Paper:
A Learning Error Analysis for Structured Prediction with Approximate Inference
In this work, we try to understand the differences between exact and approximate inference algorithms in structured prediction. We compare the estimation and approximation error of both underestimate (e.g., greedy search) and overestimate (e.g., linear relaxation of integer programming) models. The result shows that, from the perspective of learning errors, performances of approximate inference could be as good as exact inference. The error analyses also suggest a new margin for existing learning algorithms. Empirical evaluations on text classification, sequential labelling and dependency parsing witness the success of approximate inference and the benefit of the proposed margin.

Review:
This paper is on the important topic of learning with approximate
inference. Previous work, e.g., Kulesza and Pereira (2007), has demonstrated the
importance of matching parameter update rules and inference approximation
methods. This paper presents a new update rule based on PAC Bayes bounds, which
is fairly agnostic to the inference algorithm used -- it assumes a
multiplicative error bound on model score and supports both over and under
approximations.
The example given in section 3.2 is a great illustration of how approximation
error is more subtle than we might think it is. Sometimes an approximate
predictor can fit the training data better because it represents a different
family of functions!
The experiments presented in the paper are on NLP problems. However, in NLP the
"workhorse" approximate inference algorithm used is beam search. I strongly
encourage the authors to extend their experiments to include beam search
(varying the beam size to control the approximation quality).
    If, for some reason, beam search does not match their method, it should be
    made very clear.
    Related to beam search is work done by Liang Huang (e.g.,
    http://www.aclweb.org/anthology/N12-1015) about how perceptron with beam
    search can diverge and how to fix the learning algorithm (see "early update"
    and "max violation" updates).
    This paper is pretty influential in NLP.
    This is a case where the learning rule is modified. I think the NLP
    community (at least) would be very interested if beam search could actually
    work with your approach.
Overall, I like this paper and I think that it could have a real impact because
learning with approximate inference is such an important topic and the approach
describe here seems to be pretty effective. Unfortunately, the current state of
the paper is a bit too rough for me to recommend acceptance. (If only we had the
option to "accept with required revisions")
Question
========
I'm confused about Theorem 6.
  a) Isn't this bound looser than the one in theorem 4?  Should making *more*
     assumptions (about \rho *and* \tau) lead to a tighter bound?
  b) I expected \rho to be replaced with a bound based solely on \tau. What's
     the deal?
Am I missing something?
Lower-level comments
====================
The authors assume that the reader is already familiar with the concepts of
under-estimation and over-estimation. I do not believe most readers will know
about this. I strongly recommend defining these terms very early in the paper.
I would also recommend explaining early on why these two classes of
approximations have been studied in the past (and why they are typically studied
separately) -- this would be much more useful than the current discussion in the
related work section.
bottom of page 1: Please define risk? In particular is e(h) /empirical/ risk or
true risk?
39: typo? "We investigate three wildly used structured prediction models"
    -> I think you meant "widely", but I suppose "wildly" also works :P
41: At this point in the paper, I'm very surprised to see "text classification"
    on a list of things that might use approximate inference... It might be
    useful to explain how this is relevant -- it seems as if the answer is that
    it's just a synthetic example.
    There are some settings, such as "extreme classification", where the number
    of labels is huge and approximate inference might be used (e.g., based on
    locality sensitive hashing or approximate nearest neighbors).
42: typo "high order" -> "higher-order"
45: It's tough to say that "Collins [5]" was "the first" since many results for
the /unstructured/ multi-label case technically do transfer over.
47,49,50,etc: Citations are not nouns use use \citet instead of \citep and
\usepackage{natbib}.
   "[37] provided..." -> "Collins (2001) provided ..."
49: "Taskar’s bound" is undefined, presumably it's in one of the citations in
    the paragraph. (Also, you probably mean "Taskar et al.'s bound".)
There are many excellent citations in related work (section 2), but without much
discussion and clear connection to the approach in the paper.
68: "w the parameter." -> "w is the parameter vector." (probably want to say $w
\in \mathbb{R}^d$ as well)
70: clearly over and under estimation method do not partition the space of
approximate inference algorithms: what sorts of algorithms does it omit? does it
matter? Also, what is the role of \rho? Do we need to know it? does it have
constant for all x? (Looks like later, e.g., theorem 4, it's allowed to depend
on the example identifier) Why are multiplicative bounds of interest? Which
approx methods give such bounds? (Later it is generalized based on
stability. Probably worth mentioning earlier in the paper.)
The notation h[w](x) is unusual, I'd expect/prefer h_w(x), or even h(w, x).
73, 74: definitions of H, H- and H+ are a little sloppy and verbose. The
definition of H is pretty useless.
73: The typesetting of R doesn't mach line 68. Please be consistent.
section 3.1
 - L(Q,S,h[w]) seems unnecessary (it's fine to just say that the expectation is over S).
 - Also, S is undefined and the relationship between S and x_i is not explained
   (I assume x_i \in S and |S|=m). Alternatively E_{x \sim S} would do the
   trick.
Lemma 2: Missing citation to PAC-Bayes Theorem.
  Please explain the relationship between the prior p(w') and Q(w'|w). The use
  of a prior comes out of the blue for readers who are unfamiliar with PAC Bayes
  bounds.
  A few words about why I should even be thinking about priors at this point
  would be useful.
  Also, if I don't care about priors can't I optimize the prior away?
77: Should technically pass Q(.|w) to KL divergence, i.e., $D(Q(.|w) || P(.))$.
80: It might be more useful to simply define under and over estimation with this
definition of margin.
theorem 4:
 - assume h'[w] be a ..." -> "theorem 4: assume h'[w] is ..."
 - Should be a \rho_i approximation of h[w](x_i)?
 - what is \bar \cal Y ?
 - why bother subscripting \lambda_S? just use \lambda
90: a better/more-intuitive name for square root term?
107: definition 8, "contains" is not a good term since it's easy to mistaking it
for set containment or something like that. Might be better to call it
"dominates".
Section 3.2 could be summarized as approximate inference algorithms give us a
/different/ family of predictors. Sometimes the approximate inference family has
a better predictor (for the specific dataset) than the exact family.
   Giving high-level intuitions such as the one I provided with resonate more
   with the reader.
Figures 1 and 2 take a little too much effort to understand.
    Consider hoisting some of the discussion from the main text into the
    caption.
    At the very least, explain what the visual elements used mean mean (e.g.,
    What is the gray arc? What are the columns and rows?)
159: "... or by tuning." on development data or training data?
246: "non-comparableness" -> "incomparability"