Incremental Support Vector Machine Classification

Second SIAM International Conference on Data Mining Arlington, Virginia, April 11-13, 2002

Glenn Fung & Olvi Mangasarian

Data Mining Institute

University of Wisconsin - Madison

Key Contributions

Fast incremental classifier based on PSVM Proximal Support Vector Machine

Capable of modifying an existing linear classifier by both adding and retiring data

Extremely simple to implement

Small memory requirement Even for huge problems (1 billion)

NO optimization packages (LP,QP) needed

Outline of Talk

(Standard) Support vector machines (SVM) Classification by halfspaces

Proximal linear support vector machines (PSVM) Classification by proximity to planes

The incremental and decremental algorithm Option of keeping or retiring old data

Numerical results1 Billion points in 10 dimensional space classified in less than 3 hours! Numerical results confirm that algorithm time is linear in the number of data points

Support Vector MachinesMaximizing the Margin between Bounding

Planes

x0w = í + 1

x0w = í à 1

A+

A-

jjwjj22

w

Proximal Support Vector MachinesFitting the Data using two parallel

Bounding Planes

x0w = í + 1

x0w = í à 1

A+

A-

jjwjj22

w

Standard Support Vector MachineAlgebra of 2-Category Linearly Separable Case

Given m points in n dimensional space Represented by an m-by-n matrix A Membership of each in class +1 or –1 specified by:A i

An m-by-m diagonal matrix D with +1 & -1 entries

D(Awà eí )=e;

More succinctly:

where e is a vector of ones.

x0w = í æ1: Separate by two bounding planes,

A iw=í + 1; for D i i = + 1;A iw5í à 1; for D i i = à 1:

Standard Support Vector Machine Formulation

Margin is maximized by minimizing21kw;í k2

2

÷> 0 Solve the quadratic program for some :

2÷kyk2

2 + 21kw;í k2

2

D(Awà eí ) + y > ey;w;ímin

s. t.(QP)

,

, denoteswhere D ii = æ1 A+ Aàor membership.

PSVM Formulation

We have from the standard QP SVM formulation:

w;í (QP)2÷kyk2

2 + 21kw;í k2

2

D(Awà eí ) + y

min

s. t. = e=

This simple, but critical modification, changes the nature of the optimization problem tremendously!!

Solving for in terms of and gives:

minw;í 2

÷keà D(Awà eí )k22 + 2

1kw; í k22

y w í

Advantages of New Formulation

Objective function remains strongly convex.

An explicit exact solution can be written in terms of the problem data.

PSVM classifier is obtained by solving a single system of linear equations in the usually small dimensional input space.

Exact leave-one-out-correctness can be obtained in terms of problem data.

Linear PSVM

We want to solve:

w;ímin

2÷keà D(Awà eí )k2

2 + 21kw; í k2

2

Setting the gradient equal to zero, gives a nonsingular system of linear equations.

Solution of the system gives the desired PSVM classifier.

Linear PSVM Solution

H = [A à e]Here,

íw

h i= (÷

I + H 0H)à 1H 0De

The linear system to solve depends on:

H 0H(n + 1) â (n + 1)which is of size

is usually much smaller than n m

Linear Proximal SVM Algorithm

Classifier: sign(w0x à í )

Input A;D

Define H = [A à e]

Solve (÷I + H 0H) í

wh i

= v

v = H0DeCalculate

Linear & Nonlinear PSVM MATLAB Code

function [w, gamma] = psvm(A,d,nu)% PSVM: linear and nonlinear classification% INPUT: A, d=diag(D), nu. OUTPUT: w, gamma% [w, gamma] = psvm(A,d,nu); [m,n]=size(A);e=ones(m,1);H=[A -e]; v=(d’*H)’ %v=H’*D*e; r=(speye(n+1)/nu+H’*H)\v % solve (I/nu+H’*H)r=v w=r(1:n);gamma=r(n+1); % getting w,gamma from r

Incremental PSVM Classification

E = A1 à eA2 à e

ô õ) E = E 1

E 2

ô õ

) E0E = E1

E2

ô õ0

E1 E2[ ]= E01E1 + E0

2E2

íw

h i= (÷

I + E01E1+ E0

2E2)à 1(E0

1D1e+ E0

2D2e)

The linear system to solve depends on the compressed blocks:

(n + 1) â (n + 1)which are of the size

E01E1; E0

2E2

A1 2 Rm1â n A2 2 Rm2â nand

Suppose we have two “blocks” of data

Linear Incremental Proximal SVM Algorithm

InitializationE 0E = 0;d = 0;i = 1

A i; di Read from disk

E i 0

E i

di = E i 0

D ie(n + 1) â (n + 1)

(n + 1) â 1

Compute andStore in memory

i = imax?

i = i + 1

YesCompute output

w; í

E 0E = E 0E + E i 0

E i

d = d+ di

Update in memory

No

Discard:

Keep:

A i;D i;E i;di

E 0E ;d

Linear Incremental Proximal SVM Adding – Retiring Data

Capable of modifying an existing linear classifier by both adding and retiring data

Option of retiring old data is similar to adding new data

Financial Data: old data is obsolete

Option of keeping old data and merging it with the new data:

Medical Data: old data does not obsolesce.

Numerical experimentsOne-Billion Two-Class Dataset

Synthetic dataset consisting of 1 billion points in 10- dimensional input space Generated by NDC (Normally Distributed Clustered) dataset generatorDataset divided into 500 blocks of 2 million points each.Solution obtained in less than 2 hours and 26 minutes About 30% of the time was spent reading data from disk.Testing set Correctness 90.79%

Numerical Experiments Simulation of Two-month 60-Million Dataset

Synthetic dataset consisting of 60 million points (1 million per day) in 10- dimensional input space Generated using NDC At the beginning, we only have data corresponding to the first month Every day:

The oldest block of data is retired (1 Million) A new block is added (1 Million) A new linear classifier is calculated daily

Only an 11 by 11 matrix is kept in memory at the end of each day. All other data is purged.

Numerical experimentsSeparator changing through time

Numerical experiments Normals to the separating hyperplanes

Corresponding to 5 day intervals

Conclusion

Proposed algorithm is an extremely simple procedure for generating linear classifiers in an incremental fashion for huge datasets. The linear classifier is obtained by solving a single system of linear equations in the small dimensional input space. The proposed algorithm has the ability to retire old data and add new data in a very simple manner. Only a matrix of the size of the input space is kept in memory at any time

Future Work

Extension to nonlinear classification

Parallel formulation and implementation on remotely located servers for massive datasets

Real time on-line application, e.g. fraud detection