
Submitted by
Assigned_Reviewer_9
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)
Aims to improve the mixing rate of Gibbs sampling in
pairwise Ising models with strong interactions, which are known to be
"slowmixing". Several projections to "fastmixing" models are proposed;
essentially, parameters are identified which are as close as possible to
the original model, but weak enough to satisfy a spectral bound
establishing rapid mixing. Experiments show some regimes where this leads
to improved marginal estimates for a given computation time, compared to
variational methods (mean field and belief propagation variants) and Gibbs
sampling in the original model.
The technical content builds
heavily on prior results establishing conditions for rapid mixing [4,8].
But I haven't seen the idea of projecting onto the nearest rapidly mixing
model, as an approximate inference method, explored before.
On the
domain of "toy tiny Ising models", results offer some nice improvements
over a good set of baselines (standard sampling and contemporary
variational methods). My main concern is that I think there should be more
than the usual amount of skepticism as to whether these results will scale
to larger models. Experiments involve comparisons of mixing times of
samplers on different models, and it is hard to judge how these will scale
with problem size. Also there is a (from all appearances) computationally
expensive projection step required to build the fastmixing model, the
cost of which seems not to be accounted for. Finally, it is not at all
clear that generalization beyond binary states will be possible, since
establishing convergence bounds more generally is far more challenging.
CLARITY: In general the presentation is clear and reasonably
accessible, given the technical content. But there are some places where
reorganization and clarification is needed: * The paper refers several
times to "standard results" and even states these in the form of a theorem
(e.g. Theorem 6) without reference or proof sketch. Lemma 5 is left
unproven, the reference proves for a special case of zerofield. *
Discuss properties of the dependency matrix R. For instance, it does not
appear to be tractable due to the maximization, please state whether this
is the case. (I presume this is why lemma 5 is invoked.) * The second
half of Sec. 4 is nearly impossible to follow. Before Theorem 7 the text
references g, M, and Lambda before they are introduced. Then the statement
of Theorem 4 includes notation that is not really explained. If this
optimization is going to be discussed, more explanation is needed.
(Perhaps less time could be spent restating results from [8] which are not
really used.) * In Sec. 1, KL(qp) is used for both directions of KL
divergence when the notation needs to be shifted for one. In the first
paragraph, q is used for the true distribution and p for the tractable
approximation; this is the opposite of almost all related literature.
EXPERIMENTS: It appears that the time comparison in Figure 2
does *not* include the computation required for projection. A somewhat
ambiguous statement to this effect appears in Sec 6.1 but is unclear;
please clarify and if it is the case, show results with projection time
included. As it stands, the proposed methods essentially get to use the
output of a sophisticated variational optimization without penalty, which
certainly makes the improvement over standard Gibbs less convincing.
It was disappointing not to see experiments on larger models,
given that Gibbs mixing times often depend on problem size. There are
certainly options for running experiments in regimes where junction tree
is intractable, like using a model where symmetries let the true marginals
be computed, or taking the output of a very long sampling run as truth.
I understand that KL(\theta\psi) is intractable in general, but
it would still be interesting to explore here as a potential "best case"
for how sampling in an approximate model would perform. (Junction tree
could be used for the toy models used in the submission.)
Mean
field is a degenerate case of the reverse KL projection, as the paper
points out, yet there is a large difference between mean field error and
the error from reverse KL projection. This deserves discussion.
Q2: Please summarize your review in 12
sentences
The idea of approximating slowmixing models by
projecting to the closest fastmixing model is a nice one, and recent work
on mixing bounds is leveraged in an elegant way. But there are some
concerns about experimental comparisons, and the limited range of models
to which this approach is potentially applicable. Submitted by
Assigned_Reviewer_11
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 propose a method for improving the mixing
rate of Gibbs sampling in Ising models by projecting the models original
parameterization onto a new parameter setting that satisfies the Dobrushin
criterion needed for fast mixing. They formulate the projection as a
constrained optimization problem, where the constraints are needed to
ensure that the new parameters are defined over the same space as the
original parameters, and examine this projection under several
distance/divergence measures.
In my opinion, methods that combine
principles from stochastic and deterministic inference is an
underexplored area and as a result, I think this is an interesting idea.
