
Submitted by
Assigned_Reviewer_3
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper proposes parsimonious triangular model
(PTM), which constrains the O(K^3) parameter space of mixedmembership
triangular model (MMTM) to O(K) for faster inference. Authors develop a
stochastic variational inference algorithm for PTM and additional
approximation tricks to make it further scalable. It is shown from
synthetic dataset that the reduction of the number of variables may lead
to stronger statistical power, and from realworld datasets that the
proposed method is competitive with existing methods in terms of accuracy.
Quality:
PTM seems to be an interesting specialization of
MMTM, but it is questionable what is the practical advantage of achieving
good scalability in terms of K (the number of possible roles). To
empirically evaluate the value of such a method, it is critical for us to
answer "how does it help if we can learn MMTM with large K?" Since MMSB
and MMTM are mixedmembership models, using small K may not be as
troublesome as it is in singlemembership models! For real network
experiments, what would be really interesting is to see the performance of
both MMTM and PTM as a function of K: if we can really see that PTM with
large K outperforms MMTM with small K, it will show the power of PTM.
Unfortunately such experiments were not done. Actually I could not find
good description about how K's were chosen in real data experiments for
both PTM and its competitor MMSB; the method of model selection could've
biased experimental results.
Authors are also a bit reticent on
what are the impact of simplifying assumptions they make. It is clear that
we can achieve good scalability by constraining the parameter space, but
it would be also nice to hear about when would such model fail. If there
really are O(K^3) distinct values in the parameter matrix B (following
authors' notation) of MMSB and MMTM, PTM should perform poorly, right? In
this regard, it is unsatisfactory that the choice of true B in synthetic
data experiments were not clearly described; authors' should've
experimented with various B's which some are in favor of PTM and some are
against the model. It would be more beneficial for readers to know when
would PTM succeed and when would it fail, than to know only when it would
succeed.
On the other hand, authors introduce a number of tricks
to make PTM scale; while I find them to be very interesting and effective,
this paper does a poor job in describing and analyzing the impact of such
an approximation. 1) What is the impact of choosing the threshold \delta?
On realworld and synthetic data experiments, does it help to choose
larger or smaller \delta? How much can we lower \delta to improve
scalability while not sacrificing much of accuracy? 2) Authors claim that
aMMSB does not work since downsampling compromises accuracy; why would
PTM free of such problem? MMTM and PTM are actually ignoring triplets with
only one edge. Why can't we ignore nonedges in MMSB while we can in PTM?
3) What is the impact of O(K) approximation to local update (in page 5)?
What happens if we sample 2K, 4K, or 100K of triples? Would the quality of
approximation suffer when K gets larger?
To sum up, I am not
convinced experimental setup was fair to competitors, and the description
of potential weakness of the algorithm/model is poor.
Clarity: the
paper is very clearly written and wellorganized.
Originality: In
terms of modeling it is a bit incremental, but achieving scalability via
constraining parameters is an interesting and original direction to
pursue, as recently in this topic a lot of work has been focused on
generalizing existing models.
Significance: Provided simplifying
assumptions in the model and the inference algorithm do hold in practice,
it would have a good impact on the practice of latentspace models on
networks, as not many of them actually scale to large data.
Additional Comments: Personally I think the title is too broad.
There are other scalable approaches to other latent space models, and
authors propose a scalable algorithm to _a_ latent space model PTM.
Also, I am not sure that PTM is really a probabilistic method.
Authors argue in the last paragraph of Section 3 that this is same in
spirit to bagofwords assumption, but I am not convinced. When LDA
generates a bagofwords, there always exists a document corresponding to
it; so it has proper likelihood function, no matter how unrealistically
simple it might be. On the other hand, with very high probability there
does not exist any network corresponding to a bagoftriples generated by
MMTM and PTM. Therefore MMTM/PTM is not really a generative model for
networks, and majority of posterior probability mass is placed on modeling
nonnetworks! It might be dangerous to abandon basic principles of
likelihood in "probabilistic" methods (what does probability mean in
PTM?)... I am curious what authors and other reviewers
think. Q2: Please summarize your review in 12
sentences
Authors propose modeling assumptions and inference
tricks that would make inference on MMTM scalable. Sideeffects of such
decision, however, are poorly investigated. Submitted by
Assigned_Reviewer_4
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper proposes a technique for learning latent
structure in social network data. The model is fit to triangle count
statistics obtained from a preprocessing step, analogous to a bagofwords
representation for text documents. The main modeling advance is motivated
by aMMSB: rather than parameterize all between"community" interactions
they consider the configurations of the triangles to identify a set of K
roles to parameterize. This results in fewer parameters in the model and,
in turn, a more scalable implementation. They apply recent advances
stochastic variational inference  deriving a global update and local
updates for the proposed model.
