## 1 Introduction  
In recent years, with the advancement of 5G technology, the internet of things (IoT) has seen rapid development. In the world of \"internet of everything\", many sensors are deployed in the environment to collect information in real time and generate a large amount of data. The traditional machine learning (ML) framework stores data on a central node, and its all functions are implemented on this node. However, such a centralized learning framework has high latency and poses a great challenge to privacy protection. Fortunately, due to the significant increase in data processing capabilities of mobile devices, the concept of edge learning has been introduced to solve these problems, which processes data at the edge rather than at the cloud center. Federated learning (FL) is one of the most promising edge learning frameworks, where user devices (UDs) only send local models calculated by local resource to the base station (BS) without sharing local data [1, 2]. In FL, the time and energy consumed in model training are the important aspects of its performance, which depends on firmly the allocated computation resources for local model training and communication resources for model transmission between BS and UDs [3]. In addition, due to the limited computing power and energy reserve of UDs, the problem of joint computation and communication resource allocation is crucial for the deployment of FL in wireless networks. The authors proposed a resource management algorithm for the multi-access edge computing enabled FL system in [4, 5]. In [6, 7], the resource scheduling problem was studied under imperfect channel state information (CSI). Due to the development of the metamaterial technology, an emerging paradigm called intelligent reflector surfaces (IRS) was introduced into the field of wireless communications. IRS can significantly improve the wireless communication efficiency by adjusting the phase shift of the incident signal using passive reflective elements, which are low-cost and low energy [8; 9]. Thus, this attracts people to study the beneficial effects of IRS in FL [10; 11; 12]. However, these studies are based on the assumption that perfect CSI can be obtained. Unfortunately, due to the lack of the ability to receive, send and process signals for IRS, it is a challenge to estimate IRS-related channels, including the UDs-IRS channel and the IRS-BS channel, which thus have attracted widespread research attention. At present, channel estimation of IRS-related channels can be divided into two methods. One is to directly estimate IRS-related channels [13; 14]. In [13], some active channel components can be equipped on the IRS to enable it to receive, send and process signals. The authors proposed a channel estimation protocol to estimate the channel by adjusting the reflective elements ON/OFF in [14]. However, these methods will bring additional cost, such as hardware cost and power consumption cost, which will cause an unaffordable burden on the IRS. The other is to estimate the cascade UD-IRS-BS channel. The advantage of this method is that the cascaded channel can be estimated without additional hardware and power cost. Therefore, there are a lot of works dedicated to IRS cascaded channel estimation [15; 16; 17; 18]. In [15; 16], the transmission protocol was proposed to estimate the cascaded channel. The author studied the robust transmission design base on imperfect CSI in [17; 18]. Against the above background, this paper studies the computation and communication resource allocation problem in IRS-assisted FL systems based on imperfect CSI. Specifically, we aim to design the resource allocation algorithm to minimize the delay of FL. The main contributions of this paper can be summarized as follows:  
* In this paper, we study the FL system assisted by IRS under the imperfect CSI, and outage probability is introduced to characterize the impact of imperfect CSI.