While the idea of augmenting the original parameters to improve mixing
time is straightforward, I found the description of the dual of the
projection problem to be a little unclear  e.g. how do the z_ij*d_ij =0
constraints ensure that the new parameterization is over the same space
(can't it be over a smaller space)? I also was unsure of the overall
procedure  do you perform the projected gradient operations to completion
and then run a Gibbs sampler, or do you somehow interleave sampling with
the gradient updates? An algorithm description box would help to clarify.
Also, is the proposed projected gradient descent strategy guaranteed to
converge?
I also found the experiments to be a little
unconvincing. In the first set of experiments, why was the Gibbs sampler
run for 30K iterations? Since you are comparing a sampling method to
deterministic methods, a comparison on the basis of time would seem more
fair. Also, where are the error bars on these charts  the reader cannot
tell if these results are significant. The second set of results compare
the original/naive gibbs sampler with gibbs samplers under different
projections and show that the projection does lead to faster mixing.
However, it takes time to performing the projection operation and this is
not accounted for in this comparison (e.g. if the projection takes 1
minute, and the naive Gibbs sampler can generate 10k samples in that time,
then the projection might not be worthwhile). Last, why was there no
comparison to blocked Gibbs samplers or Gogate's "Lifted Gibbs Sampler"?
Q2: Please summarize your review in 12
sentences
All in all, a very nice idea. However, the development
of the projection problem and proposed method could use a little work and
a bolstered set of experiments are needed to convince me of the utility of
the method. Submitted by
Assigned_Reviewer_13
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)
Success of inference by Gibbs sampling in MRF (here,
only with two labels, ie, ising model) depends strongly on the mixing
rate of the underlying Monte Carlo Markov chain. The paper suggests
the following approach to inference:
1) Project the input
model (which possibly does not mix fast) on the set of models that do
mix fast.
2) Do inference on the obtained model that mixes fast.
The space of fastmixing models is defined by bounding the
spectral norm of the matrix of absolute values of Ising edge
strengths. "Projection" is defined by divergences of Gibbs
distribution. It is forced to preserve the graph structure. Projection
in Euclid distance is obtained by dualizing the initial task and using
LBFGSB. For other divergences (KL, piecewise KL, and reversed KL
divergences are implemented), projected gradient algorithm is used. In
reversed KL divergence, Gibbs sampling (but on a fast mixing model)
must be done to compute the projection.
Extensive experiments
on small random models are presented compare the approximated
marginals with the true marginals. The methods tested are the proposed
one (with all the above divergences) and loopy BP, TRW, MF and Gibbs
sampling on the original model. Not only accuracy but also
runtimevsaccuracy evaluation is done. The experiments show that the
proposed methods consistently outperform TRW, MF and LBP in accuracy,
and for reasonable range of runtimes also Gibbs sampling on the
original model. Of the proposed methods, the one with reversed KL
divergence performs consistently best.
Comments:
The
projected gradient algorithm from section 5.1 n fact has two nested
loops, the inner loop being LBFGSB. Pls give details on when the
inner iterations are stopped.
It is not clear what the horizontal
axis in the plots in Figure 2 (and the supplement) means. It is titled
"number of samples" but sampling is used only for reversed KL
divergence. I believe the horizontal axis should be runtime of the
algorithm. Similarly, why not to report also runtime of LBP, TRW and
MF. This would ensure fair accuracyruntime comparison of all tested
algorithms. Please, clarify this issue  without that it is hard to
interpret the experiments. Give absolute running time in seconds.
Please consider experimental comparison with larger models. An
interesting option is to use models from the paper
[Boris Flach: A
class of random fields on complete graphs with tractable partition
function, to appear in TPAMI, available online]
which allow
polynomial inference.
222: replace "onto the tree" with "onto
graph T"
226: Shouldn't we ssy "subgradient" rather than
"derivative"? Q2: Please summarize your review in 12
sentences
Interesting paper, convincing empirical results.
Practical utility can be limited though due to high runtime (this needs
clarification in rebuttal).
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.
Thanks to all the reviewers for helpful comments. We
have a couple of minor comments for each reviewer, and a longer discussion
of the efficiency of projection, which all reviewers asked about.)
(Minor Comments)
Reviewer_11
The z_ij*d_ij=0
constraints are to ensure that the new parameterization is not over a
*larger* space. Without these constraints, projection of a
parameters on a graph would yield a densely connected graph.
Convergence is guaranteed, in a certain sense (see reference 3 on
ergodic mirror descent). However, as with meanfield, a nonconvex
objective is being fit, meaning local minima are possible.
