A Robbins-Monro type multidimensional stochastic approximation algorithm which converges in mean square and with probability one to the fixed point of a locally contractive regression function is developed. The algorithm is applied to obtain maximum likelihood estimates of the parameters for a mixture of multivariate normal distributions

Lawton, W. M.

English

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MULTIuIMFN. STOCHASTIC APPROX.
USING LOCALLY CONTRACTIVE FNS.
w. M. LAwTnN, INST FUR ADV STUDY
PRINCETON, NEW JERSEY
REPORT 946 AUGUST 1915
YR
RECEIVED
NASA STi FACIL/(y,
fhpUj tim?4C?{
3801 CULLEN BLVD.
HOUSTON. TEXAS 77004
C	 MuLti.dimenzionat Stochastic Appft.oximati.on USing
Locatt y Contka.ct.i.ve
 Func.ti.on6
August, 1975
1i
Bibliography
1. C.C. Blaydon, K.S. Fu, and R.L. Kashyap, "Stochastic approxin
Adaptive, Learning and Pattern Recognition Systems, Academic
York and London, 1970, Edited.by J.M. Mendel and K.S. Fu.
2. A.S. Householder, Theory of Matrices and Numerical Analysis, Blaisdell
Publishing Co., New York, 1964.
3. B. Charles Peters, Jr. and Homer F. Walker, "An iterative procedure for
obtaining maximum-likelihood estimates of the parameters for a mixture
or normal distributions", to appear.
k
ii
Multidimensional Stochastic Approximation
Using Locally Contractive Functions
b 1. Summary. A Robbins-Monro type multidimensional stochastic approximation
algorithm which converges in mean square and with probability one to the fixed
point of a locally contractive regression function is developed. The algorithm
is applied to obtain maximum likelihood estimates of the parameters for a mixture
of multivariate normal distributions.
2. Introduction. Let E  be.real k-dimensional Euclidean space with inner
product denoted by < , > and norm denoted by 	 Corresponding to every
positive definite real k xk matrix B we define the B-inner product
<x,y>B
 = <x,By> and the B-;-norm 11 X 11 B 	<x,Bx>1/2 , for x,y e Ek . A
function F:D	 Ek, where, D is an open subset of E k , is locally contractive
at a point eo e D if there. exists a B-norm on E  and a number a,
0 < a < 1 such that
(2.1)_Ile° - F (e) 11 B <<- ^ Jl e - elIB
whenever 6 is sufficiently near a	 If the above inequality holds for every
6 in some neighborhood W of Ef , we say F is X-locally contractive at
8 0
 throughout W. Clearly, e o will be a unique fixed point in -W for T.
For amy kxk matrix A, let the spectral radius of A be denoted by
p (A) = sup{Ia i:a is an eigenvalue of Al. The Frechet derivative of F, if
it exists, will be denoted by F. The .following result, a consequence of
Taylor's theorem and the theory in [2; section 2.3], will be used in part
j
Yi
2
4 of this paper.
c
(2.2) Lemma If QF exists and is continuous in a neighborhood of 8 0 , a
necessary and sufficient condition for F to be locally contractive at eo
is that F(8°) = 8 0 and p(VF(8o)) ' < 1.
Let {Y(8):8 E D} be a family of random variables with values in E 
satisfying the following conditions
(2.3)	 sup  E( Y(8) 2 ) < - .(E denotes conditional expectation with 6 fixed)
(2.4)
	
	
the regression function of {Y(8)}, denoted by M(8) = E(Y(8)), can
be expressed as M(8) = 8 -T(0) where F(8) is locally contractive
0
at 8 E D.
In part 3 of this paper we develop an algorithm which, given the conditions
above and given a sufficiently close approximation to 60, yields a sequence
of recursively defined random variables with values in E k which converges to
8' in mean square and with probability one.
In part 4 of this paper, this algorithm is used to formulate a stochastic
analog of the iterative procedure developed by Peters and Walker in [3] for
obtaining maximum likelihood estimates of the parameters for a mixture of
normal distributions.
(3) Derivation of the Algorithm
-
(3.1) Lemma Let pl	 (p(B-1)) 1 and p2 p(B) where B is a positive
r	 definite kxk matrix, and let rl ,r2 be positive real numbers such that
r
i
i
i
i3
Arl 
< 1. Let e° , 6 E Ek such that I10r - 811 < r  and .let
c.	 2
	
P2
S = {8 E Ek :116 
	
5 r2 }. Define a function I:Ek--ter S as follows
a
0 if e E S
( )r
(1-t)0 + t0 where t
	
