程序代写代做代考 ER algorithm Untitled

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Journal of Machine Learning Research 10 (2009) 883-906 Submitted 4/08; Revised 1/09; Published 4/09

Estimation of Sparse Binary Pairwise Markov Networks using
Pseudo-likelihoods

Holger Höfling HHOEFLIN@GMAIL.COM
Department of Statistics
Stanford University
Stanford, CA 94305, USA

Robert Tibshirani TIBS@STANFORD.EDU
Depts. of Health, Research & Policy and Statistics
Stanford University
Stanford, CA 94305, USA

Editor:Michael Jordan

Abstract
We consider the problems of estimating the parameters as well as the structure of binary-valued
Markov networks. For maximizing the penalized log-likelihood, we implement an approximate
procedure based on the pseudo-likelihood of Besag (1975) and generalize it to a fast exact algo-
rithm. The exact algorithm starts with the pseudo-likelihood solution and then adjusts the pseudo-
likelihood criterion so that each additional iterations moves it closer to the exact solution. Our
results show that this procedure is faster than the competing exact method proposed by Lee, Gana-
pathi, and Koller (2006a). However, we also find that the approximate pseudo-likelihood as well
as the approaches of Wainwright et al. (2006), when implemented using the coordinate descent
procedure of Friedman, Hastie, and Tibshirani (2008b), are much faster than the exact methods,
and only slightly less accurate.
Keywords: Markov networks, logistic regression, L1 penalty, model selection, Binary variables

1. Introduction

In recent years a number of authors have proposed the estimation of sparse undirected graphical
models for continuous as well as discrete data through the use of L1 (lasso) regularization. For
continuous data, one assumes that the observations have a multivariate Gaussian distribution with
mean µ and covariance matrix Σ. Then an L1 penalty is applied to Θ = Σ−1. That is, the penalized
log-likelihood ℓ(Θ)−ρ||Θ||1 is maximized, where ℓ is the Gaussian log-likelihood, ||Θ||1 is the sum
of the absolute values of the elements of Θ and ρ ≥ 0 is a user-defined tuning parameter. Several
papers proposing estimation procedures for this Gaussian model have been published. Meinshausen
and Bühlmann (2006) develop a lasso based method for estimating the graph structure and give
theoretical consistency results. Yuan and Lin (2007), Banerjee et al. (2008) and Dahl et al. (2008)
as well as Friedman, Hastie, and Tibshirani (2008a) propose algorithms for solving this penalized
log-likelihood with the procedure in Friedman, Hastie, and Tibshirani (2008a), the graphical lasso,
being especially fast and efficient.

Here we focus on estimation of networks of discrete, more specifically binary-valued, units with
pairwise interactions. These are a special class of Markov networks. The use of L1 penalties for this

c⃝2009 Holger Höfling and Robert Tibshirani.

HÖFLING AND TIBSHIRANI

X1

X2X3

X4

Figure 1: A simple graph for illustration.

x1 x2 x3 x4
1 1 1 0
1 1 0 1
0 0 1 0
1 1 0 1
1 1 1 0
1 1 0 1
0 0 1 0
1 1 0 1
0 1 1 0
1 1 0 1

Table 1: Sample data for graph of Figure 1

special class as well as more general Markov networks was proposed by Lee, Ganapathi, and Koller
(2006a). This problem is more difficult than the continuous Gaussian version because of the need
to compute the first and second moments under the model, which are derivatives of the log-partition
function. Figure 1 shows an example, and Table 1 shows some sample data from this graph. Given
this data and a model for binary graphical data (detailed later), we would like to a) infer a structure
something like that of Figure 1, and b) estimate the link parameters itself. For the data in Table 1,
Figure 2 shows the path of solutions for varying penalty parameter. Most edges for L1-norm≤ 2 are
correctly identified as in the graph in Figure 1. However, edge (1,3) is absent in the true model but
included in the L1 penalized model relatively early.

Our main focus in this paper is to develop and implement fast approximate and exact proce-
dures for solving this class of L1-penalized binary pairwise Markov networks and compare the
accuracy and speed of these to other methods proposed by Lee, Ganapathi, and Koller (2006a) and
Wainwright et al. (2006). Here, by “exact” procedures we refer to algorithms that find the exact
maximizer of the penalized log-likelihood of the model whereas “approximate” procedures only
find an approximate solution.

In Section 2 we describe the Ising model as well as the details of the competing methods of
Lee, Ganapathi, and Koller (2006a) and Wainwright et al. (2006). Section 3 then introduces the
basic pseudo-likelihood model and outlines the computational approach to increase the speed of
the algorithm. The pseudo-likelihood is a very interesting and fast approximate method which has

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SPARSE MARKOV NETWORKS

0 1 2 3 4 5 6

−1
.0

−0
.5

0.
0

0.
5

1.
0

L1 norm

C
oe

ffi
ci

en
ts

1−2

1−3

1−4, 2−4

2−3

3−4

Figure 2: Toy example: profiles of estimated edge parameters as the penalty parameter is varied.

the added advantage that it can be used as a building block to a new algorithm for maximizing the
penalized log-likelihood exactly. The adjustments necessary to achieve this are described in Section
4. Finally, Section 5 discusses the results of the simulations with respect to speed and accuracy of
the competing algorithms.

2. The Model and Competing Methods

In this section we briefly outline the competing methods for maximizing the penalized log-likelihood.
Apart from the method proposed in Lee, Ganapathi, and Koller (2006a), which we already men-
tioned above, we also discuss a very simple solution that was presented in Wainwright et al. (2006).
We first describe the underlying model in more detail. Consider data generated under the Ising
model

p(x,Θ) = exp
!


s∈V

θsxs+ ∑
(s,t)∈E

θstxsxt −Ψ(Θ)

.

for a single observation x = (x1, . . . ,xp)T ∈ {0,1}p and model parameters θs and θst for s, t ∈ V =
{1, . . . , p}. Here, V denotes the vertices and E the edges of a graph. Ψ(Θ) is the normalization
constant, which is also known as the log-partition function. By setting θst = 0 for (s, t) ̸∈ E, and
using the fact that x is a binary vector we can write the log-likelihood as

l(x,Θ) = log p(x,Θ) =
p


s≥t≥1

θstxsxt −Ψ(Θ)

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HÖFLING AND TIBSHIRANI

where Θ is a symmetric p× p matrix with θss = θs for s= 1, . . . , p. Note that for notational conve-
nience, we do not distinguish between θst and θts and therefore we enforce symmetry of Θ although
the log-likelihood only uses the lower-triangular matrix of Θ. For the L1-penalty let R be a p× p
lower triangular matrix of penalty parameters. The penalized log-likelihood for all N observations
is

p


s≥t≥1

(XTX)stθst −NΨ(Θ)−N||R∗Θ||1

where ∗ denotes component-wise multiplication.
The algorithms that we will discuss in the following sections could be generalized to more

general categorical variables or higher order interaction terms than those included in the Ising model.
However, as we will see, solving these problems exactly is already computationally demanding in
the pairwise binary case so that we chose not to adapt the algorithms to these more general settings.

