Abstract
The tasks with continuous state and action spaces are difficult to be solved with high sample efficiency. Model learning and planning, as a well-known method to improve the sample efficiency, is achieved by learning a system dynamics model first and then using it for planning. However, the convergence of the algorithm will be slowed if the system dynamics model is not captured accurately, with the consequence of low sample efficiency. Therefore, to solve the problems with continuous state and action spaces, a model-learning-based actor-critic algorithm with the Gaussian process approximator is proposed, named MLAC-GPA, where the Gaussian process is selected as the modeling method due to its valuable characteristics of capturing the noise and uncertainty of the underlying system. The model in MLAC-GPA is firstly represented by linear function approximation and then modeled by the Gaussian process. Afterward, the expectation value vector and the covariance matrix of the model parameter are estimated by Bayesian reasoning. The model is used for planning after being learned, to accelerate the convergence of the value function and the policy. Experimentally, the proposed method MLAC-GPA is implemented and compared with five representative methods in three classic benchmarks, Pole Balancing, Inverted Pendulum, and Mountain Car. The result shows MLAC-GPA overcomes the others both in learning rate and sample efficiency.
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Acknowledgments
This paper is supported by National Natural Science Foundation of China (61702055, U1764261, 51705021, 61972059, 61773272), Natural Science Foundation of Jiangsu (BK2012616), Science technology program of Jiangsu (BK2015260), High School Natural Foundation of Jiangsu(13KJB520020), Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (93K172014K04, 93K172017K18), Suzhou Industrial application of basic research program part (SYG201422, SYG201308).
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Appendix: Recursive inference of the model parameter
Appendix: Recursive inference of the model parameter
Recall the expression of the parameter posterior: \({\hat {\beta _{t}}} = {\Phi }_{t} Q_{t}{\boldsymbol {x}_{t + 1}}\), \({\hat {\mathbf {P}}_{t}} = {\textbf {I}} - {\Phi }_{t} \boldsymbol {Q}_{t} {\Phi }_{t}^{T}, \boldsymbol {Q}_{t} = {\left ({{\mathbf {\Phi }}_t^T{{\mathbf {\Phi }}_t} + {\Sigma _t}} \right )^{- 1}}\)
The matrices Φt, Σt and \(Q_t^{- 1}\) can be written recursively as follows:
By using the matrix formula, the inversion of \(\boldsymbol Q_t^{- 1}\) can be represented as
where
The expression of \({\hat \upbeta _t}\) can be represented as
where \({d_t} = {x_{t + 1}} - \boldsymbol g_t^{T}{\boldsymbol x_t}\) and \({\boldsymbol p_t} = {\boldsymbol \phi _t} - \boldsymbol {\Phi }_{t - 1}^{}{\boldsymbol g_t}\)
Next, the simple computaion for dt, pt and st will be derived. Let us begin with dt:
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Zhong, S., Tan, J., Dong, H. et al. Modeling-Learning-Based Actor-Critic Algorithm with Gaussian Process Approximator. J Grid Computing 18, 181–195 (2020). https://doi.org/10.1007/s10723-020-09512-4
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DOI: https://doi.org/10.1007/s10723-020-09512-4