	 2 =	 if 6 S
Il a-e II
Then the following inequality holds for every e E E 
Ile - (e)i1B < Il e - 6I1B
r
Proof. We may assume t 	 2 < 1 for otherwise e E S implying
0e-eN
Ve) = 0. By the fundamental theorem of calculus:
37
d
eqB = Ike° - (e)I(B +
	
	
a—s[tie - (1-s)6 - s0`+sIds
S=t
It suffices to prove that the integrand is nonnegative for all s, t _< S <_ 1.
Consider	 j
a—s[Ile - (1-s)e se^rB] = 2<e - e,e° (1-s - se>B
= 2.s<e - 6,6	 a>B-+ 2<e	 6,e° - 6>B.
}9
Since t 5 s 5 1, by the principle axis theorem for real symmetric matrices,
a
r I C	 ',
4
the left side of the above expression is bounded below by 2 rZl`6
—e1I
 Pl'
Similarly, the right hand side of the above expression is bounded, in absolute
<. c
value, by 2 r1116 -el)p 2 . The result follows from the hypothesis concerning
r l ,r2 ,P1
 and p2.
For the remainder of part 3 we adopt the notation and assumptions stated
previously. Furthermore, we assume that F is locally contractive at 66
throughout S and that S c D. Define a family {Y(6) :6 e E k I of random
variables by
(3.2)	 Y(e) = Y(I(e)) - q(6) + 6
Then the following inequality is valid for every E > 0.
(3.3) Corollary	 inf	 E(<e-e , Y(e)> ) > 0
Proof. Since E(Y(e)) = q (6) - F(c^(e)) - ?(e ) + e, the expression above =
inf	 f<6-E° ,e—e° B+<e—e ,e F(Ve))>B 1.
 The first term above = 110—e°il B,
E _ U0-
`^
and the second term is bounded above in absolute value by
qe- Ek 11 F (1( 0)) - 0 11B
s wile— e% 111(e)	 a II B	 (by (2.1) )
s x119-6°112 	 (by (3.1) )
Therefore	 _inf	 E(<e-6* ! Y(.e) >B) z (1	 a)E2*> 0•
F-:5 de-611B
F	 ^	 1 1
r
5
(3.4) Definition. A gain sequence is a sequence {a R} of positive numbers
CO	 00
= CO and a2 < 0O•is	 satisfying	 aQ QQ-1 Q -1 6
(3.5)	 Remark.	 For any	 C > 0, {aQ = Q}
	 is a gain sequence since
•	 limit 2_k = limit c log K = OO and
a2 = Or	
.
k -} oo Q=1	 k -* co Q=1
(3.6) Theorem. Let {Y(8):8 E Ek} be as in (3.2) and let {aY,} be a gain
sequence. Then the following sequence of recursively defined random vectors
(3.7)	 8R+1 = 8Q - a.Y(8Q), 8 1 arbitrarily chosen converges in mean square
and with probability one to 8
Proof. We refer the reader to the algorithm described in [l,pp 332-333] and
the convergence proof given in the appendix to [l,pp 350-352]. Replacing
their gain sequence {pk} with the gain sequence {a,}, and replacing their
norm 11•11 and inner product <V,W> = V W where V denotes the transpose
of the vector V and V W denotes matrix multiplication) with the B-norm
and B-inner product respectively, the theorem will follow once we verify that
n	 conditions (Al) - (A3) in [l,pp 332-3331 are satisfied.
•	 (Al) Since E(Y(A)) = 8 - F(q(8)), this result follows from (3.3) since B
positive definite implies lie- ell B = 0 if and only if 8	 8'
(A2) Follows from (3.3)
(A3) E(IIY(8)I{ B) = E(<Y(1(8)) - ^(0) + 8 ^ Y(^(e)) - Ve) + 6>B)
< h(1 + [I8-AIt2) for sufficiently large h > 0 since S c D and by (2.3) 	 j
sup	 2) < 00.
eED
r
fr
6
4. Application to Maximum Likelihood Estimation
ti
Let D = {(ai' ui' E i)} i=1	 m where each ai > 0 with > ai = 1,,...,	 i=1
each ui E Rn and each E  is a positive definite real symmetric n xn matrix.
We consider D to be realized as an open subset D of E  where
k = m(n+1)2 (n+2) - 1. For each 8 = (ai' ui' E i) i=1	 m let x(6) be a random,...,
4	
variable with values in Rn and with distribution function
m
P(8,x) _
	
	
ai pi (x)	 for x E Rn
i=1
where
Pi(e,x) = ( 27r) -n/2 1Eij -1/2exp{- 2( x-ui) TE il (x_p. }
for each i = 1,...,m.
Fix e E D and let {xk }k=1,...,Nc Rn be an independent sample of observations—
of x(8'). A maximum-likelihood estimate of eo based on {xk} is a choice
Of 8 E D which locally maximizes the log-likelihood function
N
L	 log p(8,xk)
k=1
In the appendix to ['3], Peters and Walker prove there exists a'sufficiently
a
small neighborhood of 8 o , such that with probability -> l as N-* -, there
exists a unique maximum-likelihood estimate of 8 ` in that neighborhood.
Furthermore, with probability -} 1 as N 	 this estimate + 8' This
estimate will be called the consistent maximum-likelihood estimate (which we
abbreviate by c.m.l.e.)
Equating DeL = 0 and performing algebraic manipulation of the resulting
0
equations, yields the following necessary conditions for a m.l.e. e of 6
8 = Z (e) where TD -} D is defined by
((ai' ui' ^i )
 i=1, ... ,m) - (a, ui' ^i)
where, for each i = 1,...,m
	
a c
	 P ( )
	