2.1 The Lee, Ganapathi, and Koller (2006a) Method

The penalized log-likelihood is a concave function, so standard convex programming techniques
can be used to maximize it. The main difficulty is that the log-partition function Ψ(Θ) is the sum
over 2p elements, and therefore it is computationally expensive to calculate Ψ or its derivatives
in general. However, for sparse matrices Θ, algorithms such as the junction tree algorithm exist
that can calculate Ψ and its derivatives efficiently. Therefore, it is especially important to maintain
sparsity of Θ for any optimization method. Lee, Ganapathi, and Koller (2006a) achieve this by
optimizing the penalized log-likelihood only over a set F of active variables which they gradually
enlarge until an optimality criterion is satisfied.

To be more precise, they start out with a set of active variables F = F0 (e.g., the diagonal of
Θ if it is unpenalized). Using either conjugate gradients or BFGS, the penalized log-likelihood is
maximized over the set of variables F . Then one of the currently inactive variables is selected by
the grafting procedure (see Perkins et al., 2003) and added to the set F . These steps are repeated
until grafting does not add any more features. The algorithm can be used for more general Markov
networks, but for ease of implementation they choose to work with binary random variables and we
do the same as well.

Their procedure provides an exact solution to the problem when the junction tree algorithm
is used to calculate Ψ(Θ) and its derivatives. In their implementation, however, they used loopy
belief propagation, which is faster on denser matrices, but only provides approximate results. In
their method as well as ours, any procedure to evaluate Ψ(Θ) can be “plugged in” without any
further changes to the rest of the algorithm; we decided to evaluate the speed and performance of
only the exact algorithms. The relative performance of an approximate method using loopy belief
propagation would likely be similar. They also provide a proof that under certain assumptions and
using the L1 regularized log-likelihood, it is possible to recover the true expected log-likelihood up
to an error ε.

2.2 The Wainwright et al. (2006) Method

Wainwright et al. (2006) propose estimation of the Markov network by applying a separate L1-
penalized logistic regression to each of the p variables on the remaining variables. For every s ∈V
regress xs onto x\s = (x1, . . . ,xs−1,xs+1, . . . ,xp)T . Let the p× p matrix Θ̃ denote the estimate of Θ.

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SPARSE MARKOV NETWORKS

Then set θ̃ss = β0, the intercept of the logistic regression, and θ̃st = βt , where βt is the coefficient
associated with xt in the regression.

In the outline above, we assumed that Θ is a symmetric matrix. However, due to the way it was
constructed, Θ̃ is not necessarily symmetric. We investigate two methods for symmetrizing Θ̃. The
first way is to define Θ as

