Deep Q-learning: A robust control approach

This work aims at constructing a bridge between robust control theory and reinforcement learning. Although, reinforcement learning has shown admirable results in complex control tasks, the agent's learning behavior is opaque. Meanwhile, system theory has several tools for analyzing and controlling dynamical systems. This article places deep Q-learning is into a control-oriented perspective to study its learning dynamics with well-established techniques from robust control. An uncertain linear time-invariant model is formulated by means of the neural tangent kernel to describe learning. This novel approach allows giving conditions for stability (convergence) of the learning and enables the analysis of the agent's behavior in frequency-domain. The control-oriented approach makes it possible to formulate robust controllers that inject dynamical rewards as control input in the loss function to achieve better convergence properties. Three output-feedback controllers are synthesized: gain scheduling Script capital H2$$ {\mathscr{H}}_2 $$, dynamical Script capital H infinity$$ {\mathscr{H}}_{\infty } $$, and fixed-structure Script capital H infinity$$ {\mathscr{H}}_{\infty } $$ controllers. Compared to traditional deep Q-learning techniques, which involve several heuristics, setting up the learning agent with a control-oriented tuning methodology is more transparent and has well-established literature. The proposed approach does not use a target network and randomized replay memory. The role of the target network is overtaken by the control input, which also exploits the temporal dependency of samples (opposed to a randomized memory buffer). Numerical simulations in different OpenAI Gym environments suggest that the Script capital H infinity$$ {\mathscr{H}}_{\infty } $$ controlled learning can converge faster and receive higher scores (depending on the environment) compared to the benchmark double deep Q-learning.