The Role of Target Update Frequencies in Q-Learning

TL;DR

This paper provides a theoretical analysis of target update frequencies in Q-learning, revealing how adaptive schedules outperform constant ones by reducing sample complexity and improving convergence.

Contribution

It introduces a principled, finite-time convergence analysis of target update frequencies, showing how to optimally set and adapt this hyperparameter during learning.

Findings

Optimal target update frequency increases geometrically during training.

Constant update schedules incur unnecessary logarithmic sample complexity overhead.

Adaptive schedules significantly improve sample efficiency and convergence.

Abstract

The target network update frequency (TUF) is a central stabilization mechanism in (deep) Q-learning. However, their selection remains poorly understood and is often treated merely as another tunable hyperparameter rather than as a principled design decision. This work provides a theoretical analysis of target fixing in tabular Q-learning through the lens of approximate dynamic programming. We formulate periodic target updates as a nested optimization scheme in which each outer iteration applies an inexact Bellman optimality operator, approximated by a generic inner loop optimizer. Rigorous theory yields a finite-time convergence analysis for the asynchronous sampling setting, specializing to stochastic gradient descent in the inner loop. Our results deliver an explicit characterization of the bias-variance trade-off induced by the target update period, showing how to optimally set this critical hyperparameter. We prove that constant target update schedules are suboptimal, incurring a logarithmic overhead in sample complexity that is entirely avoidable with adaptive schedules. Our analysis shows that the optimal target update frequency increases geometrically over the course of the learning process.