism (e.g. use parallel BLAS)

• They avoid (some) local minima and regularize ill defined problems

in an intuitively clear fashion

• They are fast (no Monte Carlo)

• They determine number of clusters automatically (Most important?) • Support iterative visualization approaches as increment number of

clusters systematically

n Developed first by Durbin as Elastic Net for TSP

n Extended by Rose (my student then; now at UCSB)) and Gurewitz

(visitor to C3_{P) at Caltech for signal processing and applied later to}

many optimization and supervised and unsupervised learning methods.

n See K. Rose, "Deterministic Annealing for Clustering, Compression,

Classification, Regression, and Related Optimization Problems,"

Proceedings of the IEEE, vol. 80, pp. 2210-2239, November 1998 Cited by 286

n A deterministic annealing approach to clustering K Rose, E Gurewwitz,

G Fox - Pattern Recognition Letters, 1990 Cited by 159

n Statistical mechanics and phase transitions in clustering K Rose, E

High Level Theory

n Deterministic Annealing can be looked at from a Physics,

Statistics and/or Information theoretic point of view

n In physics view, you get result by using “mean field

approximation” but simplest is information theoretic

n Let E be function that one wishes to minimize

n Then we wish to find minima that is most probable given density

of states.

n One minimizes Free Energy F= E-TS where S is Entropy and S is

just sum over plnp for each degree of freedom with probability

p

• At large T, Entropy dominates while at small T Energy dominates • Annealing lowers temperature so solution tracks continuously

n Key idea is to only apply this to “missing data” i.e. to hidden

degree of freedom k that labels cluster or mixture

n So S is a sum over these discrete degrees of freedom and one can

Deterministic Annealing for Clustering II

n This is an extended K-means algorithm or a simplified Gaussian

mixture

n Start with a single cluster giving as solution Y₁ as centroid n For some annealing schedule for T, iterate above algorithm

testing correlation matrix in Xi about each cluster center to see if

“elongated”

n Split cluster if elongation “long enough”; splitting is a phase

transition in physics view

n You do not need to assume number of clusters but rather a final

resolution T or equivalent

n Minimum evolving as temperature decreases n Movement at fixed temperature going to local

minima if not initialized “correctly

Solve Linear Equations for each temperature

Nonlinearity effects mitigated by approximating with solution at previous higher temperature

Deterministi

Annealing

F({y}, T)

Views from

Past on

Physical

Parallelism

n

All the methods – Deterministic Annealing, Mixture

Models, GTM, PCA and Newton’s method involve

n

Sums over data points

x

=1..N to calculate values and

derivatives

• One can divide points up between cores and efficiently

parallelize

• A little tricky to add results from separate data sets running

in parallel (problem is Cache coherence)

n

Matrix algebra

such as finding eigenvalues

• Well known how to parallelize but need low latency i.e.

Where are we for Clustering?

n We have deterministically annealed clustering running well on

8-core (2-processor quad 8-core) Intel systems using C# and Microsoft Robotics Studio CCR/DSS

n Could also run on multicore-based parallel machines but didn’t

do this (is there a large Windows quad core cluster on TeraGrid?)

• This would also be efficient on large problems

n Applied to Geographical Information Systems (GIS) and census

data

• Could be an interesting application on future broadly deployed PC’s • Visualize nicely on Google Maps (and presumably Microsoft Virtual Earth)

n Applied to several Cheminformatics problems and have parallel

efficiency but visualization harder as in 150-1024 (or more) dimensions

n Will develop a family of such parallel (annealing) data-mining

tools where basic approach known for

• Clustering and Gaussian Mixtures (Expectation Maximization) • Mapping High Dimensional Spaces – MDS and GTM

Clustering Data

n Cheminformatics was tested successfully with small datasets and

compared to commercial tools

n Cluster on properties of chemicals from high throughput

screening results to chemical properties (structure, molecular weight etc.)

n Applying to PubChem (and commercial databases) that have

6-20 million compounds

• Comparing traditional fingerprint (binary properties) with real-valued

properties

n GIS uses publicly available Census data; in particular the 2000

Census aggregated in 200,000 Census Blocks covering Indiana

• 100MB of data

n Initial clustering done on simple attributes given in this data

• Total population and number of Asian, Hispanic and Renters

n Working with POLIS Center at Indianapolis on clustering of

SAVI (Social Assets and Vulnerabilities Indicators) attributes at http://www.savi.org) for community and decision makers

