# On the Rates of Convergence in Learning of Optimal Temporally Fair Schedulers

## TL;DR

This paper analyzes the convergence rates of threshold-based scheduling strategies under temporal fairness constraints, showing how empirical threshold estimates approach optimality and their impact on system utility.

## Contribution

It provides the first analysis of convergence rates for TBS thresholds in learning-based scheduling under fairness constraints, including their effect on utility.

## Key findings

- Threshold estimates converge at a rate of at least ω(1/√t).
- Schedulers can temporarily exceed optimal utility by violating fairness.
- Simulation results confirm theoretical convergence and utility behavior.

## Abstract

Multi-user schedulers are designed to achieve optimal average system utility (e.g. throughput) subject to a set of fairness criteria. In this work, scheduling under temporal fairness constraints is considered. Prior works have shown that a class of scheduling strategies called threshold based strategies (TBSs) achieve optimal system utility under temporal fairness constraints. The optimal TBS thresholds are determined as a function of the channel statistics. In order to provide performance guarantees for TBSs in practical scenarios --- where the scheduler learns the optimal thresholds based on the empirical observations of the channel realizations --- it is necessary to evaluate the rates of convergence of TBS thresholds to the optimal value. In this work, these rates of convergence and the effect on the resulting system utility are investigated. It is shown that the best estimate of the threshold vector is at least $\omega(\frac{1}{\sqrt{t}})$ away from the optimal value, where $t$ is the number of observations of the independent and identically distributed channel realizations. Furthermore, it is shown that under long-term fairness constraints, the scheduler may achieve an average utility that is higher than the optimal long-term utility by violating the fairness criteria for a long initial period. Consequently, the resulting system utility may converge to its optimal long-term value from above. The results are verified by providing simulations of practical scheduling scenarios.