Enhancing Robustness in Deep Reinforcement Learning: A Lyapunov Exponent Approach

Deep reinforcement learning agents achieve state-of-the-art performance in a wide range of simulated control tasks. However, successful applications to real-world problems remain limited. One reason for this dichotomy is because the learnt policies are not robust to observation noise or adversarial attacks. In this paper, we investigate the robustness of deep RL policies to a single small state perturbation in deterministic continuous control tasks. We demonstrate that RL policies can be deterministically chaotic, as small perturbations to the system state have a large impact on subsequent state and reward trajectories. This unstable non-linear behaviour has two consequences: first, inaccuracies in sensor readings, or adversarial attacks, can cause significant performance degradation; second, even policies that show robust performance in terms of rewards may have unpredictable behaviour in practice. These two facets of chaos in RL policies drastically restrict the application of deep RL to real-world problems. To address this issue, we propose an improvement on the successful Dreamer V3 architecture, implementing Maximal Lyapunov Exponent regularisation. This new approach reduces the chaotic state dynamics, rendering the learnt policies more resilient to sensor noise or adversarial attacks and thereby improving the suitability of deep reinforcement learning for real-world applications.

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Peer review

Reviewer 2W377/10 · confidence 4/52024-06-23

Summary

This work studies adversarial RL and policy stability from the perspective of the Lyapunov spectrum. A regularization term is introduced to encourage more robust policies.

Strengths

- A novel idea being introduced from classical literature and implemented in the deep RL setting - The regularization introduced is effective at smoothing out trajectories generated by learned policies - This can have important real-world effects in safety-critical / sensitive applications - Opens work for future (theory/computational/applications) studies to build off

Weaknesses

The weaknesses I've found are mostly minor, but I still hope you can address: - The correlation between Figure 2 and 3 is not immediately clear to me; can you please discuss this further? - On that note, the improvement in terms of reward is not clear (I know this is not necessarily "the point" but given the discussion of reward trajectories, it would be nice to see further iumprovemetn here - How error-prone is the calculation of LEs discussed in Sec 2.2? - It's not very clear to me the use of "no actions" in the figures. Can you explain how this "benchmark" should be interpreted? How about in environments where there is no such "do nothing" action? - Can you include non-scalar comparisons of algorithms (cf. https://github.com/google-research/rliable) - The appendix is a bit sparse. Can you provide any further experiment details?

Questions

- Can you please provide legends in the figures? If you have the data, can you include a third panel in Fig. 6 corresponding to the regularized policy? (otherwise this figure is ambiguous without comparison) - Maybe I missed it, but what is $S$ in Eq. 4? - Can you measure the divergence in trajectories such as in Fig. 4 & 8 and compare to the calculated LE? - By how much do you weight the regularization in Eq. 5? - Can you discuss the use of hidden states $h$? It isn't clear to me how this affects the loss beyond what the first term contributes These questions may open new areas of research, and I don't intend for them to be answered within this paper, but I'm curious to hear your thoughts on the following: - How does this work relate to the idea of imposing Lipschitz constraints / regularization on DNNs? I see it as an orthogonal direction, but can they solve the same problem? If so, can they be compared? - Can we use pre-trained policies, then finetune them wrt the LE regularization to make them more robust? - In 2.3, can you discuss the case of $\gamma \to 1$? minor typographical: - Eq. 1, I believe you need a conditional on the state $s$ and an expectation over initial states. - L161: "...low levels of chaos **as** they have..." - Eq. 5 use parentheses over both t-dep terms - Please increase fontsizes in Fig 7

Rating

7

Confidence

4

Soundness

4

Presentation

3

Contribution

3

Limitations

Can you please include limitations in Sec 7?

Reviewer pxFr6/10 · confidence 4/52024-07-02

Summary

Deep RL methods are usually lack of robustness in control tasks whose dynamics are chaotic, thereby having positive maximal Lyapunov exponents (MLEs). This paper proposes an approach that improves the stability of trained deep RL controller through MLE regularization.

Strengths

* The problem studied in this paper is well-motivated and important. * Characterizing the robustness using maximal Lyapunov exponents is promising. * The presentation is straightforward.

Weaknesses

* This paper dedicates many pages to the chaotic phenomena in RL (Sections 3 and 4), which have been addressed in previous works as introduced in Sections 1 and 2. The core contribution, which I believe is the regularization technique, needs more elaboration and analysis. * There is no concrete algorithm provided that the audience can follow to easily implement the approach proposed in this paper. Imagine a person with little or no knowledge of dynamical systems. Can they manage to implement the algorithm after reading the paper? * (minor) Some information is missing. For example, what are $S$, $H$ and $v_\phi(s_t)$ in equation (4)?

Questions

* Does the policy network architecture affect the performance of your approach? * Is it possible to find two trajectories with equal rewards, where one is stable and the other is chaotic? In other words, the reward itself may not be sufficient to reflect the stability of a trajectory. * How does the approach perform in the case that it is unable to estimate the MLE accurately in practice?

Rating

6

Confidence

4

Soundness

3

Presentation

2

Contribution

2

Limitations

It has discussed its empirical limitations in Section 6.

Reviewer TDUf6/10 · confidence 4/52024-07-12

Summary

To address the issue of stability of deep reinforcement learning, the authors first gauge the chaotic behavior of various state-of-the-art deep reinforcement learning policies in continuous control environments, and quantify the stability of those policies with significant impact of their applicability to real world problems. Then, the authors proposed an improvement based on implementing a Maximal Lyapunov Exponent regularization in the RL architecture, and demonstrate the improvement with examples.

Strengths

Lyapunov exponent is an important concept developed in dynamical system. There are many related sophisticated research conducted. It is beneficial to both machine learning and dynamical system to have used it in the analysis of deep RL. Sec 3 & 4 present a reasonable analysis on the stability study of deep RL.

Weaknesses

The proposed method on maximal Lyapunov exponent regularization needs to be further explored and studied, both in terms of the theory involved and amount of experiments conducted. The idea is novel, but not fully explored and explained. The current set of numerical experiments are limited and not truly convincing.

Questions

Could more explanation of all the terms and parameters in Equation (4) be provided? Also, how is the proposed regularizing term in (5) is related to (4)?

Rating

6

Confidence

4

Soundness

3

Presentation

3

Contribution

3

Limitations

The authors adequately addressed the limitations.

Reviewer pxFr2024-08-09

Reply to rebuttal

I would like to thank the authors for their rebuttal and it has resolved my concerns. I still recommend the authors to add a formal algorithm table for their approach so that the audience can easily implement it. I am raising my score to 6.

Authorsrebuttal2024-08-09

Thank you very much for your positive feedback and for raising your score. We are delighted to hear that we have effectively addressed your concerns and appreciate your suggestion to add a formal algorithm. We agree that it will enhance the clarity of our work and will, therefore, incorporate this into our camera-ready version.

Reviewer TDUf2024-08-10

Response to the rebuttal

Thank you the the rebuttal. They address some of my concerns. I would also like to raise my score to (6).

Reviewer 2W372024-08-12

I appreciate the authors' responses, thank you for addressing the questions raised. I believe the revised version will make for a great paper. I will maintain the current score as it appears the reviewers have converged.

Program Chairsdecision2024-09-25

Decision

Accept (poster)

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