Paper ID: | 8832 |
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Title: | Robust exploration in linear quadratic reinforcement learning |

The paper is very well written and organized and its contributions are quite original as it proposes a novel coarse-ID method for robust model-based reinforcement learning in which both exploration AND exploitation are optimized jointly (which was not the case in previous similar works). The method proposed to solve the robust Reinforcement Learning problem is all the more original as it does not rely on Stochastic Dynamic Programming, but rather on Semidefinite Programming. Concerning clarity, the only element that is not clear for me is related to equation (1) in page 2: do you consider in the system model some uncertainty in the measurements of the states x ? For example, it is said in the supplemental material that the velocity of the servo-motor of your second experiment is estimated using a high pass-filter, and is hence not perfectly known. If it is modeled, is it included in the process noise w or how do you deal with it ? This seems important as it is rather common not to have access to perfect measurements in controlled systems. Furthermore, in page 3 you restrict the study to static-gain policies. I believe that it would be useful for non-expert readers to comment on how restrictive this assumption is. You could say for example whether it is a common assumption and give one or two examples of systems which can be properly controlled that way. I also believe that the authors should comment the approximations made in section 4.1 leading to problem (7), on how the solutions of this convex surrogate problem are close or not to those of the original problem (3). Indeed, this seems to be an important element of the theoretical analysis here, as the method proposed allows to solve (7), while previous methods (DP) seem to solve a gridded version of problem (3). <--- I apologize for this comment. I reckon after reading the authors rebuttal that this question was indeed treated in the manuscript and that I just did not see it. ---> Moreover, no comparisons are made what so ever between the proposed approach and the classical Dynamic Programming approach (or other known Optimal Control methods). For example, in page 7, a complexity analysis of the RRL method proposed is given, but it is never compared to the complexity of DP. This seems crucial as the authors justify RRL in page 3 by the fact that DP is "computationally intractable for systems of even modest dimension ». As far as I’m aware, semidefinite programming problem also suffers heavily from dimensionality, which can indeed be seen in the complexity presented in page 7. In the same direction, experimental results in section 5 never compare RRL to SoTA methods. Also, how much time do the computations take ? Again, the motivation for not solving (3) with a known method seems to be that known methods are not efficient/tractable, and yet no results points toward the conclusion that the proposed method is efficient/practical in higher dimension. Also concerning experimental results: 1. In figure 2a, what happened in the experiments corresponding to the outliers of the RRL results plot? These seem to be worth investigating/commenting as the RRL method is supposed to deliver robust solutions. 2. In figure 2c, shouldn’t greedy method interpolate between RRL and Nominal, as it is first equal to Nominal than shifted towards RRL ? 3. More experiments with more real-word data would make the paper way more convincing. All in all, the ideas in this submission seem quite interesting and it could likely be significant for other researchers in the field, but this is somehow mitigated by the lack of experimental and theoretical comparisons to SoTA methods supporting the claimed strength of the new approach proposed. ----- I have read the authors rebuttal and I am mostly satisfied by their clarifications. I am hence increasing my score under the condition that they add indeed the new experiments and complexity analysis. PS: I don't understand however why the execution times that the authors are willing to add to their V2 was not included in the rebuttal.

Originality: Convex approximations to LQR-like problems have been studied extensively in the literature, and the authors are up-front about this. However, the bound on the estimation error, the convex reformulation of the infinite horizon LQR problem, and the convex semi-definite program formulation are all novel. Quality: The paper is of extremely high quality, and provides thorough proofs and analysis of all claims. The authors could spend more time comparing their algorithm to other benchmarks tasks (if only to situate the algorithm with others in the space), but the provided analysis seems more than sufficient for publication. Clarity: The work is quite clear. I particularly enjoyed the comparisons demonstrating the higher information gleaned by greedy algorithms (yet concomitant lower performance) because of their failure to automatically balance exploration/exploitation. Significance: While I am not overwhelmingly familiar with the space, this work seems like an extremely general tool for use on LQR type problems. I would appreciate more discussion of how this method compares with others in the space (if only for this reviewer's edification).

Originality ----------- The paper seems to be significantly novel, and the related work section touches on many related areas and prior work. Admittedly, I am no expert on the field of robust control, however my impression is that the authors have done their due diligence in citing prior work and delineating their contributions from these works. Quality ------- The quality of the work is perhaps my greatest concern. As I have mentioned, experiments were only conducted on numerical simulations and a simple hardware setup, and though the analysis on these two domains are decently thorough, it is rather unclear from just these domains how well the method scales or whether it provides tangible benefits on systems of actual interest. I would request that the authors do two things: (1) Elaborate further on the real hardware experiment. Is there a goal for controlling this system, i.e., why is this system of interest beyond demonstrating the capabilities of the method? The physical servo is connected to a synthetic linear system, which seems to make the experiment more toy as this limits the realisticness of the setup. Finally, is there any reason this system in particular benefits from robust control, in that there may be danger or costs associated with running the system? (2) Conduct further experiments, ideally on real robotic systems. As the authors do not assume that the underlying dynamics are known, it seems feasible to run this algorithm on, e.g., robot arms or locomotion platforms where the true dynamics are not linear. These systems seem much closer to reality and can benefit much more convincingly from robust control, as safe control is certainly desirable. If real robots are not an option, additional simulated systems could be useful, even extremely simple domains such as a 2D point mass navigation domain where the system truly is linear quadratic. This would have the benefit that the domain can potentially be engineered with obstacles, traps, etc., that make robust control extremely desirable. Clarity ------- In general, the paper is well-written except for some minor issues. Most egregiously, there is no conclusion, which makes it hard for readers to understand the key takeaways and limitations of the work. A couple of minor nits: - line 52, “essential for implementation on physical systems”: I would argue that this is only true for some physical systems - proposition 2.1: it may be useful to explain that the “uniform prior over the parameters” is an improper prior, or otherwise remove the need for a degenerate prior altogether from the analysis Significance ------------ As I have already discussed most of my concerns, I will simply state that the aforementioned improvements to the quality and clarity of the work would greatly improve the significance as well.