θst = θts =

#

θ̃st if |θ̃st | > |θ̃ts|
θ̃ts if |θ̃st |≤ |θ̃ts|

which we call “Wainwright-max”. Similarly, “Wainwright-min” is defined by

θst = θts =

#

θ̃st if |θ̃st | < |θ̃ts| θ̃ts if |θ̃st |≥ |θ̃ts| . Wainwright et al. (2006) mainly intended their method to be used in order to estimate the presence or absence of an edge in the underlying graph of the model. They show that under certain assumptions, their method correctly identifies the non-zero edges in aMarkov graph, asN→∞ even for increasing number of parameters p or neighborhood sizes of the graph d, as long as N grows more quickly than d3 log p (see Wainwright et al., 2008). Due to the simplicity of the method it is obvious that it could also be used for parameter estimation itself and here we will compare its performance in these cases to the pseudo-likelihood approach proposed below and the exact solution of the penalized log-likelihood. Furthermore, as an important part of this article is the comparison of the speeds of the underlying algorithms, we implement their method, using the fast coordinate descent algorithm for logistic regression with a lasso penalty (see Friedman, Hastie, and Tibshirani, 2008b). 3. Pseudo-likelihood Model In this section we first introduce an approximate method to infer the structure of the graph that is based on pseudo-likelihoods (Besag, 1975). As we will see in the simulations section, the results are very close to the exact solution of the penalized log-likelihood. In the next section, we use the pseudo-likelihood model to design a very fast algorithm for finding an exact solution for the penalized Ising model. The main computational problem in the Ising model is the complexity of the partition function. One possibility in this case is to solve an approximate version of the likelihood instead. Approaches of this kind have been proposed in various papers in the statistical literature before, for example the pseudo-likelihood approach of Besag (Besag, 1975) and the treatments of composite likelihoods in Lindsay (1998) and Cox and Reid (2004) among others. Here, we want to apply the approximation proposed in Besag (1975) to our problem. This approach is also related to the method of Wainwright et al. (2006), however instead of performing separate logistic regressions for every column of the parameter matrix Θ, the pseudo-likelihood approach here allows us to estimate all parameters at the same time. This way, the resulting matrix Θ is symmetric and no additional step like the max or min-rule described above is necessary. The log-pseudo-likelihood is then given by l̃(Θ|x) = p ∑ s=1 log p(xs,Θ|x\s) with log p(xs,Θ|x\s) = xi(θss+∑ t ̸=s xtθst)−Ψs(x,Θ). 887 HÖFLING AND TIBSHIRANI Here,Ψs(x,Θ) is the log-normalization constant when conditioning xs on the other variables, which is exactly the same as in logistic regression with a logit link-function, that is, Ψs(x,Θ) = log(1+ exp(θss+∑ t ̸=s xtθst)) where as above for notational convenience we set θst = θts for t ̸= s. Putting all this together, the pseudo-likelihood for a single observation x is given by l̃(Θ|x) = p ∑ s=1 p ∑ t=1 xsxtθst − p ∑ s=1 Ψs(x,Θ). In the usual way, the pseudo-likelihood for all N observations is given by the sum of the pseudo- likelihood of the individual observations l̃(Θ|X) = N ∑ k=1 l̃(Θ|xk) where xk is the kth row of matrix with observations X ∈RN×p. As this is just a sum of logistic likelihoods, the pseudo-likelihood is a concave function and therefore the L1-penalized pseudo-likelihood N ∑ k=1 l̃(Θ|xk)−N||S∗Θ||1 is concave as well. Here S = 2R− diag(R) and is chosen to be roughly equivalent to the penalty terms in the penalized log-likelihood. The penalty term is doubled on the off-diagonal, as the derivative of the pseudo-likelihood on the off-diagonal is roughly twice as large as the derivative of the log-likelihood (see Equation 1). 3.1 Basic Optimization Algorithm Due to its simple structure, a wide range of standard convex programming techniques can be used to solve this problem, although the non-differentiability of the L1 penalty poses a problem. Here, we want to use a local quadratic approximation to the pseudo-likelihood. As the number of variables is p(p+ 1)/2, the Hessian could get very large, we restrict our quadratic approximation to have a diagonal Hessian. In order to construct the approximation, we need the first and second derivative of l̃ w.r.t θst . These are ∂l̃ ∂θst = 2(XTX)st − (p̂Ts X)t − (p̂ T t X)s s ̸= t, (1) ∂l̃ ∂θss = (XTX)ss− N ∑ k=1 p̂sk where 1− p̂sk = 1/(1+ exp(θss+∑s̸=t Xktθst)). The second derivatives are ∂2 l̃ (∂θst)2 = −(XTdiag(p̂s)diag(1− p̂s)X)tt− (XTdiag(p̂t)diag(1− p̂t)X)ss s ̸= t, ∂2 l̃ (∂θss)2 = − N ∑ k=1 p̂sk(1− p̂sk). 888 SPARSE MARKOV NETWORKS Algorithm 1: Estimation for L1 penalized pseudo-likelihood if Θ(0) not given then Set Θ(0) = diag(logit(p̂(0))) where p̂(0)s = 1N ∑ N k=1 xks end Set k:=0; while not converged do With current estimate Θ(k), define local approximation f Θ(k) (Θ) to l̃−N||S∗Θ||1; Find solution Θ∗ of f Θ(k) (Θ); Perform backtracking line search on the line from Θ(k) to Θ∗ to find Θ(k+1); Set k:=k+1 end Assume that at the k-th step the parameter estimate is Θ(k). Then define the local approximation to l̃(Θ|X)−N||S∗Θ||1 as f Θ(k) (Θ) =C+∑ s≥t ∂l̃ ∂θst (θst −θ (k) st )+ 1 2 ∂2 l̃ (∂θst)2 (θst−θ (k) st ) 2−N||S∗Θ||1 where C is some constant. As stated above, this is just a quadratic approximation with linear term equal to the gradient of l̃ and a diagonal Hessian with diagonal elements equal to the diagonal of the Hessian of l̃. The main reasons for using this simple structure are that it keeps the computation complexity per iteration low and it is very easy to solve this L1 penalized local approximation. Let Θ̃ be the minimizer of the unpenalized f Θ(k) (Θ), then θ̃st = θ (k) st − ∂l̃ ∂θst ∂2 l̃ (∂θst)2 . As the Hessian is diagonal, the L1-penalized solution Θ∗ of fΘ(k) (Θ) can be obtained by soft thresh- olding as θ∗st = sign(θ̃st) $ θ̃st − sst/ ∂2 l̃ (∂θst)2 % + . Using Θ∗, the next step Θ(k+1) can now be obtained by, for example, a backtracking line search. The whole algorithm can be seen in Algorithm 1 and a proof of convergence that closely follows Lee, Lee, Abbeel, and Ng (2006b) is given in the appendix. 3.2 Speed Improvements In practice, there are several things that can be done to speed up the algorithm given above. First of all, as Θ will in general be sparse, all computations to calculate p̂ should exploit this sparse structure. However, the sparseness of Θ can also be used in another way: 3.2.1 USING ACTIVE VARIABLES As the solutions are usually sparse, calculating the gradient for all variables in every step is wasteful. Most variables are zero and will not change from one iteration to the next. In order to be more 889 HÖFLING AND TIBSHIRANI Algorithm 2: Pseudo-likelihood algorithm using active variables if Θ(0) not given then Set Θ(0) = diag(logit(p̂(0))) where p̂(0)s = 1N ∑ N k=1 xks end Set k:=0; Set A = {(s, t) : s≥ t,θst ̸= 0} as active variables; repeat while not converged over variables in A do Find Θ(k+1) using local approximation over variables in A ; Set k:=k+1; end Set A = & (s, t) : θst ̸= 0 or ' ' ' ∂l̃ ∂θst ' ' ' > sst

(

;
until A did not change ;

efficient, it is possible to move variables that are zero only once in a while. Several different kinds of
methods have been proposed to exploit this situation, for example grafting (Perkins et al., 2003) and
the implementation of the penalized logistic regression in Friedman, Hastie, and Tibshirani (2008b)
among others. In our case here, we use an outer and an inner loop. The outer loop decides which
variables are active. The inner loop then optimizes over only the active variables until convergence
occurs. Active variables are those that are either non-zero, or that have a gradient large enough so
that they would become non-zero in the next step. More details are given in Algorithm 2.

When using this method, convergence is still guaranteed. In the outer loop, the criterion chooses
variables to be active that are either non-zero already or will be non-zero after one step of the local
approximation over all variables. Therefore, if the active set stays the same, no variables would be
moved in the next step as all active variables are already optimal and therefore, we have a solution
over all variables. However, if the active set changes, the inner loop is guaranteed to improve the
penalized pseudo-likelihood and find the optimum for the given set of active variables. As there are
only a finite number of different active variables sets, the algorithm has to converge after a finite
number of iterations of the outer loop.

