We study the problem of multiset prediction. The goal of multiset prediction is to train a predictor that maps an input to a multiset consisting of multiple items. Unlike existing problems in supervised learning, such as classification, ranking and sequence generation, there is no known order among items in a target multiset, and each item in the multiset may appear more than once, making this problem extremely challenging. In this paper, we propose a novel multiset loss function by viewing this problem from the perspective of sequential decision making. The proposed multiset loss function is empirically evaluated on two families of datasets, one synthetic and the other real, with varying levels of difficulty, against various baseline loss functions including reinforcement learning, sequence, and aggregated distribution matching loss functions. The experiments reveal the effectiveness of the proposed loss function over the others.
Summary: 
The paper considers the prediction problem where labels are given as multisets. The authors give a definition of a loss function for multisets and show experimental results. The results show that the proposed methods optimizing the loss function perform better than other alternatives.
Comments: 
The problem of predicting multisets looks challenging and interesting. The experimental results look nice. On the other hand, I have several concerns about writing and technical discussions. 
First of all, the goal of the problem is not exactly stated. After I read the experimental section, I realized that the goal is to optimize the exact match score (EM) or F1 measure w.r.t. the ground truth multisets. This goal should be explicitly stated in the paper. Now then, the approach of the paper is to design surrogate loss functions to optimize these criteria. 
The technical discussions for defining the proposed loss function seems not reliable for the reasons below. Therefore, I do not understand the rationale of the definition of the proposed loss function.:  
- An exact definition of the term multiset is not given. If I understand it correctly, a multiset is a “set” of instances allowing duplicated ones. 
- There is no definition of Prec or Rec (which look like Precision and Recall) in Remark 1. The definitions appear in Appendix, which might not be well-defined. For example, let y, Y be mutisets , y=[a, a, a] and Y = [a, b]. Then, by definition, Prec(y,Y)=3/3 =1. Is this what you meant? (Maybe, the ill-definedness  comes from the lack of definition of inclusion in a mutiset.) 
- I cannot follow the proof of Remark 1 since it does not seem to take account of the randomness by the distribution \pi^*. 
- I do not understand the definition of the oracle policy exactly. It seems to me that, the oracle policy knows the correct label (multi-set) \calY for each instance x and use it to construct \calY_t. But, this implicit assumption is not explicitly mentioned. 
- In (1), (2) and Definition 3, what defines \calY_t? If \calY_t is determined by some “optimal” oracle, you cannot define the loss function in Def. 3 since it is not known a priori. Or, if the learner determines \calY_t, I don’t understand why the oracle policy is optimal since it depends on the learner’s choices. 
Also, I expect an investigation of theoretical properties of the proposed loss function, e.g., relationship to EM or F1 or other loss functions. Without understanding the theoretical properties and the rationale, I cannot judge the goodness of the experimental results (look good though). In other words, I cannot judge the paper in a qualitative perspective, not in a quantitative view. 
As a summary, I think the technical contribution of the paper is marginal because of the lack of reliable mathematical discussion or investigation.