= i L	 i xk(4.1)
	
ai 
N k=1 xk p (x,,)
1	
1 
N	 pl(xk)	
1 
N pi(xk)
(4.2) 'Pi - {N	
xk P( )}	 {N	 P( )}
1 N T pi(xk)	 1 SSN p,(xk)(4.3)	 Ei = {N l ( — i) (xk- i)	 P(xk) }	 {N	 p(xk) }
i
a
a
3
where each p i
 and p is evaluated with respect to the parameters
E) = (ai $PV E )i=1	 m j will be called the likelihood function.
In [3], Peters and'Walker-develop an iterative procedure which, starting
with any initial estimate e' which is sufficiently close to 8` , yields a
sequence in D converging to the c.m.l.e. of e" based on {xkj=1	 N'
Their technique consists in proving that, for s <	 4	 ' the functionm(n+1) (n+2) '.
(4.4)	 E(e) = (1-6)e + e	 (e)
is locally contractive (at the c.m. l. e. of e ) throughout a neighborhood of
a
i
8j^ 	 6	 Thus, for any e' in this neighborhood of 6 , the sequence {e^} defined
z
c	 recursively by	 j
(4.5)	 ek+1 -	 E (e^)., e l = e'
a
converges to the c.m.l.e. of 6 .
In concluding, they discuss the computational advantages of this procedure
as compared to classical numerical techniques such as Newton's method or the
method of scoring. In particular, the procedure satisfies the following con-
ditions.
(4.6) At each stage of the..iteration in (4,5), the constraints on the
parameters in e are satisfied.
`
(4.7) The 'step size' € depends only on n and m and not on 6
e .
(4.8) The procedure does not.require the inversion, at each stage of the
i
iteration, of a kxk matrix.
i
We will present a stochastic approximation analog of the iterative procedure
defined by (4.5). In contrast to the classical stochastic method of scoring,
our procedure satisfies the conditions *(4.6) and (4.7). A step size E
R will not appear explicitly in our algorithm.
a
Fix e e D and let{ }	 be an infinite sequence of independent
xk k=1 , ., ,^
samples of observations of x(e°). For any function g from R n to any real
vector space V, let
1
3
E (g)
	
	
g(x)p(6',x)dx
Rn
i
t	 I 1	 r	 i
1
9
e
be the expectation, if it exists, of gox(e°). (o denotes composition of
:I	 functions). Then, by the strong law of large numbers, with probability one
^ b
as N 4. 00 , the equations (4.1) - (4.3) converge to
P
R	 (4.9)	 ai = aiE(pi)
Pi	(4.10)	 ui = E(x p)/E(Pi)
Pi	(4.11)
	 ^i = E((x-pj) (x-p i) T P)/E( pi) .
We denote the corresponding limiting value of the likelihood function by 	 .
Clearly ,T(e) is a continuously differentiable function of e and( 8u )	 8`;
also by (4. 4)
	 E (e) _ (1-e)e + E	 (e) is locally contractive at A0	 By
(2.2), p(0 Z E (e o )) < 1, implying that the eigenvalues of P 1(0 0 ) have real
parts strictly less than 1. Now define a function d:D + D by
(4.12)
	
, m)	 (cci ,otiui'ai^i i=1, ... ,m.
Clearly d is a differentiable function from D -> D such that d-1 exists
and is differentiable. Define a function' 	 :D } D by 	 (e) d	 d l(8).
By the chain, rule for Frechet Derivatives,
-_.	 V f' (d(e°))	 [vd(e°) ] [a j ( e°) ] [od(e °) ]-1
i
_f
si
r
10
hence the function
(4.13)	 ^'	 _ ( 1-e) © + e' (6)E (6)
is locally contractive throughout some neighborhood	 W of	 d(6').
Define a family	 {Y(6):6 E D}	 of random variables with values in	 E 	 by
Pi' Pi
11
ui ) T pl)Pi,(4.14)	 Y(6)	 = e(a.-a. 	 }t. -a.x(6^)	 ,	 E .-a. (x(6°i	 i 	 i	 i )-	 (xA- .p	 p ai ai	 i.=1,p	 ... ,m
where	 pi = pi (d 1(e),x(e"))	 and	 p' = p (d 1(6) ,x(6p))•
Then	 E(Y(6)) = 6	 Therefore, the family {Y(6):6 E D} satisfies
conditions	 (2.3)	 and	 (2.4).
Let	 {Y ( 6) :e E EkI	 be constructed from	 {Y(6):6 E D}	 as in	 (3.2) and
let	 {aQ}	 be any gain sequence. 	 Then by	 (3.6),	 the sequence in (3.7)
0
converges in mean square and with probability one to 	 d(6 ).	 Since {eaQ }	 is
a gain sequence whenever 	 {a!C}	 is a gain sequence,	 a need not appear explicitly
in the sequence in(3.7).	 -
s
i
i
,
3
r
k


Multidimensional stochastic approximation using locally contractive functions

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