3.2.2 SUBSTITUTING THE LINE SEARCH

Calculating the pseudo-likelihood is computationally expensive compared to the cost of the local
approximation. Therefore, we save time by not performing a line search after every step. Instead,
we fix a step size γ and skip the line search. However, with this method, the algorithm may diverge.
In order to detect this, we calculate the penalized pseudo-likelihood every 100 iterations and check
that we improved during the last 100 steps. If yes, the step size remains the same. If no, reset the
variables to where they were 100 steps ago and divide the step size by 2. If the step size drops below
a pre-specified threshold, we revert to the original line-search algorithm. This way, in most cases,
we do not have to perform the line-search. However, the algorithm is still guaranteed to converge
as it automatically detects convergence problems when the fixed step size is used and reverts to the
line-search algorithm, which as stated above is guaranteed to converge.

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SPARSE MARKOV NETWORKS

4. Exact Solution Using Pseudo-likelihood

We now turn to our new method for optimizing the penalized log-likelihood. As the log-likelihood
is a concave function, there are several standard methods that can be used for maximizing it, for
example, gradient descent, BFGS and Newton’s method among others. For each step of these
methods, the gradient of the log-likelihood has to be calculated. This requires the evaluation of the
partition function and its derivatives, which is computationally much more expensive than any other
part of the algorithms. Taking this into considerations, the standard methods mentioned above have
the following drawbacks:

Gradient descent: It can take many steps to converge and therefore require many evaluations of
the partition function and its derivatives (using the junction tree algorithm). Also, it does not
control the number of non-zero variables in intermediate steps well to which the runtime of
the junction tree algorithm is very sensitive. Therefore, intermediate steps can take very long.

BFGS: Takes less steps than gradient descent, but similar to gradient descent, intermediate steps
can have more non-zero variables than the solution. Thus, same as above, computations of
intermediate steps can be slow.

Newton’s method: In order to locally fit the quadratic function, the second derivatives of the log-
likelihood are needed. Computing these is computationally prohibitive.

Lee, Ganapathi, and Koller (2006a) use the BFGS method only on a set of active variables onto
which additional variables are “grafted” until convergence. This mitigates the problem of slow
intermediate steps and makes using BFGS feasible. However, this comes at the expense of an
increased total number of steps, as only one variable at a time is being grafted. Here, we want to
use the pseudo-likelihood in a new algorithm for maximizing the penalized log-likelihood.

The functional form of the pseudo-likelihood is by its definition closely related to the real like-
lihood and in Section 5 we will see that its solutions are also very similar, indicating that it approxi-
mates the real likelihood reasonably well. Furthermore, we can also maximize the pseudo-likelihood
very quickly. We want to leverage this by using a quasi-Newton method in which we fit a “tilted”
pseudo-likelihood instead of a quadratic function. Specifically, among all functions of the form

f
Θ(k)

=
1
2
l̃+∑

s≥t
ast(θst−θ

(k)
st )− γ∑

s≥t
(θst−θ

(k)
st )

2−N||R∗Θ||1,

we fit the one that at Θ(k) is a first order approximation to the penalized log-likelihood l. Essen-
tially, f

Θ(k)
is an L1-penalized pseudo-likelihood with an added linear term as well as a very simple

quadratic term for some γ> 0. Here, ast will be chosen so that the sub-gradient is equal to the penal-
ized log-likelihood l at Θ(k). The additional quadratic term with coefficient γ is only being included
to ensure the existence of a global maximum. In practice, we can set γ = 0, unless a convergence
problem occurs. In our simulations, γ= 0 always worked very well.

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HÖFLING AND TIBSHIRANI

In particular, for the approximation at Θ(k), choose ast such that

f
Θ(k)

=
1
2

)

l̃(Θ|X)+∑
s>t

(θst−θ
(k)
st )
*

(p̂(Θ(k))Ts X)t +(p̂(Θ
(k))Tt X)s−2 ·N ·wst(Θ

(k))
+

+

+∑
s
(θss−θ

(k)
ss )

)


k
p̂sk(Θ(k))+(XTX)ss−2 ·N ·wss(Θ(k))

,,

−∑
s≥t
γ(θst−θ

(k)
st )

2−N||R∗Θ||1

where W is a matrix with elements wst(Θ) = ∂Ψ∂θst (Θ) = EΘ(xsxt) is the derivative of the partition
function and p̂s is as defined in the pseudo-likelihood section.

For the algorithm, we need an initial parameter estimate Θ(0), which we pick as follows: Let
Z(θ) = e

θ

1+eθ be the logistic function and let Z
−1 denote its inverse. Then choose

θ
(0)
st =

#

0 if s ̸= t
Z−1

– 1
N ∑

N
k=1 xks

.

if s= t
.

In this case we than have p̂sk = 1N ∑
N
k=1 xks ∀k and also

Wst =

#

– 1
N ∑

N
k=1 xks

.- 1
N ∑

N
k=1 xkt

.

if s ̸= t
– 1
N ∑

N
k=1 xks

.

if s= t

and together with γ= 0 we then get that

f
Θ(0)

=
1
2
l̃(Θ|X)−N||R∗Θ||1

and thus just a regular pseudo-likelihood step. The only slight difference is that in this case the
penalty term on the diagonal is twice as large as in the pseudo-likelihood case presented above.
However, in practice we recommend not to penalize the diagonal at all, so that this difference
vanishes. Therefore, this algorithm is a natural extension of the pseudo-likelihood approach that
starts out by performing a regular pseudo-likelihood calculation and then proceeds to converge to
the solution by a series of adjusted pseudo-likelihood steps.

The algorithm for maximizing the penalized log-likelihood is now very similar to the one pre-
sented for maximizing the penalized pseudo-likelihood. Assume our current estimate is Θ(k). Then
approximate the log-likelihood locally by f

Θ(k)
and find the maximizer Θ∗ of f

Θ(k)
. This is essen-

tially the pseudo-likelihood and the algorithm presented above can easily be adjusted to accommo-
date the additional terms. Now, using a line search on the line between Θ(k) and Θ∗, find the next
estimate Θ(k+1). This algorithm is guaranteed to converge by the same argument as the pseudo-
likelihood algorithm. The proof that f

Θ(k)
approximates l at Θ(k) to first order can be found in

Appendix B, which is a prerequisite for the convergence proof of the algorithm in Appendix A.

4.1 Speed Improvement

As for the pseudo-likelihood algorithm, we can again save computations by using the active vari-
ables technique presented above. Here, the savings in time are especially large due to a special fea-
ture of the junction tree algorithm that we use to calculate the derivatives of the partition function.

