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We have investigated the stability and convergence of policy-based robust adversarial Reinforcement learning, on the fundamental linear quadratic setup in continuous control

On the Stability and Convergence of Robust Adversarial Reinforcement Learning: A Case Study on Linear Quadratic Systems

NIPS 2020, (2020)

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Abstract

Reinforcement learning (RL) algorithms can fail to generalize due to the gap between the simulation and the real world. One standard remedy is to use robust adversarial RL (RARL) that accounts for this gap during the policy training, by modeling the gap as an adversary against the training agent. In this work, we reexamine the effectivene...More

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Introduction
  • Reinforcement learning (RL) can fail to generalize due to the gap between the simulation and the real world.
  • The authors develop an update-initialization pair that provably guarantees both robust stability and convergence.
  • The authors identify several stability issues of the popular RARL scheme in the LQ setup, showing that guaranteeing robust stability during learning requires a non-trivial intertwinement of update rules and controller initializations.
Highlights
  • Reinforcement learning (RL) can fail to generalize due to the gap between the simulation and the real world
  • To achieve the goal of learning a policy that is robust against a family of possible model uncertainties, robust adversarial RL (RARL) jointly trains a protagonist and an adversary, where the protagonist learns to robustly perform the control tasks under the possible disturbances generated by its adversary
  • Motivated by the deep connection between RARL and robust control theory, this paper reexamines the effectiveness of RARL under a fundamental robust control setting: the linear quadratic (LQ) case
  • We identify several stability issues of the popular RARL scheme in the LQ setup, showing that guaranteeing robust stability during learning requires a non-trivial intertwinement of update rules and controller initializations
  • We have investigated the stability and convergence of policy-based robust adversarial RL, on the fundamental linear quadratic setup in continuous control
  • We believe that researchers of reinforcement learning (RL), especially those who are interested in the theoretical foundations of robust RL, would benefit from this work, through the new insights and angles we have provided regarding robust adversarial RL (RARL) in linear quadratic (LQ) setups, from a rigorous robust control perspective
Results
  • The stability issues above demonstrate the significance of both the initialization and the update rule (properly chosen (NK , NL)), in developing policy-based LQ RARL algorithms.
  • The authors show that this algorithm can guarantee both the stability of the policy pair (K, L) and the robust stability of K, along the optimization process, and provably converges to (K∗, L∗) if initialized at a robustly stabilizing policy K0.
  • The authors first show that the outer-loop iterate Kn of the algorithm is guaranteed to satisfy the robust stability condition, if K0 is robustly stabilizing.
  • The inner-loop algorithm is initialized at some stabilizing L (L = 0 suffices) and applies the NPG update as follows
  • With stepsize ηL ≤ 1/(2 Rw − C PKn,LC ), the inner-loop NPG update (4.2) is guaranteed to be stabilizing and converges to L(Kn) at a linear rate.
  • The numerical results indicate that the algorithms even with NL = 1 work well if the initial policy K0 satisfies the robust stability condition.
  • The authors have identified examples where these algorithms fail to converge when the initial policy does not satisfy the robust stability condition.
  • These interesting findings reaffirm the complicated intertwinement between update rule and initialization, in order to guarantee the stability and convergence of LQ RARL in general.
  • The robust stability condition on K is provably significant for the double-loop algorithm, and empirically useful for other variants such as alternating or multi-step update rules.
  • The authors have investigated the stability and convergence of policy-based robust adversarial RL, on the fundamental linear quadratic setup in continuous control.
Conclusion
  • Several stability issues of LQ RARL have been identified, illustrating the intertwinement of both the initialization and update rule in developing provably convergent RARL algorithms.
  • Through the lens of robust control, the authors have proposed a provably stable and convergent initialization-update pair, and developed H∞-based approaches to robustify the initializations.
  • Interesting future directions include developing robustly stable RARL methods against some structured uncertainty, extending the robust control view to RARL in nonlinear systems, investigating the global convergence of descent-ascent methods, and studying the theoretical guarantees of the robustification approach.
Summary
  • Reinforcement learning (RL) can fail to generalize due to the gap between the simulation and the real world.
  • The authors develop an update-initialization pair that provably guarantees both robust stability and convergence.
  • The authors identify several stability issues of the popular RARL scheme in the LQ setup, showing that guaranteeing robust stability during learning requires a non-trivial intertwinement of update rules and controller initializations.
  • The stability issues above demonstrate the significance of both the initialization and the update rule (properly chosen (NK , NL)), in developing policy-based LQ RARL algorithms.
  • The authors show that this algorithm can guarantee both the stability of the policy pair (K, L) and the robust stability of K, along the optimization process, and provably converges to (K∗, L∗) if initialized at a robustly stabilizing policy K0.
  • The authors first show that the outer-loop iterate Kn of the algorithm is guaranteed to satisfy the robust stability condition, if K0 is robustly stabilizing.
  • The inner-loop algorithm is initialized at some stabilizing L (L = 0 suffices) and applies the NPG update as follows
  • With stepsize ηL ≤ 1/(2 Rw − C PKn,LC ), the inner-loop NPG update (4.2) is guaranteed to be stabilizing and converges to L(Kn) at a linear rate.
  • The numerical results indicate that the algorithms even with NL = 1 work well if the initial policy K0 satisfies the robust stability condition.
  • The authors have identified examples where these algorithms fail to converge when the initial policy does not satisfy the robust stability condition.
  • These interesting findings reaffirm the complicated intertwinement between update rule and initialization, in order to guarantee the stability and convergence of LQ RARL in general.
  • The robust stability condition on K is provably significant for the double-loop algorithm, and empirically useful for other variants such as alternating or multi-step update rules.
  • The authors have investigated the stability and convergence of policy-based robust adversarial RL, on the fundamental linear quadratic setup in continuous control.
  • Several stability issues of LQ RARL have been identified, illustrating the intertwinement of both the initialization and update rule in developing provably convergent RARL algorithms.
  • Through the lens of robust control, the authors have proposed a provably stable and convergent initialization-update pair, and developed H∞-based approaches to robustify the initializations.
  • Interesting future directions include developing robustly stable RARL methods against some structured uncertainty, extending the robust control view to RARL in nonlinear systems, investigating the global convergence of descent-ascent methods, and studying the theoretical guarantees of the robustification approach.
Related work
  • Exploiting an adversary to tackle model-uncertainty and improve sim-to-real performance in RL dates back to [1], which, interestingly, stemmed from the H∞-robust control theory. Actor-critic robust RL algorithms were proposed therein, though without theoretical guarantees for either convergence or stability. This minimax idea was then carried forward in the popular RARL scheme [2], with great empirical successes, which has then been followed up and improved in [22, 23]. The policy-based RARL algorithms therein serve as the starting point for our work. Following the same worst-case modeling idea for uncertainty, robust RL has also been investigated in the realm of robust Markov decision processes (MDPs) [24, 25, 26]. Our LQ RARL setup can be viewed as a specification of robust MDP in the continuous control context. Other recent advances on robust RL for continuous control include [4, 5, 27, 6]. An increasing attention has also been paid to ensuring robustness and stability in general data-driven control [28, 29, 30].
Funding
  • Acknowledgments and Disclosure of Funding The research of K.Z. and T.B. was supported in part by the US Army Research Laboratory (ARL) Cooperative Agreement W911NF-17-2-0196, and in part by the Office of Naval Research (ONR) MURI Grant N00014-16-1-2710
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Author
Kaiqing Zhang
Kaiqing Zhang
Bin Hu
Bin Hu
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