Parallel Multicore

Deterministic Annealing Clustering

“Constant1”

Increasing number of clusters decreases communication/memory bandwidth overheads

Parallel Overhead for large (2M points) Indiana Census clusterin on 8 Threads Intel 8

Parallel Multicore

Deterministic Annealing Clustering

“Constant1”

Increasing number of clusters decreases communication/memory bandwidth overheads

Parallel Overhead for subset of PubChem clustering on 8 Threads (Intel 8b

The fluctuating overhead is reduced to 2% (as bits not doubles

Visualizing High Dimensional Spaces

n For GIS we have a 2D or 3D underlying space and visualization

of data correlations is clear

n For cheminformatics we have hundreds (continuous) to

thousands (binary) degrees of freedom and difficult to evaluate any data mining – in particular clustering

n Need to map high dimensional space into lower (here two)

dimensional space with some constraints

n This field has been well studied and there are at least two

approaches

• SOM (Self Organizing Maps) and GTM (Generative Topographic

Mapping) which are really clustering algorithms with a built in 2D organization for clusters

• MDS (Multi Dimensional Scaling) which “just views this as an

optimization problem”.

• Principal Component Analysis (PCA) is here viewed as a special case of

MDS Multi Dimensional Scaling

n This is rather straightforward (and perhaps good for that

reason).

n Consider n points Y which are vectors in a high dimensional

space that we want to map into n points X in a low dimensional space (bad notation)

n Let us minimize

Stress (X) = i<j=1n weight(i,j) (ij- d(Xi , Xj))2

n _ij is distance between original vectors Y in high dimensional

space but it can be any scoring of discrepancy between points i

and j.

n d(X_i , X_j) is distance between mapped vectors

n Here simplest choice is weight(i,j) = 1 but one can also look at

choices like Sammon’s mapping

PCA Principal Component Analysis I

n This is closely related to analysis used in WNPF approach to

nonlinear fitting and DA method for determining whether to split clusters. We form DD matrix PCA which depends on

vectors Yk in original space and a “mean” M about which we do

the expansio

n For cluster splitting the mean M is a cluster center and there is

a kernel which is probability that Yk belongs to a given cluster

n For MDS, the kernel is often absent and the mean M is the

PCA Principal Component Analysis II

n

The PCA approach finds the eigenvalues and

eigenvectors of the D



D matrix

PCA

and maps point

point Y

into its expansion in the first 2 (or choose here

lower dimensional space) PCA eigenvectors with

largest eigenvalues

n

This is optimal solution to minimizing stress under two

conditions

• One only looks at mappings Y_k  X_k that are orthogonal

projections

• Choice of stress as _i<j=1n (_ij2- d(X_i , X_j)2)

n

For example there are “better” linear transformations

that add scaling of PCA vectors

Classical MDS: SMACO

Scaling by minimizing a complicated function

n This is a variant of steepest descent approach to minimizing

Stress. One sets two matrices V and B in nL dimensional space (L=2 is dimension of target space of mapping).

• n is number of vectors Y_k to be mappe

n Then one can establish a rigorous bound to show that one always

decreases Stress (X) by iteration labeled by t i

More obvious MDS for Clustering or

Mixture Model Mapping

n

The number of parameters in MDS is typically modest as in

simple cases just

2

(dimension of target space)



Number of

Clusters

(or Number of points to be mapped)

n

Thus Newton’s method should work well and it is trivial to

derive first and second order derivatives for

(X)

n

As well as “general” approaches to regularization, one can

enhance convergence by

•

Initializing optimization

with SMACOF iterations

•

Noting that DA clustering algorithm gives us results with

number of clusters increasing by one each time. Thus we

can initialize optimization for n clusters with results

GTM Generative Topographic Mapping

n This addresses a slightly different problem of mapping ALL

points in a high dimensional space. It approaches this by simultaneously clustering in high dimensional space and mapping cluster (centers) to low dimension space