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SPARSE MARKOV NETWORKS

Algorithm 3: Likelihood algorithm using active variables
if Θ(0) not given then

Set Θ(0) = diag(logit−1(p̂(0))) where p̂(0)s = 1N ∑
N
k=1 xks

end
Set k:=0;
Set A = {(s, t) : s≥ t,θst ̸= 0} as active variables;
repeat
while not converged over variables in A do

CalculateW only over variables in A ;
Find Θ(k+1) using local approximation over variables in A ;
Set k:=k+1;

end
Calculate the whole matrixW;
Set A =

&

(s, t) : θst ̸= 0 or

∂l
∂θst


> rst

(

;
until A did not change ;

In order to calculate derivatives with respect to non-zero variables, only one pass of the junction
tree algorithm is necessary. However, p passes of the junction tree are needed in order to get the full
matrix of derivativesW. Therefore, depending on the size of p, using only active variables can be
considerable faster. For details, see Algorithm 3.

4.2 Graphical Lasso for the Discrete Case

In the case of Gaussian Markov networks, the graphical lasso algorithm (see Friedman, Hastie, and
Tibshirani, 2008a) is an very efficient method and implementation for solving the L1-penalized log-
likelihood. In order to leverage this speed, we extended the methodology to the binary case treated
in this article. However, the resultant algorithm was not nearly as fast as expected. In the Gaussian
case, the algorithm is very fast as the approach to update the parameter matrix Θ one row at a time
allows for a closed form solution and efficient calculations. In the binary case on the other hand, the
computational bottleneck is not all of the calculation involved in the update of Θ, but specifically
by a large margin the evaluations of the partition function itself. Therefore, any fast algorithm for
solving the penalized log-likelihood exactly has to use as few evaluations of the partition function
as possible. The graphical lasso approach is thus not suitable for the binary case as it takes a lot of
small steps towards the solution.

This observation also explains the improvement in speed of the pseudo-likelihood based exact
algorithm over the specific methods proposed in Lee, Lee, Abbeel, and Ng (2006b). Standard
convex optimization procedures often rely on either the first or second derivatives of the functions
they seek to optimize. Newton-like procedures that use the second derivative often converge in very
few steps, however these cannot be used here as it is prohibitively expensive to evaluate the second
derivative of the partition function, even in small examples. Approaches like conjugate gradients of
BFGS as proposed in Lee, Lee, Abbeel, and Ng (2006b) are somewhat less efficient and take more
steps. This is where the advantage of using the pseudo-likelihood as a local approximation comes
into play. It usually only takes very few steps to converge and is therefore faster than standard
methods.

893

HÖFLING AND TIBSHIRANI

5. Simulation Results

In order to compare the performance of the estimation methods for sparse graphs described in this
article as well as Lee, Ganapathi, and Koller (2006a) andWainwright et al. (2006), we use simulated
data and compare the speed as well as the accuracy of these methods.

5.1 Setup

Before simulating the data it is necessary to generate a sparse symmetric matrix Θ. First, the di-
agonal is drawn uniformly from the set {−0.5,0,0.5}. Then, using the average number of edges
per node, upper-triangular elements of Θ are drawn uniformly at random to be non-zero. These
non-zero elements are then set to either −0.5 or 0.5, again each uniformly. In order for Θ to by
symmetric, the lower triangular matrix is set equal to the upper triangular matrix. The actual data is
generated by Gibbs sampling using Θ as described above.

With respect to the penalty parameters that we use for the different methods, we always leave
the diagonal unpenalized and all off-diagonal elements have the same parameter ρ, that is we set

rst =

#

0 if s= t
ρ otherwise

and the penalty term matrix for the pseudo-likelihood S = 2R−diag(R) as defined above. For the
Wainwright-methods, the penalty parameter is ρ with no penalty on the intercept. Although the
log-likelihood functions that are being penalized are somewhat different, this choice of parameters
makes them perform roughly equivalent, as can be seen in Figure 3. The number of edges is plotted
against the penalty parameter used and all methods behave very similar. However, in order not to
confound some results by these slight differences of the effects of the penalty, all the following plots
are with respect to the number of edges in the graph, not the penalty parameter itself.

5.2 Speed Comparison

First, we compare the speed of the four methods for the L1-penalized model. We used an an-
nealing schedule for Lee, Ganapathi, and Koller (2006a) to improve convergence as suggested in
their article. Plots of the speeds of the exact methods can be seen in Figure 4 and the approxi-
mate methods are shown in Figure 5. Each plot shows the time the algorithm needed to converge
versus the number of edges in the estimated graph. As can be seen, the pseudo-likelihood based
exact algorithm described above is considerable faster than the one proposed in Lee, Ganapathi, and
Koller (2006a). For the approximate algorithms, we can see that the p logistic regressions in the
Wainwright et al. (2006) algorithm take roughly the same amount of time as the pseudo-likelihood
algorithm presented above. This is not surprising due to the similarity of the optimization methods
and any difference that can be observed in Figure 5 is mostly due to the specific implementations
used. Furthermore, we would like to note that we decided to plot the speed against the number of
edges in the graph instead of the penalty parameter, as the runtime of the algorithm is very closely
related to the actual sparseness of the graph.

Overall, when comparing the computation times for the exact algorithms to the approximate
algorithms, we can see that the approximate methods are orders of magnitude faster and do not suffer
from an exponentially increasing computation time for decreasing sparsity as the exact methods.
Therefore, if an exact solution is required, our exact algorithm is preferable. On the other hand, the

894

SPARSE MARKOV NETWORKS

0.02 0.05 0.10 0.20

0
10

20
30

40

P=20, N=200, Neigh=3

Penalty parameter

N
um

be
r o

f e
dg

es
in

G
ra

ph Exact
Wain−Min
Wain−Max
Pseudo

0.05 0.10 0.20

0
20

60
10

0

P=40, N=200, Neigh=4

Penalty parameter

N
um

be
r o

f e
dg

es
in

G
ra

ph

0.05 0.10 0.20

0
20

40
60

80

P=50, N=300, Neigh=4

Penalty parameter

N
um

be
r o

f e
dg

es
in

G
ra

ph

0.05 0.10 0.20

0
20

40
60

80

P=60, N=300, Neigh=4

Penalty parameter

N
um

be
r o

f e
dg

es
in

G
ra

ph

Figure 3: Number of edges in the graph vs. penalty parameter for different problem sizes, averaged
over 20 simulations.

superior speed of the pseudo-likelihood and Wainwright et al. (2006) algorithm warrants a closer
look at the trade-off with respect to accuracy.