Lifted
Gibbs Sampling doesn't seem applicable here since we aren't in a Makov
Logic Network setting, but one could certainly apply Blocked Gibbs
sampling. As we note in the paper, the mixing time bound used here is
sufficient to guarantee fast mixing for block Gibbs. (Though it could be
loose.) We agree that extension of the mixingtime bounds and projections
to these more complex samplers is important, but also quite challenging.
Reviewer_13
We have done some experiments comparing to
large (e.g. 40x40) zerofield planar Ising models, which has results
similar to those shown. (Although one must measure pairwise marginal
accuracy, since true univariate marginals are always exactly uniform.)
Thank you for the reference on regular fullyconnected graphs. Along with
attractive zerofield planar Ising models, this provides a very
interesting class of largescale models with tractable exact marginals.
Reviewer_9
It is indeed nonobvious how to generalize
these results beyond the Ising model. We quickly outline how this might be
done on lines 418420. After writing this paper, we spent several months
working on this generalization, and found that it is possible to do these
types of projections with general MRFs, although the more general
projection algorithm is considerably more complex.
We actually ran
all the experiments including projection under KL(\theta\psi) (with
marginals computed via the junction tree algorithm) but removed it out of
a concern it might be confusing, since it is intractable on large graphs.
Still, as you'd expect, it does perform better than the "reversed" KL
divergence, so we'd be happy to put the results back in.
We were
also at first surprised by the difference of performance between
meanfield and reverse KL divergence. As one increases the allowed
spectral norm from zero (mean field) to 2.5, the quality of results
smoothly interpolates, confirming the performance difference is just due
to optimizing the divergence over a larger constraint set. Presumably,
this continues as the norm increases further, but of course Gibbs sampling
could be exponentially slow.
(Efficiency of Projection)
The experiments do not currently include the computational effort
deployed to find the projected parameters (also mentioned on lines
354358). Particularly with the SGD algorithm, finding the most efficient
optimization parameters (sample pool size, # Markov transitions, LBFGS
convergence threshold, gradient step size) is challenging. We did not
attempt to find the fastest parameters, instead opting for conservative
(slow, reliable) parameters to ensure transparency of the results. Our
thinking was, first, the main idea is to show that strong fastmixing
approximations exist, and second that projection could be deployed
"offline". (e.g. one might project once, and then do inference after
conditioning on various inputs.)
That being said, in hindsight it
seems obvious that readers would also be interested in the efficiency of
projection as compared to Gibbs or variational methods, and the paper
should certainly make such comparisons clear. Measurement is somewhat
complicated by unoptimized implementation in an interpreted language, but
the computational bottlenecks are (a) Gibbs sampling, and (b) Singular
Value Decomposition, as called when optimizing the dual in Theorem 7. As
these are implemented efficiently in C/Fortran, we report their times
below for the 8x8 grid with attractive strength 3 interactions.
30,000 iterations of Gibbs sampling: 0.18s A single SVD of a
dependency matrix: 0.0026s Euclidean projection: (93 SVDs): 0.15s
Piecewise1 projection: (399 SVDs): 1.1s KL via Stochastic
Gradient Descent: (540 SVDs + 30,000 Gibbs steps): 1.4s+0.2 = 1.6s
Variational Methods: less than 0.001s
Here are the same
measurements for strength 1.5 interactions:
Euclidean projection:
(23 SVDs): 0.061s Piecewise1 projection: (206 SVDs): 0.51s KL via
Stochastic Gradient Descent: (252 SVDs + 30,000 Gibbs steps): 0.7s+0.2 =
0.9s Variational Methods: less than 0.001s
Note here that,
though SGD uses 60 Euclidean projections, the cost is much less than 60x
as much, due to the use of warmstart.
Roughly speaking: with
strength 3, a single Euclidean costs the same as 30,000 iterations of
Gibbs, and KL projection costs the same as 300,000 iterations of Gibbs.
With strength 1.5, Euclidean costs as 10,000 iterations of Gibbs, and KL
costs the same as 30,000 iterations. The paper should definitely include a
table of such timing measurements, and run Gibbs for 300,000 iterations
(one more order of magnitude) to be equal to the cost of KL projection.
Note that Gibbs will still clearly be inferior to projecting with strong
interaction strengths and/or dense graphs. (Observe the essentially
horizontal lines for original parameters in Figs 58.) A online hybrid
sampling/projection algorithm would presumably be superior in this
setting, but we believe this goes beyond what can be done in one
paper.
 