  Federated learning (FL) enables multiple clients to collaboratively learn a
shared model without sharing their individual data. Concerns about utility,
privacy, and training efficiency in FL have garnered significant research
attention. Differential privacy has emerged as a prevalent technique in FL,
safeguarding the privacy of individual user data while impacting utility and
training efficiency. Within Differential Privacy Federated Learning (DPFL),
previous studies have primarily focused on the utility-privacy trade-off,
neglecting training efficiency, which is crucial for timely completion.
Moreover, differential privacy achieves privacy by introducing controlled
randomness (noise) on selected clients in each communication round. Previous
work has mainly examined the impact of noise level ($\sigma$) and communication
rounds ($T$) on the privacy-utility dynamic, overlooking other influential
factors like the sample ratio ($q$, the proportion of selected clients). This
paper systematically formulates an efficiency-constrained utility-privacy
bi-objective optimization problem in DPFL, focusing on $\sigma$, $T$, and $q$.
We provide a comprehensive theoretical analysis, yielding analytical solutions
for the Pareto front. Extensive empirical experiments verify the validity and
efficacy of our analysis, offering valuable guidance for low-cost parameter
design in DPFL.