5.3 Accuracy Comparisons

In this subsection we compare the algorithms mentioned above with respect to the accuracy with
which they recover the original model. As both our L1 penalized exact algorithm and the one by Lee,
Ganapathi, and Koller (2006a) find the exact maximizer of the L1-penalized log-likelihood, we only
use our algorithm in the comparison. The other 3 methods we compare to are “Wainwright-min”,
“Wainwright-max” and the pseudo-likelihood.

895

HÖFLING AND TIBSHIRANI

0 10 20 30 40 50 60

0
5

10
15

20

P=20, N=200, Neigh=3

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
) Lee−BFGS

Pseudo−Exact

0 20 40 60 80 120

0
20

0
60

0
10

00

P=40, N=200, Neigh=4

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
)

0 20 40 60 80 120

0
20

0
60

0
10

00

P=50, N=300, Neigh=4

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
)

0 20 40 60 80 120

0
20

0
60

0
10

00

P=60, N=300, Neigh=4

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
)

Figure 4: Computation time of the exact algorithms versus the number of non-zero elements in Θ.
Values are averages over 20 simulation runs, along with ±2 standard error curves. Also,
p is the number of variables in the model, N the number of observations and Neigh is the
average number of neighbors per node in the simulated data. Here, Pseudo-Exact refers
to the the exact solution algorithm that uses adjusted pseudo-likelihoods as presented in
Section 4.

First we investigate how closely the edges in the estimated graph correspond to edges in the true
graph. In Figure 6, ROC curves are shown, plotting the false positive (FP) rates against true positive
(TP) rates for edge identification, for various problem sizes. Note that only partial ROC curves are
shown since our method cannot estimate non-sparse graphs due to very long computation times.
Overall, we see that all approximate algorithms match the results of the exact solution very closely
and in some of the plots, the curves even lie almost on top of each other.

896

SPARSE MARKOV NETWORKS

0 10 20 30 40 50 60

0.
0

0.
4

0.
8

1.
2

P=20, N=200, Neigh=3

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
) Wainwright

Pseudo

0 20 40 60 80 120

0.
0

1.
0

2.
0

P=40, N=200, Neigh=4

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
)

0 50 100 150

0.
0

1.
0

2.
0

3.
0

P=50, N=300, Neigh=4

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
)

0 50 100 150 200 250

0
1

2
3

4

P=60, N=300, Neigh=4

Number of edges

C
om

pu
ta

tio
n

tim
e

(s
)

Figure 5: Computation time for the approximate algorithms versus the number of non-zero ele-
ments in Θ. Values are averages over 20 simulation runs, along with ±2 standard error
curves. Also, p is the number of variables in the model, N the number of observations
and Neigh is the average number of neighbors per node in the simulated data.

Apart from the accuracy of edge identification, we also consider other statistics. The unpenal-
ized log-likelihood is a measure of how well the estimated model fits the observed data (higher
values are better). Again, the approximate solutions are all very close to the exact solution (see
Figure 7) and the differences are always smaller than 2 standard deviations. In Figure 8, we plot
the difference of the log-likelihood with respect to the exact solution. Also in this plot, no clear
”winner” can be identified.

We also use the Kullback-Leibler divergence DKL(P||Q), which is a measure of difference be-
tween a true probability distribution P and an arbitrary other distribution Q. Here, for the true

897

HÖFLING AND TIBSHIRANI

0.00 0.05 0.10 0.15 0.20 0.25

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

P=20, N=200, Neigh=3

False positive rate

Tr
ue

p
os

iti
ve

ra
te

Exact
Wain−Min
Wain−Max
Pseudo

0.00 0.05 0.10 0.15

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

P=40, N=200, Neigh=4

False positive rate

Tr
ue

p
os

iti
ve

ra
te

0.00 0.04 0.08

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

P=50, N=300, Neigh=4

False positive rate

Tr
ue

p
os

iti
ve

ra
te

0.00 0.02 0.04 0.06

0.
0

0.
2

0.
4

0.
6

0.
8

1.
0

P=60, N=300, Neigh=4

False positive rate

Tr
ue

p
os

iti
ve

ra
te

Figure 6: ROC curves: false positive versus true positive rate for edge identification. Values are
averages over 20 simulation runs.

probability distribution we use the distribution of the binary Markov network using the true Θ0-
matrix that was used to generate the simulated data. The distribution Q in our case is the binary
Markov network using the estimated Θ̂-matrix. We can compute DKL(P||Q) as

DKL(P||Q) =∑
x

P(x) log
P(x)
Q(x)

=Ψ(Θ0)−Ψ(Θ̂)+∑
x

P(x)tr

xxT (Θ̂−Θ0)
.

=

=Ψ(Θ0)−Ψ(Θ̂)+ tr

EP(xxT )(Θ̂−Θ0)
.

.

If the distributions P andQ are identical, then DKL(P||Q) = 0. In our simulations, the exact solution
has lower KL-divergence than the other methods, however the differences are very small. For a plot
of the KL-divergence against the number of edges in the model see Figure 9. Again, all approximate

898

SPARSE MARKOV NETWORKS

0 10 20 30 40 50 60

−1
3.

1
−1

2.
9

−1
2.

7

P=20, N=200, Neigh=3

Number of edges

lo
g−

lik
el

ih
oo

d

0 50 100 150

−2
5.

4
−2

5.
0

−2
4.

6

P=40, N=200, Neigh=4

Number of edges

lo
g−

lik
el

ih
oo

d

0 50 100 150

−3
2.

0
−3

1.
6

−3
1.

2

P=50, N=300, Neigh=4

Number of edges

lo
g−

lik
el

ih
oo

d

0 50 100 150

−3
8.

4
−3

8.
0

−3
7.

6

P=60, N=300, Neigh=4

Number of edges

lo
g−

lik
el

ih
oo

d

Exact
Wain−Min
Wain−Max
Pseudo

Figure 7: Log-likelihood of the estimated model vs number of edges in the graph for different
problem sizes, averaged over 20 simulations.

methods match the exact solution very closely and any differences are well within the 2 standard
deviation error band. In Figure 10 the differences of the KL-divergence of the approximate to the
exact method can be seen. Again, all methods are very close with the pseudo-likelihood approach
performing the best in this case.