The novel modeling contribution is
a straightforward combination of the aMMSB with the MMTM, though somewhat
tedious bookkeeping was required. The presentation of this material could
be improved.
Though most of the learning algorithm is a direct
application of stochastic variational inference, the authors provide an
intuitive and intriguing approximation for the local update rule.
Given the prevalence of directed networks, a detailed discussion
of how these methods might extend would be welcome.
Ongoing
concerns:  "our Parsimonious Triangular Model (PTM) is essentially a
latent space counterpart to ERGMs, that represents the processes of
generating triangular motifs". This seems like a curious statement: 1)
ERGMs can include much more varied statistics than just triangles and 2) I
don't see how the PTM is a "latent space counterpart", unless all latent
space network models are "latent space counterparts".  Are the
inferred latent space interpretable? If so, what do we learn about the
similarities and differences among these graphs? The difference
between Delta^1 and Delta^2 is not clear (at least to this reader). 
It is not made clear (in this paper) why subsampling the triangles for
each node (using delta) provides an adequate approximation.  If the
goal is to recover the "set of nonzero roles" then perhaps there should
be more of a binary notion of "role" in the model.  How robust are
the latent space recovery results to the threshold of .125 (which seems
somewhat arbitrary)?  A variety of latent space models can produce
powerlaw networks. Which one did you use for the experiments?  Why
is PTM CGS have K^3 complexity? Didn't you argue that there are O(K)
parameters rather than O(K^3)?  One concern is how this will perform
when fewer edges are observed. In the experiments they only hold out 10%
of the edges, presumably because the distribution of adjacent triangles
will be quite biased if too many edges are removed prior to the
preprocessing step.  It is counterintuitive that the method beats
MMSB on a synthetic network generated via MMSB (Figure 1, left 4 panels).
Q2: Please summarize your review in 12
sentences
This work uses stochastic variational inference to
learn a mixedmembership blockmodel (with a restricted set of parameters)
on the triangle counts in a graph. Their modeling choices and
approximations allow for a more scalable O(NK)
algorithm. Submitted by
Assigned_Reviewer_5
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
The authors develop a scalable approximate inference
algorithm for the "triangle model" of networks. They develop a
powerful subclass of the model that shares parameters in a clever way,
and they use stochastic variational inference to infer the posterior
with very large networks. The paper is well written, and well
exexcuted. I think it is acceptable.
p1: What is "parsimonious"
modeling? I don't think this is a runofthemill expression in the
context of probabilistic models. (I might be wrong about that.) I
suggest unpacking that term in a sentence since it is important to the
paper.
p1: From my reading of Gopalan et al., downsampling the
zeros does not comprimise accuracy. Your sentence in the introduction
is misleading.
p4: I'm sympathetic to the triangle model being
OK from a pragmatic perspective, but I don't think appealing to LDA is
fair. If we run the generative process of LDA, we will obtain
something that "is" a document, albeit a silly one, even if it's not
the original. Running your model, however, can produce objects that
are not networks. Again, this does not bother meit's a good and
practical modelbut the defense by means of LDA is not appropriate.
p5: You should cite Hoffman et al. with Amari and Sato. In your
scenario, where there are "local variables" (in Hoffman's language),
Sato's algorithm does not apply. The needed theory is in Hoffman.
p6: How did you generate power law networks in the latent space?
The paper is vague around this point.
p6: Earlier you cite
Gopalan et al. as a way of speeding up MMSB. You should compare to
their algorithm, in addition to (or instead of) MMSB batch.
Small comments
p2: mixturemembership >
mixedmembership
p8: I suggest removing the last sentence. The
secondtolast sentence is a stronger way to end the paper.
In
the bibliography, cite the JMLR version of "Stochastic Variational
Inference".
Q2: Please summarize your review in
12 sentences
The authors develop a scalable approximate inference
algorithm for the "triangle model" of networks. They develop a
powerful subclass of the model that shares parameters in a clever way,
and they use stochastic variational inference to infer the posterior
with very large networks. The paper is well written, and well
exexcuted. I think it is acceptable.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank all reviewers for their valuable comments and
appreciation of our work's clarity, originality and significance. We also
gladly acknowledge R4 & R5 for sharing our view on how the paper
should be received. For networks with large scale N>1m and many
overlapping communities K>100 (Yang & Leskovec, 2012;
snap.stanford.edu), having O(NK) runtime is crucial. We have developed a
highly scalable O(NK) technique targeted at such networks, enabling latent
space inference at unprecedented scale (N>1m, K>100), while the
stateoftheart (MMTM, aMMSB) only scales to N~=100k. Our algorithm
delivers a muchneeded technique for meaningful inference on large
networks, while small N, K are not the target of our work.
