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)
The authors consider robust polynomial optimization
where the uncertainty parameter is a probability distribution estimated
from data samples. The solution is shown to be the limit of a sequence of
SDP relaxations and finitesample consistency guarantees are given.
Numerical results on a water network problem are provided.
Quality: The paper is technically sound. The proofs are
provided in the supplementary material.
minor comments: 1) A
more detailed explanation (example) on constructing distributionally
uncertainty sets from sampled information may be helpful.
2)
"Table 6" on page 7 should be "Table 2". It seems to be hard to justify
the advantage of the proposed robust approach from the given example.
Clarity: The paper is wellorganized.
Originality: Forming
distributionally robust polynomial optimization into polynomial
optimization problems and constructing distributionally uncertainty sets
from sampled information sound interesting. The asymptotic guarantees in
Theorems 4.1 and 5.1 follow from the references [1] and [22].
Significance: Polynomial optimization arises from many fields. A
practically useful robust polynomial optimization approach with certain
theoretical guarantees is important. Q2: Please
summarize your review in 12 sentences
Robust polynomial optimization using probability
distribution estimated from data samples is interesting. The asymptotic
guarantees seem to be extensions of the results in the
literature. 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)
The authors considered robust optimization for
polynomial optimization problems where the uncertainty set is a set of
possible distributions of the parameter. In specific, this set is a ball
around a density function estimated from data samples. The authors showed
that this distributionally robust optimization formulation can be reduced
to a polynomial optimization problem, hence computationally the robust
counterpart is of the same hardness as the nominal (nonrobust) problem,
and can be solved using a tower of SDP known in literature. The authors
also provide finitesample guarantees for estimating the uncertaity set
from data. Finally, they applied their methods to a water network problem.
This seems to be an interesting paper. Distributionally robust
optimization is now topical in operations research. Most previous
literature is restricted to the case that the distribution is
characterized by (first and second) moments, which has the apparent
disadvantage that consistency is impossible regardless of the number of
samples obtained. In contrast, the proposed approach overcomes this
difficulty, as when more samples are obtained, we can shrink the
uncertainty set until it becomes a singleton, and hence achieving
consistency. It is also nice to see that, at least for the polynomial
optimization problems, the resulting robust counterpart does not incur
additional computational difficulty.
Some minor points:
1.
Assumption 4.1 is a bit confusing. In particular, I am not sure how the
sentence "and there exist u\in R[x]... compact." is related to the set X
and S. 2. In line 194, "\hat{S} satisfies Assumption 4.1". Why? 3.
L236, "We consider the multidimensional case in the next section" appears
to be wrong. 4. L375 "Table 6". There is no table 6 in the submission.
5. L379 "Unbiased". I am not sure why the uncertainty set are
unbiased (in fact I am even sure what does it mean that a set is
unbiased). 6. L148, "A notion of distributional uncertainty set has
also been studied in the setting of Markov decision problems [14]". I
think [14] does not consider distributional uncertainty. To my knowledge,
distributional uncertainty in MDP is investigated in Xu and Mannor (2012).
Ref: H. Xu, S. Mannor, "Distributionally Robust Markov
Decision Processes", Mathematics of Operations Research, 37(2), pp288300,
2012.
Q2: Please summarize your review in 12
sentences
This seems to be an interesting paper.
Distributionally robust optimization is now topical in operations
research. Most previous literature is restricted to the case that the
distribution is characterized by (first and second) moments, which has the
apparent disadvantage that consistency is impossible regardless of the
number of samples obtained. In contrast, the proposed approach overcomes
this difficulty, as when more samples are obtained, we can shrink the
uncertainty set until it becomes a singleton, and hence achieving
consistency. It is also nice to see that, at least for the polynomial
optimization problems, the resulting robust counterpart does not incur
additional computational difficulty.
I have taken into
consideration the rebuttal. 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 paper introduces the notion of data driven robust
optimisation(DRO) to solve optimization problems with inherent
uncertainty. In particular it investigates a minmax problem over the
expectation of a polynomial function. The max is taken over all
allowed probability densities and the min is over the decision
variable. The results appear to be correct and could be of significant
interest to robust optimization community. It would have been nice if
the paper could discuss a relevant ML problem in the context of DRO.
Quality: The paper scores high in mathematical rigour. .
Clarity: The paper is clearly written and the results are properly
stated.
Originality: The results are novel.
Significance:
The results in this paper could be of significant interest to Operations
Research community. Q2: Please summarize your review in
12 sentences
The main results of this paper may not be of interest
for a ML community.
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.