6. Discussion

When we embarked on this work, our goal was to find a fast method for maximizing the L1 penal-
ized log-likelihood of binary-valued Markov networks. We succeeded in doing this, and found that
the resulting procedure is faster than competing exact methods. However, in the course of this work,

899

HÖFLING AND TIBSHIRANI

0 10 20 30 40 50 60

−0
.0

5
0.

05
0.

10

P=20, N=200, Neigh=3

Number of edges

D
iff

er
en

ce
o

f l
l t

o
ex

ac
t Wain−Min

Wain−Max
Pseudo

0 50 100 150

−0
.1

0
0.

00
0.

10

P=40, N=200, Neigh=4

Number of edges

D
iff

er
en

ce
o

f l
l t

o
ex

ac
t

0 50 100 150

−0
.0

4
−0

.0
2

0.
00

0.
02

P=50, N=300, Neigh=4

Number of edges

D
iff

er
en

ce
o

f l
l t

o
ex

ac
t

0 50 100 150

−0
.1

5
−0

.0
5

P=60, N=300, Neigh=4

Number of edges

D
iff

er
en

ce
o

f l
l t

o
ex

ac
t

Figure 8: Difference of log-likelihood of the estimated model to the exact model vs number of
edges in the graph for different problem sizes, averaged over 20 simulations.

we also learned something surprising: several approximate methods exist that are much faster and
only slightly less accurate than the exact methods. In addition, when a dense solution is required,
the exact methods become infeasible while the approximate methods can still be used. Our imple-
mentation of the methods of Wainwright et al. (2006) uses the fast coordinate descent procedure
of Friedman, Hastie, and Tibshirani (2008b), a key to its speed. The pseudo-likelihood algorithm
also uses similar techniques, which make it very fast as well. We conclude that the Wainwright and
pseudo-likelihood methods should be seriously considered for computation in Markov networks.

In this article, we treated the case of pairwise Markov networks with a binary response variable.
We think these methods can also be extended to more general cases. With respect to the response
variables, a multinomial instead of a binary response could be used. In addition to this, it would

900

SPARSE MARKOV NETWORKS

0 10 20 30 40 50 60

0.
18

0.
22

0.
26

P=20, N=200, Neigh=3

Number of edges

K
L−

di
ve

rg
en

ce

0 50 100 150

0.
48

0.
52

0.
56

P=40, N=200, Neigh=4

Number of edges

K
L−

di
ve

rg
en

ce

0 50 100 150

0.
50

0.
60

0.
70

P=50, N=300, Neigh=4

Number of edges

K
L−

di
ve

rg
en

ce

0 50 100 150

0.
65

0.
75

0.
85

P=60, N=300, Neigh=4

Number of edges

K
L−

di
ve

rg
en

ce

Exact
Wain−Min
Wain−Max
Pseudo

Figure 9: Kullback-Leibler divergence of the estimated model vs. number of edges in the graph for
different problem sizes, averaged over 20 simulations.

also be possible to generalize the graph structure by introducing higher order interaction terms.
Apart from these extensions, an interesting possibility for future work would also be to prove the
theoretical results of Wainwright et al. (2006) for the pseudo-likelihood model. Furthermore, we
believe that both the exact and fast approximate methods can also be applied to the learning of
multilayer generative models, such as restricted Boltzmann machines (see Hinton, 2007).

An R language package for fitting sparse graphical models, both by exact and approximate
methods, will be made available on the authors’ websites.

901

HÖFLING AND TIBSHIRANI

0 10 20 30 40 50 60

0.
00

0
0.

01
0

P=20, N=200, Neigh=3

Number of edges

D
iff

er
en

ce
o

f K
L

to
e

xa
ct Wain−Min

Wain−Max
Pseudo

0 50 100 150
0.

00
0

0.
01

5
0.

03
0

P=40, N=200, Neigh=4

Number of edges

D
iff

er
en

ce
o

f K
L

to
e

xa
ct

0 50 100 150

0.
00

0
0.

01
0

0.
02

0

P=50, N=300, Neigh=4

Number of edges

D
iff

er
en

ce
o

f K
L

to
e

xa
ct

0 50 100 150

0.
00

0
0.

00
6

0.
01

2

P=60, N=300, Neigh=4

Number of edges

D
iff

er
en

ce
o

f K
L

to
e

xa
ct

Figure 10: Difference of Kullback-Leibler divergence of the approximate methods to the exact so-
lution vs. number of edges in the graph for different problem sizes, averaged over 20
simulations.

902

SPARSE MARKOV NETWORKS

Acknowledgments

Hoefling was supported by a Stanford Graduate Fellowship. Tibshirani was partially supported by
National Science Foundation Grant DMS-9971405 and National Institutes of Health Contract N01-
HV-28183. We would like to thank Trevor Hastie and Jerome Friedman for code implementing the
fast L1 logistic regression and Daniela Witten for helpful comments on a draft of this article. We
also want to thank the editor and two anonymous reviewers for their work and helpful reviews.

Appendix A. Proof of Convergence for Penalized Pseudo-likelihood and Penalized
Log-likelihood Algorithms

Lee, Lee, Abbeel, and Ng (2006b) gives a proof of convergence for an algorithm that solves an L1
constrained problem by quadratic approximation of the objective function. Here, we will follow
this proof very closely and make few changes to accommodate that we are only using a first order
approximation and are working with the Lagrangian form of the L1 constrained problem instead of
the standard form.

Assume that g(Θ) is a strictly convex function with a global minimum that we want to mini-
mize. Furthermore, let fΘ0(Θ) be a first order approximation of g at Θ0 and assume that fΘ0(Θ)
is strictly convex, has a global optimum and is jointly continuous in (Θ0,Θ). Here, by first order
approximation at Θ0, we mean that fΘ0 −g is twice continuously differentiable with derivative 0 at
Θ0. Assume that our algorithm works as follows:

initialize Θ(0);
Set k:=0;
while not converged do

With current estimate Θ(k), define local approximation f
Θ(k)

(Θ) to g;
Find solution Θ∗ of f

Θ(k)
(Θ);

Perform backtracking line search on the line from Θ(k) to Θ∗ to find Θ(k+1);
Set k:=k+1;

end

Then Θ(k) converges to the global minimizer of g(Θ). In order to show this, we first need the
following lemma:

Lemma 1 Let Θ0 be any point that is not the global optimum. Then there is an open subset SΘ0
and a constant KΘ0 such that for every Φ0 in SΘ0 every iteration of the algorithm starting at Φ0
will return a point Φ1 that improves the objective by at least KΘ0 , that is, g(Φ1) ≤ g(Φ0)−KΘ0 .

