__ Summary and Contributions__: The paper presents improved bounds for learning in fully-observable, tabular MDPs with side observations using a feedback graph.

__ Strengths__: The regret and sample-complexity bounds are tighter than previous best in a similar setting. The feedback graph has potential as a new analysis tool both for defining bounds and developing new algorithms.

__ Weaknesses__: Tabular MDPs, accurate side information. I have no objection in principle to analysis that starts with the simplest case but the paper is careless in postulating that it will generalize to a realistic setting.
Would have liked to see more intuition developed around the feedback graph. Particularly because I misunderstood the first time around and thought it depended on the side information provided with the currents state being transitions from the state.
-=- Misunderstanding kept for posterity -=-
It is essentially an expanded MDP diagram, where each vertex is a {s, a} pair and each edge indicates a valid transition. This is an easy construct to think about. For example, Mas-number is then the longest possible history that doesn't repeat itself, independence number is the maximum number of state-action pairs that don't connect directly to each other, and domination number is the minimal set of {s, a} pairs whose neighbours cover the space. It would be vastly easier to follow the implications if these kinds of common-sense ties were explicitly provided in the paper rather than left as an exercise to the reader.
Appendix F? Perhaps this work would be better served as a journal submission, if 48 pages are required to present it clearly.

__ Correctness__: Empirical methodology irrelevant as not necessary for the value of this work.
Claims are sound and well supported.
-=- Kept for posterity -=-
Self-loops being implicit and thus ignored seemed incorrect, given the analysis and bounds depend on mas-number. If self-loops exist for all {s, a} pairs then there are no acyclic subgraphs.
There is no empirical analysis, although I don't believe that is strictly necessary for this kind of work.

__ Clarity__: I found the paper incredibly difficult to follow with too much reliance on appendices.
Update: However, given the authors' response I do think the readability will be improved for the unfortunates like myself who do not have the relevant background information easy to hand. Material in the appendix helps greatly and if some can be moved into the body of the paper that could resolve this issue.

__ Relation to Prior Work__: The paper rigorously compares this approach to related bandit analysis.
I would have liked to see this related to Dyna-style approaches that use side information drawn from trajectories, and a tighter exploration of how generated/estimated side information would impact the analysis.

__ Reproducibility__: Yes

__ Additional Feedback__: I was quite intrigued by the core idea and would like to see it explored in either a forum that allows for the level of detail required or split into more easily-digestible pieces.

__ Summary and Contributions__: This paper addresses the setting where an RL agent receives additional observations, after executing every action, which provide it with information about possible transitions that it could have experienced. These side observations may be generated, for instance, by auxiliary sensors. The authors formalize this setting by constructing a feedback graph based on additional observations. Feedback graphs may be used by model-based RL algorithms to learn more efficiently. In particular, the authors show that the regret of the resulting model-based algorithm is bounded by certain properties of the graph, instead of depending on the number of states and actions that exist in the original problem (without side observations).

__ Strengths__: This is a well-written theoretical paper. Although it does not demonstrate empirically that feedback graphs work in real-life problems, the underlying theory seems sound and the formal results that are presented are certainly not trivial.

__ Weaknesses__: The main weakness of this paper is that it lacks an empirical evaluation section demonstrating that the theory is actually applicable to interesting problems. Also, the results depend on a few assumptions that may not always be realistic: tabular model-based algorithms, or multitask settings where the rewards of all but one task are known, and where the dynamics are equal across tasks.

__ Correctness__: The theory underlying the results seems correct. The paper has no empirical results.

__ Clarity__: Yes.

__ Relation to Prior Work__: Related work is briefly but appropriately discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: This paper addresses the problem of an RL agent that receives additional observations, after executing every action, which provide it with information about possible transitions that it could have experienced. These side observations might be generated, for instance, by auxiliary sensors. The authors formalize this setting by can defining a feedback graph based on the additional observations. Feedback graphs may be used by model-based RL algorithms to learn more efficiently. In particular, the authors show that the regret of the resulting model-based algorithm is bounded by certain properties of the graph, instead of depending on the number of states and actions that exist in the original problem (without side observations).
This is a well-written theoretical paper. Although it does not demonstrate empirically that feedback graphs work in real-life problems, the underlying theory seems sound and the formal results that are presented are not trivial.
I have just a few questions and comments:
1) Is it possible to formally characterize the relation between the results that you present and possible bounds on the performance achievable by off-policy learning algorithms? There might exist a relation, given that at each step, these algorithms may also run updates over (past) experiences other than the current experience.
2) Could your analyses be used to formally characterize the regret of algorithms that operate over a replay buffer? I believe that it may be possible to interpret many modern algorithms that use a buffer of past experiences (and that run updates over them) as algorithms that, besides that current experience, are also given, at each step, a set of additional observations. I wonder if your results could be extended to show regret bounds on off-policy algorithms of this type.
3) Intuitively, what are the main challenges involved in adapting your findings to model-free algorithms?
4) Could you please better motivate the key assumption made when analyzing extended MDPs? In particular, in what type of multitask settings would it be realistic to assume that the rewards of all but one task are known, and that the dynamics are equal across tasks? This does not seem to fit into the usual multitask setting, where the dynamics may differ from task to task, and where the rewards are typically assumed to be unknown.
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POST-REBUTTAL COMMENTS
I have read the authors' rebuttal and thank them for addressing my questions and suggestions. I believe that they have satisfactorily addressed the main points that I brought up.