R3:
There exist networks with many overlapping/mixed communities K
(Yang & Leskovec, 2012), for which algorithms must scale in K. MMSB
variational & MMTM CGS have O(K^2) & O(K^3) runtime, too slow for
large K (Fig 1). Since PTM is intended for the large K setting, comparing
largeK PTM against smallK MMTM is beyond the scope of this paper. K can
be tuned via crossvalidation on the heldout likelihood, though it is not
our focus. In our linkprediction experiments on real world networks, the
same K=20 (the MMSB computational limit from Fig 1) was used for MMSB and
PTM, so there is no model selection bias.
The reviewer suggested
the experiments favor PTM. This is untrue. The true B's for the synthetics
are in the appendix (p3): “... under the MMSB model, we used B with
diagonal elements 0.2, and offdiagonal elements 0.001”. This matches the
MMSB generative assumption while mismatching PTM, so it actually favors
MMSB. The other setting (powerlaw) does not favor any methods. Both
synthetic and real experiments show MMSB variational beats PTM on small
networks (Fig. 1, Table 2); we have shown PTM’s weaknesses.
One
cannot experiment with O(K^3) distinct values in PTM’s B tensor, because
PTM is a probabilistic model for triangular motifs, not a network
simulation tool. Per conventional wisdom, we believe that relying on a
simple, parsimonious model leads to better speed and generalization on
large data.
Approximation quality is important, but is not the
focus of this paper (some issues were addressed in Ho et. al., 2012). Our
paper is about scalable network modeling rather than generalpurpose
approximate inference; such preferential focus exists in many variational
approximation papers, e.g. mean field for Ising models and LDA. Hence we
devoted the limited space to explaining our scalable approximation scheme
(rather than its tuning details), while providing substantial evidence
that it works well on synthetic and real networks. Moreover, we provide
some guidance on choosing delta (footnote 2, p6), while the tradeoff
between scalability and accuracy was investigated by Ho et. al. (2012).
We did not mean to claim “aMMSB does not work”; we meant there
might be a variance issue with the downsampling strategy, that can
compromise accuracy (see R5 response). We also wish to clarify that
downsampling and ignoring 0edges are not the same: the former means
giving preferential attention to the subset of node pairs with edges when
forming node pair minibatches in the SVI algorithm; the latter means
0edges are simply not modeled. aMMSB can downsample 0edges, but does
not ignore them. PTM models the distribution of open and closed triangles,
while ignoring node triples with one or no edges. The key insight is that
ignoring node triples with 0/1 edges leaves two types of triples (2/3
edges), whose utility was justified in Ho et. al. (2012). However,
ignoring 0edges in MMSB leaves us with only 1edges, leading to the B
matrix being estimated as all 1's, an uninteresting result.
R4:
The ERGM statement was admittedly obscure. We meant that PTM is
inspired by ERGM triangle count statistics. We will fix these comments.
The PTM latent space is interpretable: as a starting point,
compare ratios of closed to open triangles in each B_xxx. A high ratio
implies a legitimate, wellconnected community, while a low ratio may
signify a bipartite subgraph (e.g. scammer network).
The delta on
p5 is the variational parameter; it is unrelated to the subsampling delta
(p2). delta^2 (p6) is the subsampling delta squared.
deltasubsampling approximately preserves local cluster properties
at each node, allowing much faster inference at a small accuracy cost (Ho
et. al., 2012).
The NMI metric requires binary roles. The 0.125
threshold is half of 1/4 (because our synthetics use <=4 roles per
node). We also used mixed roles, in heldout link prediction.
Powerlaw model: see appendix and R5 response.
Like MMTM
CGS, PTM CGS blocksamples variables s (not B, which is size O(K)), hence
O(K^3) runtime.
Holding out 10% edges for prediction is typical
(p7, Gopalan et. al., 2012). Each node gets approx. (110%)^2=81% of
adjacent triangles.
On large networks, MMSB has far more latent
variables than PTM, making inference harder and yielding poor results. On
small networks, MMSB performs better (Fig. 1, Table 2).
R5:
The downsampling statement was admittedly misleading. We meant
that though the stochastic gradient expectation remains equal to the true
gradient, the variance might become worse and compromise accuracy. We will
fix the statement.
We did not mean to defend PTM by appealing to
LDA; instead, we wanted to motivate PTM by drawing an analogy between our
bagoftriangles network representation and the bagofwords document
representation. We will fix those statements.
Thanks for pointing
out Hoffman et al.’s theory, and we will gladly cite its JMLR version.
Details for the powerlaw model are in the appendix (p3). It
closely follows the MMTM paper (Ho et. al., 2012).
Because aMMSB
has no public code, while the paper had insufficient details (e.g.
initialization strategy) to fairly reproduce the algorithm, we could not
perform a fair comparison.
 