Proof First, let fΘ0 be an approximation to g at Θ0 with global optimum Θ1. Then set

δ= fΘ0(Θ0)− fΘ0(Θ1)

and we know that δ > 0 as Θ0 is not the global optimum. Now, there exists an ε > 0 such that for
Φ0 ∈ SΘ0 := {Θ : ||Θ−Θ0||2 < ε} the following holds: |g(Θ0)−g(Φ0)|≤ δ 8 903 HÖFLING AND TIBSHIRANI and | fΦ0(Φ1)− fΘ0(Θ1)|≤ δ 4 where Φ1 is the global minimum of fΦ0 . The existence of this ε follows from the continuity and convexity of f and g. Then fΦ0(Φ1) ≤ fΘ0(Θ1)+ δ 4 ≤ fΘ0(Θ0)− 3δ 4 = = g(Θ0)− 3δ 4 ≤ g(Φ0)− δ 2 . With step size 0< t < 1 in the line search, and using the previous result, it holds that fΦ0(Φ0+ t(Φ1−Φ0)) ≤ (1− t) fΦ0(Φ0)+ t fΦ0(Φ1) ≤ g(Φ0)− t δ 2 For the next step observe that the minimizer Φ1 of fΦ0 is a continuous function of Φ0 due to the convexity of f in the second argument and the continuity in both arguments. Then, as SΘ0 is a compact set, there exists a compact set TΘ0 with Φ1(Φ0) ∈ TΘ0 for all Φ0 ∈ SΘ0 . Thus, as fΘ0 is a first order approximation of g at Θ0, there exists aC such that for all Θ ∈ TΘ0 g(Θ) ≤ fΘ0(Θ)+C||Θ−Θ0|| 2 2. Therefore g(Φ0+ t(Φ1−Φ0)) ≤ g(Φ0)− t δ 2 +Ct2||Φ1−Φ0||22 ≤ ≤ g(Φ0)− t δ 2 + t2CD2 where D is the diameter of TΘ0 . Now set t ∗ = min * 1, δ4CD2 + and thus we know that it exists a Φ∗ such that g(Φ∗) ≤ g(Φ0)− t∗ δ 2 + t∗2CD2. Setting KΘ0 = t ∗ δ 2 − t ∗2CD2 > 0 now finishes the proof.

Now, using the lemma the rest of the proof is again very similar as in Lee, Lee, Abbeel, and Ng
(2006b) and we only repeat it here for completeness.

Theorem 1 The algorithm converges in a finite number of steps.

Proof Pick δ > 0 arbitrary. Let Θ∗ the global optimum and Θ0 the starting point of the algorithm.
Then there exists a compact set K such that g(Θ) > g(Θ0) for every Θ ̸∈ K. Define Pδ = {Θ :
||Θ−Θ∗|| ≥ δ}∩K. We will show convergence by showing that the algorithm can only spend a
finite number of steps in Pδ. For every Θ in Pδ there exists an open set SΘ. So

Pδ ⊆ ∪Θ∈PδSΘ

904

SPARSE MARKOV NETWORKS

As Pδ is compact, Heine-Borel guarantees that there is a finite set Qδ such that

Pδ ⊆ ∪Θ∈QδSΘ.

Furthermore, as Qδ is finite, define
Cδ = min

Θ∈Qδ
KΘ.

As the lemma guarantees that every step of the algorithm inside Pδ improves the objective by at least
Cδ and a global optimum exists by assumption, the algorithm can at most spend a finite number of
steps in Pδ. Therefore, the algorithm has to converge in a finite number of steps.

For the penalized pseudo-likelihood algorithm, by definition of the approximation it is evident
that it is a first order approximation. The situation for the penalized log-likelihood algorithm is a
little more complicated and it will be shown in the next section of the appendix that the proposed
approximation is to first order and therefore satisfies the assumptions of the proof.

Appendix B. First Order Approximation of Log-likelihood

In Section 4, we defined a function f
Θ(k)

to calculate the next estimate Θ(k+1). The convergence
proof in Appendix A requires that f

Θ(k)
is a first order approximation of the objective l(Θ|X)−

N||R ∗Θ||1. Here, we want to show that this is in fact the case. For this, we need to show that
f
Θ(k)

− l(Θ|X)+N||R∗Θ||1 is twice continuously differentiable with derivative 0 at Θ(k).
First, inserting f

Θ(k)
from Section 4 yields

d
Θ(k)

= f
Θ(k)

− l(Θ|X)+N||R∗Θ||1 =

=
1
2

)

l̃(Θ|X)+∑
s>t

(θst −θ
(k)
st )
*

(p̂(Θ(k))Ts X)t +(p̂(Θ
(k))Tt X)s−2 ·N ·wst(Θ

(k))
+

+

+∑
s
(θss−θ

(k)
ss )

)


k
p̂sk(Θ(k))+(XTX)ss−2 ·N ·wss(Θ(k))

,,

−∑
s≥t
γ(θst−θ

(k)
st )

2− l(Θ|X)

which has derivative

2
∂d
Θ(k)

∂θst
= 2(XTX)st− (p̂(Θ)Ts X)t− (p̂(Θ)

T
t X)s+(p̂(Θ

(k))Ts X)t +(p̂(Θ
(k))Tt X)s−

−2 ·N ·wst(Θ(k))−4γ(θst−θ
(k)
st )−2(XTX)st +2 ·N ·wst(Θ)

for s ̸= t and

2
∂d
Θ(k)

∂θst
= (XTX)ss−∑

k
p̂sk(Θ)+∑

k
p̂sk(Θ(k))+(XTX)ss−2 ·N ·wss(Θ(k))−

−4γ(θss−θ
(k)
ss )−2(XTX)ss+2 ·N ·wss(Θ)

for s = t. These are clearly continuous and differentiable. Furthermore, inserting Θ = Θ(k) yields
that the derivative is 0. Therefore, f

Θ(k)
is a first order approximation and our proof holds.

905

HÖFLING AND TIBSHIRANI

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