Abstract: From a statistical viewpoint, it is essential to perform statistical inference in federated learning to understand the underlying data distribution. Due to the heterogeneity in the number of local iterations and in local datasets, traditional statistical inference methods are not competent in federated learning. This paper studies how to construct confidence intervals for federated heterogeneous optimization problems. We introduce the rescaled federated averaging estimate and prove the consistency of the estimate. Focusing on confidence interval estimation, we establish the asymptotic normality of the parameter estimate produced by our algorithm and show that the asymptotic covariance is inversely proportional to the client participation rate. We propose an online confidence interval estimation method called separated plug-in via rescaled federated averaging. This method can construct valid confidence intervals online when the number of local iterations is different across clients. Since there are variations in clients and local datasets, the heterogeneity in the number of local iterations is common. Consequently, confidence interval estimation for federated heterogeneous optimization problems is of great significance.