__ Summary and Contributions__: This paper considers reinforcement learning with feedback graphs --- there is a directed graph with state-action pairs as nodes, and whenever the agent is in state s and takes action a, it observes not only the next state and reward, but also samples of transitions from the state-action pairs that have an edge from (s,a). The main contributions of this paper are to give lower and upper bounds on the regret and sample complexity of the problem, as a function of various properties of the feedback graph, namely the size of the maximal acyclic subgraph, the size of the largest independent set, and the size of the smallest dominating set.
The first result shows that for standard model based algorithms, the dependence of the regret on the number of states and actions can be replaced by the m.a.s. number μ. In the lower bounds, SA can be replaced by γ, the size of the maximal independent set. For undirected graphs, μ = γ, so the bounds are tight. The improvement comes purely from the extra state transitions available to the learner.
The other major contribution is to improve the dependence on γ to α, the minimal dominating set. This is known to be possible for bandit settings, since taking dominating actions is enough to learn about all the other actions. In MDPs, however, there is an additional factor depending on how frequently it is possible to visit dominating states. Furthermore, the authors have a lower bound showing that dependence on α cannot be completely eliminated in the lower-order terms. Furthermore, the authors show that the algorithm has a negligible penalty when the dominating set is not known a priori, since the algorithm can visit states and thereby recover the full feedback graph.

__ Strengths__: This paper is a natural extension of the feedback graph idea from bandits to Markov decision processes. In bandits, it gives us a better understanding of the interplay between taking informative vs. rewarding actions, as a step towards the more general partial monitoring setting. It appears that some of these insights are valuable in the MDP setting as well, particularly the role of the m.a.s. vs independence numbers and the improvement to the dominating sets. The proof techniques used may also be interesting to others.
This paper is also extremely comprehensive, with useful discussion in the appendix and many generalizations considered, including feedback that is only available sometimes and/or biased.

__ Weaknesses__: While the paper is motivated based on extending feedback graphs from bandits to MDPs, it would be useful to give examples of applications where such structured feedback is likely to be available. In practice, MDPs and bandits are applied in different contexts, so a problem formulation that is convincing for one is not necessarily so for the latter. While I appreciated the theoretical contributions in this paper, I was left wondering how they could be applied.

__ Correctness__: I did not have time to check the correctness of all the proofs, but I did skim through the major ones and checked that the results were reasonable. It is quite possible that I have missed errors.

__ Clarity__: The paper is, on the whole, very clearly written. It suffers a little from being forced into a conference format, because much of the discussion in the (extremely long) appendix would have been valuable in the main text. In particular, I appreciated the illustrative examples of feedback graphs with their associated properties and characteristic numbers. Some minor comments are in the "additional feedback."

__ Relation to Prior Work__: To my knowledge, the paper is correct in claiming to be the first exploration of MDPs with side-information in the form of feedback graphs. While this has been explored a fair bit in the stochastic bandits literature, there are additional non-obvious insights to be gained in the MDP settings. I think this paper is a useful and novel advance over prior work.

__ Reproducibility__: Yes

__ Additional Feedback__: * Not being familiar with the "IPOC" framework, I found Remark 2 out-of-place and hard to parse. It either needs some explanatory text, or to be moved to the appropriate part of the appendix.
* In the caption of Table 1, please include the definition of Ŝ. Also, while the definition of p₀ is deferred to section 5 where it naturally belongs, it would be helpful to have a brief note on what it means earlier in the paper.
* Footnote 3 is important and should be in the main text, since a natural question is how to deal with not knowing the dominating set.
Questions:
* The paper comments on the gap between the upper and lower bounds in the case of directed graphs, but it would be useful to have a qualitative discussion of whence this gap arises and whether it's likely to be "real". I do not have much intuition for what directed feedback graphs would mean for minimizing regret, so it would be useful for the authors to share theirs. Such a discussion is provided in the appendix for the necessity of the independence number in the lower bound, and is very useful.
* Loosely speaking, in other RL work some sort of state-action representation scheme is used to generalize learned values between states. Then one may consider the hardness of planning with a given representation, vs. the hardness of learning a good representation (which is still largely unknown). The idea of a feedback graph blurs that line. I would like to see a broader comparison of the feedback-graph approach to structured MDPs vs. other more common assumptions in the literature.
*** Post-rebuttal update
Thanks to the authors for their response. I'm looking forward to seeing the suggested changes made in the paper, including more illustrative examples/applications and moving the intuitive explanations from the appendix to the main paper.

__ Summary and Contributions__: Thanks for the reply and the authors answered my question in the author feedback. The paper has a major structural issue. A significant portion of the material is presented in the appendix. This not only makes reviewing process difficult, but also bypass the page limit. I believe this paper is more suited for a journal submission.
This paper considers a tabular MDP problem with feedback graphs. Authors prove that existing model-based RL algorithms achieve regret and sample complexity bounds do not scale with SA. Lower bounds of the regret and the sample complexity are also presented to show the optimality of their analysis. Authors also present an algorithm that achieves a sample complexity bound that scales with the more favorable domination number.

__ Strengths__: The problem of online MDPs with feedback graph is interesting and promising. The theoretical analysis is rigorous and valid.

__ Weaknesses__: 1. In algorithm 3, you assume the dominating set is known to the agent. I am not sure this is true is most reinforcement learning applications. In the robot moving example, the transition model is unknown which means the agent does not know which states are in the same line. Then you don’t know the dominating set in this case.
2. In algorithm 3, authors use the extended MDP to simultaneously learn policies to reach the goal and the points in the dominating set. However, the sample complexity analysis does not show the benefit of the ‘multi-task’ learning process.

__ Correctness__: Yes. Through regret analysis, this paper shows that the proposed algorithm utilizes the feedback graph information and achieves a better regret bound.

__ Clarity__: Yes. The problem is well explained. The proofs are easy to read and follow.

__ Relation to Prior Work__: Yes. Related works are well discussed and

__ Reproducibility__: Yes

__ Additional Feedback__: