NeurIPS 2020

### Review 1

Summary and Contributions: The authors proposed a method for designing robust-adversarial LQR controllers using a double-loop policy gradient ascent/descent method. The main contribution is that the controller is designed to be robustly stabilizing during adaptation, which is very necessary for practical applications.

Strengths: The paper is clear and its novelty is in a direction well-aligned with NeurIPS community. Assumptions make sense, and claims are sound. Good empirical evidence is provided. Examples are illustrative and excellent for understanding. Great paper.

Weaknesses: Two limitations: (1) globally linear dynamics are rare in practice: this reviewer would like to know whether this could potentially be used for systems with local linear dynamics and what the problems arising in such scenarios could be? and (2) I believe the authors know the system matrices and description etc. Can then classical robust control methods be used to come up with LQ policies rather than RARL? Also, authors should mention that this is model-based approach.

Correctness: I believe so. Code is presented and looks fine.

Clarity: Yes, I had little to no trouble understanding the main paper. The supplementary is a bit haphazard and could benefit from a table of contents or organizational paragraph.

Relation to Prior Work: Reminiscent of this work, a double-loop approach to LQ control with state and input constraint satisfaction has recently been proposed where the authors are also keen to ensure recursive feasibility and stability properties: [A] https://arxiv.org/pdf/1906.11369.pdf. The current paper deals with the robust/disturbance case but some of the data-driven transformation of model-based optimization problems considered in [A] might be interesting to the authors. There is also work in the approximate dynamic programming literature by Frank Lewis and others in robust/H-infinity RL for linear systems - this could be relevant to the current paper too perhaps.

Reproducibility: Yes

Additional Feedback: - Which matrices are known at design time? Is this model-based RL? This needs clarification early on e.g. after 2.1 - Is there a reason the authors use min sup instead of min max or inf sup? --- POST REBUTTAL --- My opinion remains the same. Good paper, authors rebuttal improved things even further. - Equation (2.5) is not a Lyapunov equation (A'PA - P = -Q is). This is closer to a Riccati equation. - L. 136-137 For self-containment of the paper, please add how the gradients are computed, briefly. - In the thought experiment, there is no need to assume a K exists. If (A,B) is controllable and (A, Q^1/2) is observable for psd Q, such a K exists for L=0.

### Review 2

Summary and Contributions: The paper extends the analysis of RARL (Robust Adversarial RL), an algorithm used to train robust policies in RL, to the state-feedback control of discrete-time dynamical systems (specifically, the discrete LQR case). The authors first discuss some stability issues that can arise from blindly applying RARL to dynamical systems, and help the reader better understand how to use RARL. Then, the authors provide an algorithm that has stability and convergence guarantees. An extensive numerical section is provided in the appendix.

Strengths: I believe the paper to be clear and well written. The claims seem sound, and the results provided in the paper shed a light on the applicability of RARL. The various results presented in the paper, together with the numerical section, greatly help with the understanding of the method.

Weaknesses: - A proper conclusion is missing. - The related work section can be improved. The authors can also better explain the difference with the work in [10] -Unfortunately the paper analyses only state-feedback control. It would be nice to see also an analysis of output feedback control. -Theoretically speaking, LQR is a good algorithm to analyse. Despite that, it is almost never used in practice (from my experience). It is hard to specify design requirements using LQR, especially in the frequency domain. It would be nice to analyse RARL in case there are some control requirements.

Correctness: The method seems correct.

Clarity: I believe the paper to be clear and well written

Relation to Prior Work: Relation to prior work is discussed, though can be made clearer.

Reproducibility: Yes

Additional Feedback: - How did you choose the A,B,C matrices in the numerical sections? == Post rebuttal I have read the authors rebuttal. I will not change my score.

### Review 3

Summary and Contributions: The authors consider the Robust Adversarial Reinforcement Learning (RARL) approach in the Linear Quadratic (LQ) setting and investigate the stabilization capabilities of the resulting linear static feedback controller. First the setting, similar to e.g. [44], is described and relevant concepts are introduced and connections to related work are pointed out. As a first contribution the authors demonstrate with several examples, using numerical simulations, that RARL in the LQ setting can suffer from stability problems. One source of these problems are bad initializations of the policy, and the authors propose a simple criterion to avoid such problems. Another source of stability problems is a bad choice of the number of iterations performed in the algorithm. Next, the authors propose a new concrete RARL algorithm for the LQ setting and provide convergence and stability results for the algorithm, as well as numerical experiments demonstrating the algorithm on simple examples. Furthermore, two variants of the algorithm are presented and investigated with numerical examples, though no theoretical results are provided in these cases. Finally, a new robustification method based on robust control techniques is described, that is used to generate suitable initializations of the RARL algorithms. Overall, the paper makes a solid technical contribution. The findings, methods and theoretical results are interesting and relevant. However, the authors consider a rather specific approach (general approach from [30], setting from [44] and [10]) and the methods and theoretical findings appear somewhat incremental rather than novel or a break-through. While this work definitely should be published, it is questionable whether the contributions are considered sufficiently novel for NeurIPS.

Strengths: * The authors demonstrated stability issues of RARL in the LQ setting. Since RARL seems to be popular and the LQ setting is of fundamental importance (in particular as a benchmark), this is a relevant contribution to the reinforcement learning community. * The authors propose a simple concept for good policy initializations in order to avoid stability problems (Def 3.3). This is then used later on to provide stability guarantees for a new RARL algorithm. A strength of this approach is its conceptual simplicity and the clear connection to established robust control theory (made explicit in Lemma 3.4) * A new, simple RARL algorithm for the LQ setting is proposed (Section 4.1) with stability and convergence guarantees (Sections 4.2, 4.3). The algorithm itself seems to be just a minor extension of prior work, in particular [44], however, the particular stability and convergence guarantees seem to be new, and hence this can be viewed as a relevant contribution. The techniques used for the theoretical results are standard, though one could argue that this is actually a strength, since the authors demonstrate that established methods can be successfully used to tackle problems of reinforcement learning. According to the authors similar results are derived in [10], but the latter work contains a technical error (and uses a different technique anyway), details are provided in Remark B.1 in the supplementary. * All claims in the paper are supported also by numerical experiments on synthetic examples. It seems that all the necessary Matlab code for reproducing all experiments is provided (though I haven't run all the scripts). The code is rather scarcely documented, but it is enough to understand everything.

Weaknesses: * Only sublinear convergence can be guaranteed for the algorithm proposed in Section 4.1 It would have been nice if the authors discussed this in more detail, in particular also reporting the run times of the numerical experiments and discussing the current limits imposed by the convergence rates. Note that in Remark 4.3 and Section B.4 faster (local) convergence rates are discussed. However, as noted by the authors acceleration with Gauss-Newton method cannot be done in a model-free manner. Since this paper is aimed at the reinforcement learning community this seems to be a major weakness. This aspect makes the discussion of the slow guaranteed convergence mentioned above even more important. * The discussion of the stability problems in Section 3.2 is very vague. In particular, it is not clear how Example 3.5 is related to the (very short) discussion preceding it. Reading Section C.1.2 makes things a bit clearer, but Section 3.2 should be improved. Furthermore, it would be nice if Section 3 contains a summary like in lines 718-720 in the supplementary.

Correctness: * Theory: The theory in Section 2, 3 and Section A in the supplementary is standard (basic robust control theory and game theory, some recent results from [44]) and hence can be assumed to be correct. New theoretical contributions are in Section 4 and Section B in the supplementary and seem to be correct (though I have not recalculated every step). * Empirical results: All experimental setups are reasonable and seem to be correctly implemented and the results transparently reported. However, it would have been nice to get some information on runtime as well as the underlying standard methods used in Matlab. * In lines 13-15 the authors claim "We explain the difference between robust stability and robust performance, and then discuss several ways to learn robustly stabilizing initializations". This statement seems misleading since robust performance (in the sense of robust control) is not discussed at all (in fact, the term robust performance does not appear outside the abstract and introduction at all).

Clarity: * The theoretical contributions (including proofs) as well as the experiments (including results) are clearly described. * While the technical contributions are clearly written, in general the paper is a bit difficult to read. In particular the Introduction seems to lack a clear structure and Section 2 is difficult to understand without reading Section A in the supplementary or being already familiar with this particular line of work. Interestingly, while the paper indeed tells a coherent story (problems of RARL in LQ setting -> approach to overcome this -> theoretical guarantees) this is not reflected in the text and it feels like a collection of different pieces. Again, this is an issue with the writing and not with the content. * Furthermore, it would have been nice to have a proper Conclusion (in fact the authors put a short conclusion in the Broader Impact section)

Relation to Prior Work: * The authors state that RARL [30] is popular, however, there are only few references to substantiate this claim. * The connection to reinforcement learning (in particular [30]) and previous work (mostly [10], [44]) is clearly described. However, the related work section is very short. In particular, a brief description of other recent usages robust control methods and theory for reinforcement learning would have been nice. Furthermore, although many references for estimating the $H_\infty$ norm from data are provided in Section 5, almost no pointers to other approaches of model-free robust control methods are given (there has been quite some activity, cf. e.g. [R1]) * In lines 331-332 the authors write that "We note that such a model-free robustification approach seems novel even in the robust control area, to the best of our knowledge." This could potentially be a bit misleading since model-free approaches for robust controller synthesis (which can be interpreted as a form of robustification) are subject of recent investigation, cf. e.g. [R2] for an LMI-based approach and [R3] for MPC approaches. However, I would consider this only a minor issue. [R1] De Persis, Claudio, and Pietro Tesi. "Formulas for data-driven control: Stabilization, optimality, and robustness." IEEE Transactions on Automatic Control 65.3 (2019): 909-924. [R2] Berberich, Julian, et al. "Robust data-driven state-feedback design." arXiv preprint arXiv:1909.04314 (2019). [R3] Berberich, Julian, et al. "Data-driven model predictive control with stability and robustness guarantees." IEEE Transactions on Automatic Control (2020).

Reproducibility: Yes

Additional Feedback: * More details should be provided for Lemma 3.4 and Lemma 5.2, at least the concrete version of the Bounded Real Lemma used (Reference [45] has many versions) Minor language issues * l40: "on solid grounding" instead of "under" * l41: "robustness performance" unclear, probably robust performance is meant * l147: "if it exists" and "be worsened" * The paragraph starting from l153 should not be called a thought experiment. It is more like a discussion. ================================= POST REBUTTAL Thank you for the clarifications, esp. wrt the contributions and relations to [10], [44], which was convincing. In light of the rebuttal and discussion, I've increased my score +1.

### Review 4

Summary and Contributions: This paper studies robust adversarial RL (RARL) under LQ setting. It argues that conventional RARL scheme that greedily alternates agents’ updates can easily destabilize the system. Authors propose an algorithm to resolve such issue.

Strengths: This paper discusses the stability issues in policy gradient updates of LQ RARL. Authors propose a double loop algorithm to achieve stability.

Weaknesses: This work is limited in linear quadratic case, and PG under LQ case is well studied. So the technical challenges are not clear.

Correctness: The method seems correct. There is no empirical study in the main paper.

Clarity: This writing is understandable, however more details such as assumptions should be discussed more clearly.

Relation to Prior Work: No, this paper mentions very limited related work.

Reproducibility: Yes