• We ignore clustering and view as a way to getting mapping by

looking at all points in space not just cluster centers

n The key idea is a nonlinear mapping

• Y(k) = _m=1M W_m_m(X(k))

• _m(X) = exp( - 0.5 (X-_m)2/2 ) for fixed _m and 

n Y(k) are cluster centers in high dimensional space which map

from X(k) in lower dimensional space

n If  largish compared to separation between basis vectors X(k)

GTM for visualizing 2 Clusters in 155 Dimensions

n Here GTM just

used to visualize.

n Clustering done

separately

n Deliberately “easy”

problem!

n Note although

formal clustering gives 2 clusters

n GTM visualization

uses

n K=225 clusters n M=64 basis

functions

Basic Gaussian Mixture Models

n Mixture models write the probability of a point X(x) a

n And set the Liklihood L = _x=1N p(x,) where parameters  are centers

Y(k), Probability Pk that point generated by center k and covariance matrix

(k)2

n Expectation Maximization is (from 1977) iterative solution to minimizing –

lnL

n It starts

by estimatin

n And using this to calculate al

the  using formulae like:

n Note similarity to deterministic annealing cluster formulae

n Deterministic annealing EM algorithm - N Ueda, R Nakano - Neural Networks, 1998

Cited by 124

n A different approach which does not seem general is in

n Deterministic annealing for density estimation by multivariate normal mixture

DA / EM / DAEM / GTM I

n The General Formula for “F = - lnL” or “Free Energy” F = E-TS is

n Note x is a label and E(x) is data n For Deterministic Annealing

• a(x) = 1/N or generally p(x) with  p(x) =1

• g(k)=1

• s(k)=0.5

• T is annealing temperature varied down from  with final value of 1

• Vary Y(k) but can calculate P_kand(k) (even for matrix (k)) using IDENTICAL

formulae for Gaussian mixtures; iteration formula identical to Expectation Maximization EM which is steepest descent

• K starts at 1 and is incremented by algorithm

n For traditional Gaussian mixture models simplified to spherical distributions ((k) is

really a k  k symmetric correlation matrix)

• a(x) = 1

• g(k) = P_k/(2(k)2)D/2 where space D dimensional • s(k) = (k)2

• T = 1

DA / EM / DAEM / GTM II

n Note x is a label and E(x) is data

n For Deterministic Annealing Gaussian mixture models simplified

to spherical distributions ((k) is really a DD symmetric correlation matrix)

• a(x) = 1

• g(k)={P_k/(2(k)2)D/2}1/T • s(k)= (k)2

• T is annealing temperature varied down from  with final

value of 1

• Vary Y(k) P_k and (k)

DA / EM / DAEM / GTM III

n Note x is a label and E(x) is data

n For GTM Generative Topographic Mapping

• a(x) = 1

• g(k) = (1/K)( /2)D/2 where space D dimensional • s(k) = 1/ 

• T = 1

• Y(k) = _m=1M W_m_m(X(k))

• Fix _m(X) = exp( - 0.5 (X-_m)2/2 ) but other choices possible • Vary W_m and but fix values of M and K a priori

n Y(k) E(x) W_m are vectors in original high D dimensional space n X(k) and _m are vectors in 2 dimensional mapped space

There is presumably a version of GTM using deterministic annealing using either a pure cluster basis – anneal in  o

Some links for Parallel Computing

n

See

http://www.connotea.org/user/crmc

for

references

-- select tag

oldies

for venerable links; tags like

MPI

Applications Compiler

have obvious significance

n

h

ttp://www.infomall.org/salsa f

or recent work

including publications

n

My tutorial on parallel computing

Parallel Programming Model

n If multicore technology is to succeed, mere mortals must be able to

build effective parallel programs on commodity machines

n There are interesting new developments – especially the new Darpa

HPCS Languages X10, Chapel and Fortress

n However if mortals are to program the 64-256 core chips expected in 5-7

years, then we must use near term technology and we must make it easy

• This rules out radical new approaches such as new languages

n Remember that the important applications are not scientific computing

but most of the algorithms needed are similar to those explored in scientific parallel computing

n We can divide problem into two parts:

• “Micro-parallelism”: High Performance scalable (in number of

cores) parallel kernels or libraries

• Macro-parallelism: Composition of kernels into complete

applications

n We currently assume that the kernels of the scalable parallel

algorithms/applications/libraries will be built by experts with a

n Broader group of programmers (mere mortals) composing library

Multicore SALSA at CGL

n Service Aggregated Linked Sequential Activities

n Aims to link parallel and distributed (Grid) computing by

developing parallel applications as services and not as programs or libraries

• Improve traditionally poor parallel programming

development environments

n Developing set of services (library) of multicore parallel data

mining algorithms

n Looking at Intel list of algorithms (and all previous experience),

we find there are two styles of “micro-parallelism”

• Dynamic search as in integer programming, Hidden Markov Methods

(and computer chess); irregular synchronization with dynamic threads

• “MPI Style” i.e. several threads running typically in SPMD (Single

Program Multiple Data); collective synchronization of all threads together

n Most Intel RMS are “MPI Style” and very close to scientific

Scalable Parallel Components

n There are no agreed high-level programming environments for

building library members that are broadly applicable.

n However lower level approaches where experts define

parallelism explicitly are available and have clear performance models.

n These include MPI for messaging or just locks within a single

shared memory.

n There are several patterns to support here including the

collective synchronization of MPI, dynamic irregular thread parallelism needed in search algorithms, and more specialized cases like discrete event simulation.

n We use Microsoft CC

There is MPI style messaging and ..

n OpenMP annotation or Automatic Parallelism of existing

software is practical way to use those pesky cores with existing code

• As parallelism is typically not expressed precisely, one needs luck to get

good performance

• Remember writing in Fortran, C, C#, Java … throws away information

about parallelism

n HPCS Languages should be able to properly express parallelism

but we do not know how efficient and reliable compilers will be

• High Performance Fortran failed as language expressed a subset of

parallelism and compilers did not give predictable performance

n PGAS (Partitioned Global Address Space) like UPC, Co-array

Fortran, Titanium, HPJava

• One decomposes application into parts and writes the code for each

component but use some form of global index

• Compiler generates synchronization and messaging

• PGAS approach should work but has never been widely used – presumably

Summary of micro-parallelism

n

On

new applications

, use MPI/locks with explicit

user decomposition

n

A subset of applications can use “

data parallel

”

compilers which follow in HPF footsteps

•

Graphics Chips and Cell processor motivate such

special compilers but not clear how many

applications can be done this way

n

OpenMP and/or Compiler-based Automatic

Composition of Parallel Components

n The composition (macro-parallelism) step has many excellent solutions

as this does not have the same drastic synchronization and correctness constraints as one has for scalable kernels

• Unlike micro-parallelism step which has no very good solutions

n Task parallelism in languages such as C++, C#, Java and Fortran90; n General scripting languages like PHP Perl Python

n Domain specific environments like Matlab and Mathematica n Functional Languages like MapReduce, F#

n HeNCE, AVS and Khoros from the past and CCA from DoE

n Web Service/Grid Workflow like Taverna, Kepler, InforSense KDE,

Pipeline Pilot (from SciTegic) and the LEAD environment built at Indiana University.

n Web solutions like Mash-ups and DSS

n Many scientific applications use MPI for the coarse grain composition

as well as fine grain parallelism but this doesn’t seem elegant

n The new languages from Darpa’s HPCS program support task

parallelism (composition of parallel components) decoupling

Integration of Services and “MPI”/Threads

n Kernels and Composition must be supported both inside chips (the multicore

problem) and between machines in clusters (the traditional parallel computing problem) or Grids.

n The scalable parallelism (kernel) problem is typically only interesting on true

parallel computers (rather than grids) as the algorithms require low communication latency.

n However composition is similar in both parallel and distributed scenarios and

it seems useful to allow the use of Grid and Web composition tools for the parallel problem.

• This should allow parallel computing to exploit large investment in service

programming environments

n Thus in SALSA we express parallel kernels not as traditional libraries but as

(some variant of) services so they can be used by non expert programmers

n Bottom Line: We need a runtime that supports inter-service linkage and

micro-parallelism linkage

n CCR and DSS have this property

• Does it work and what are performance costs of the universality of

runtime?

• Messaging need not be explicit for large data sets inside multicore node.