Paper:
Efficient Learning by Directed Acyclic Graph For Resource Constrained Prediction
We study the problem of reducing test-time acquisition costs in classification systems. Our goal is to learn decision rules that adaptively select sensors for each example as necessary to make a confident prediction. We model our system as a directed acyclic graph (DAG) where internal nodes correspond to sensor subsets and decision functions at each node choose whether to acquire a new sensor or classify using the available measurements. This problem can be posed as an empirical risk minimization over training data. Rather than jointly optimizing such a highly coupled and non-convex problem over all decision nodes, we propose an efficient algorithm motivated by dynamic programming. We learn node policies in the DAG by reducing the global objective to a series of cost sensitive learning problems. Our approach is computationally efficient and has proven guarantees of convergence to the optimal system for a fixed architecture. In addition, we present an extension to map other budgeted learning problems with large number of sensors to our DAG architecture and demonstrate empirical performance exceeding state-of-the-art algorithms for data composed of both few and many sensors.

Review:
Resource constraint prediction has to select the smallest/ cheapest set of sensors in order to make accurate decisions. A new approach to this problem is proposed in this paper where the authors basically define a deterministic Markov decision process and learn its policy applying cost sensitive learning in every state of the MDP. Given a new testing example to classify, the policy has to decidewhether a new sensor should be queried (if yes, which sensor), or whether the prediction should be made with the current set of sensors measured.In line 44, the authors mention the "expected budget constraint". Please be clear about that because your paper is not budgeted learning; budgeted learning is more challenging, please see, e.g. papers of Russell Greiner (University of Alberta) on this topic.The DAG used in this paper defines an MDP. Perhaps a closer connection with MDPs could be made, e.g., you could mention that this is a special MDP that can be solved efficiently using your method. Also, it would be good to explain what are the key techniques/assumptions that allow solving this MDP efficiently. Putting your method in a broader MDP context would be very useful.Line 122, a quotation mark is missing.I have some problems understanding equation (5). If (s_j,s_k) is and edge, how can I read from the first case in the equation (5) that this represents the cost of acquiring a new sensor? Please explain. Shouldn't there be $t\in s_k \ s_j$ ?One of my key concerns is R that is used in Lemma 2.1. In [2], the authors use importance-weighted binary classification. This is basically classification that considers misclassification cost. What I am missing in Lemma 2.1 and its proof is a clear link between misclassification cost in [2] and the cost of features (or sets of features) in Lemma 2.1. Please explain. I have another problem with lemma 2.1. If node j leads to a leaf then the cost of sensors is zero because there is no need to pay for any sensors; why then, the cost of an edge j->k is considered in the proof of lemma 2.1.Vectors w are not defined in eq. 8. Well, (2) in Alg. 1 says "construct the weight vector w of edge costs per action". Please explain how to do that.It looks that the key element in the analysis in section 2.3 is to optimise at each stage over all examples in the training set. It sounded weird to me initially, but I believe that that's correct. My second main concern are Lemma 3.1 and theorem 3.2. As long as sub-modularity in Lemma 3.1 is correct I believe, I am not sure why we should not consider sub-modularity in the theorem 3.2 as well. The risk in the theorem is monotonically decreasing indeed, but we should really talk about diminishing returns which don't exist because adding a new feature f_1 may not change anything until we add another features f_2 when those two features interact. So, there are not diminishing returns and sub-modularity does not apply. Additionally, in the proof of theorem 3.2 the authors consider adding a new subset (line 34 in the appendix), but doing that will eventually violate the constraint in e.q. (1) in the appendix. I believe that this is another reason why this problem is not sub-modular, and the greedy approach can be very bad.Also, "the reward" in lemma 3.1 shouldbe explained.