Managing Temporal Resolution in Continuous Value Estimation: A Fundamental Trade-off

Zichen Zhang, Johannes Kirschner, Junxi Zhang, Francesco Zanini, Alex Ayoub, Masood Dehghan, Dale Schuurmans

Keywords: Temporal Discretization, Continuous Time, Langevin System, LQR, Policy Evaluation

TL;DR: By analyzing Monte-Carlo value estimation for LQR systems we uncover a fundamental trade-off between approximation and statistical error in value estimation.

Abstract: A default assumption in reinforcement learning and optimal control is that experience arrives at discrete time points on a fixed clock cycle. Many applications, however, involve continuous systems where the time discretization is not fixed but instead can be managed by a learning algorithm. By analyzing Monte-Carlo value estimation for LQR systems in both finite-horizon and infinite-horizon settings, we uncover a fundamental trade-off between approximation and statistical error in value estimation. Importantly, these two errors behave differently with respect to time discretization, which implies that there is an optimal choice for the temporal resolution that depends on the data budget. These findings show how adapting the temporal resolution can provably improve value estimation quality in LQR systems from finite data. Empirically, we demonstrate the trade-off in numerical simulations of LQR instances and several non-linear environments.

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