Paper:
Policy-Conditioned Uncertainty Sets for Robust Markov Decision Processes
What policy should be employed in a Markov decision process with uncertain parameters? Robust optimization's answer to this question is to use rectangular uncertainty sets, which independently reflect available knowledge about each state, and then to obtain a decision policy that maximizes the expected reward for the worst-case decision process parameters from these uncertainty sets. While this rectangularity is convenient computationally and leads to tractable solutions, it often produces policies that are too conservative in practice, and does not facilitate knowledge transfer between portions of the state space or across related decision processes. In this work, we propose non-rectangular uncertainty sets that bound marginal moments of state-action features defined over entire trajectories through a decision process. This enables generalization to different portions of the state space while retaining appropriate uncertainty of the decision process. We develop algorithms for solving the resulting robust decision problems, which reduce to finding an optimal policy for a mixture of decision processes, and demonstrate the benefits of our approach experimentally.

Review:
The authors consider distributionally robust finite MDPs over a finite horizon.  The transition probabilities conditionally to a state-action pair should remain at L1-bounded distance from a base measure, which is feasible as being generated using a given reference policy.  This is a nice idea. A few comments are mentioned next.  * One aspect I did not understand is how the reference policy should be chosen.  Related to that question, why the requirement of staying "close" to this policy would be beneficial. In safe RL for instance, we could want to remain close to "safe" states, but I can't make the connection with this and the identification of a single "safe" reference policy. In the first experiment, (line 253) the authors use a uniform policy as the reference. In the second experiment, there are only 2 actions and the authors use a state-independent randomized policy (line 301). What is the rationale? Is the choice of a "uniform" reference policy specific to these two problems?  * How should we read the results of Table 1, when p varies ? Is there a value of p that leads to a better policy? In which sense?  * I liked (line 274)  that the authors provided a sensitivity analysis of the stepsize they use to discretize a [0,1]-valued extra state variable.   * What is the exact definition of the softmin in equations (6) and (11)  * There are too few details to understand the benchmark policies such as LR, RECT, MLE to be able to reproduce the experiments (depicted in Figure 1). I would think that those methods have more parameters than those given in the supplementary material.  * (line 225) "converges when the statistics match", but "when statistics match" is unclear, can the authors explain.  * Section 2 introduces the notion of "Directed information theory for processes", but I interpret this as describing the fact that policies should be in the class of progressively measurable processes in discrete-time -- which is a normal math framework for history-dependent policies.  * The significance is somewhat difficult to assess because the test problems seem relatively low-dimensional, which, granted, facilitates benchmarking, but does not allow to demonstrate scalability of the proposed method.