FedRec: Federated Learning of Universal Receivers over Fading Channels

Fading in wireless communications encapsulates the fact that the relationship between the transmitted signal and the received one is determined by the propagation of electromagnetic waves, which is typically dynamic and subject to different forms of randomness induced by the environment. Various distributions have been proposed to represent the statistical behavior of fading channels, including the Rayleigh, Rice, and Nakagami-$m$ models [1], where each approximates the propagation profile in different settings [2].

The inherent randomness of fading channels makes symbol detection a challenging task. The common strategy is to periodically transmit a-priori known pilot signals for the receiver to estimate the channel, which in turn is utilized for detection [3]. The main drawback of this approach is that pilots must be transmitted anew each time the channel changes, i.e., on each coherence duration, inducing notable overhead in rapidly-changing channels. Alternatively in fast fading conditions, one can utilize a single detection rule for all channel conditions, which accounts for its statistical model [4]. However, such model-based detection relies on the knowledge of the fading distribution, and tends to be inaccurate when the assumed distribution does not faithfully capture the real statistical propagation profile, or in the presence of a model mismatch.

An alternative strategy, which does not require the knowledge of the underlying statistical model, is to learn the detection mapping from data. In particular, neural networks have demonstrated unprecedented success over recent years in learning complex mappings in a data-driven fashion [5]. Consequently, a multitude of NN-based receiver architectures that can operate without the prior knowledge of the underlying statistical model have been proposed, see, e.g., [6, 7, 8, 9, 10, 11]. NN-based receivers require labeled data to learn their mapping. If one has prior knowledge of the channel conditions and can generate such data artificially, the NN can be trained offline. However, this may not be the case in many practical scenarios, where labeled data must be obtained from pilot transmissions.

When the channel conditions change rapidly, NN-based receivers must be frequently retrained with new pilot signals, inducing significant computation and communication overhead. NN-based receivers can track dynamic channel conditions by periodic online training combined with methods to reduce the training complexity as in [12, 13, 14, 15, 16]. Alternatively, one can train a single NN that would work for a broad range of channel conditions, by learning a universal rule based on the fading distribution rather than its specific realization [17, 18]. Nonetheless, for a NN to learn the subtleties of a fading distribution, which may be complex and intractable, the training data must contain a sufficiently large number of channel realizations. This may be difficult to achieve even with long pilot sequences, motivating the design of universal NN-based detectors for fading channels with limited pilots.

In this work, we propose FedRec, which is a data-driven universal symbol detection scheme for multi-user downlink fading channels, designed to learn its mappings from a limited amount of pilots. FedRec is comprised of two algorithmic components: The first is the NN-based symbol detection architecture, which uses sufficient statistics from the maximum a-posteriori probability (MAP) symbol detection rule as input features. This allows FedRec to utilize a relatively compact NN, which can be trained with a smaller number of samples. The second component is the training mechanism, which exploits the fact that, in a wireless network of many users, while each user observes only a limited number of channel realizations, the realizations observed by the overall network are expected to be sufficiently diverse. FedRec builds upon this insight to have the users collaborate for training via federated learning (FL) [19, 20, 21], allowing to train a single NN over a diverse dataset without additional pilots, at the cost of several iterations of parameter exchanges with the base station (BS). Our numerical results show that FedRec yields an accurate symbol detector, with a performance approaching that of a MAP detector, and outperforms the model-based approach in the presence of inaccurate knowledge of the fading distribution. Moreover, FedRec induces substantially less communication overhead compared to learning a NN-based symbol detector in a centralized fashion.

The rest of this paper is organized as follows: Section 2 presents the system model and the problem formulation. Section 3 details the proposed FedRec receiver. Numerical examples are presented in Section 4. Finally, Section 5 concludes the paper. FedRec codes are available at: https://github.com/MahdiBoloursazMashhadi/FedRec

We consider a downlink communication scenario, where a BS serves $U$ users indexed by $u\in\mathcal{U}\triangleq\{1,\ldots,U\}$. Although we focus on single-antenna terminals in this paper, our approach can be extended to multiple-antenna terminals. Letting $x(t)\in\mathcal{C}$ be the baseband continuous-time (CT) channel input transmitted by the BS at time instance $t$, the corresponding channel output at the $u$th user is

where $\{w_{u}(t)\}$ are independent identically distributed (i.i.d.) Gaussian noise signals with unit power spectral density (PSD), while $\{h_{u}(t)\}$ are i.i.d. flat fading coefficients, following a common distribution $p_{h}(\cdot)$. Various different models exist for $p_{h}(\cdot)$, which approximate the statistical behavior under different channel conditions [22, Ch. 2], but we do not assume prior knowledge of $p_{h}(\cdot)$.

Downlink communication is typically comprised of pilot and data transmission. During downlink data transmission to user $u$ commencing at time instance $t_{\rm d}$, the BS encodes the message $s_{u}$, uniformly distributed over $\mathcal{M}\triangleq\{1,\ldots,M\}$, into a signal of temporal duration of $T_{\rm s}$ seconds, denoted by $x_{s_{u}}(t)$, $t\in t_{\rm d}+[0,T_{\rm s})$. Each user $u\in\mathcal{U}$ uses its channel output $r_{u}(t)$, obtained via (1), to recover $s_{u}$. During pilot transmission commencing at $t=0$, the BS broadcasts a sequence of $N_{T}$ a-priori known pilot symbols, denoted by $\{s^{p}_{i}\}_{i=1}^{N_{T}}$, over a duration of $N_{T}\cdot T_{\rm s}$ seconds.

We focus on regimes in which the number of pilot symbols $N_{T}$ is relatively small; and hence, it is unlikely to span a sufficient amount of different realizations of the fading coefficient at a single user. While the number of pilots is limited, we allow the users to collaborate and share their detection mappings, in order to jointly learn a unified symbol detector, i.e., one that is universally applicable not only to the users participating in the training, but also to any arbitrary user, by learning a broad fading distribution from the pilots received at all the users and the corresponding messages, i.e., $\{\mathcal{D}_{u}\}_{u=1}^{U}$, where $\mathcal{D}_{u}=\{s^{p}_{i},\mathcal{R}_{i}^{u}\}_{i=1}^{N_{T}}$, with $\mathcal{R}_{i}^{(u)}\triangleq\{r_{u}(t)|t\in[(i-1)T_{\rm s},iT_{\rm s})\}$.

The resulting symbol detector at an arbitrary user $u^{\prime}$ recovers the message $s_{u^{\prime}}$ from the received ${r}_{u^{\prime}}(t)$, $t\in t_{\rm d}+[0,T_{\rm s})$, assuming only the knowledge of the channel input-output relationship (1), but not the fading distribution $p_{h}(\cdot)$. Our proposed NN-based symbol detection scheme, which combines model knowledge of (1) with data-driven tools to utilize $\{\mathcal{D}_{u}\}_{u=1}^{U}$ in a collaborative fashion to train such a universal receiver, is detailed in the following section.

The need for a universal symbol detector applicable to a diverse range of channel fading statistics without the exact knowledge of the underlying distribution motivates using NNs, which were empirically shown to operate reliably in complex unknown statistical environments [5]. However, since NNs require large volumes of diverse training data, having each user train a local NN based on its limited pilot observations is expected to yield a less reliable model with limited generalization performance. To overcome this, we design FedRec based on the following guidelines: (i) The fact that the channel is modeled via (1) is exploited as partial domain knowledge for feature extraction, inspired by the MAP rule for such channels. This approach facilitates utilizing compact NNs (avoiding feature extraction layers), which are trainable using relatively small datasets. (ii) While each local training dataset $\mathcal{D}_{u}$ encapsulates a relatively small number of fading channel realizations, the diversity among these sets at different users is exploited to obtain a unified model for all the users by training in a federated manner.

In this section we compare FedRec with model-based detection and learning-based schemes for Rayleigh fading channels. We consider two cases with i.i.d. and non-i.i.d. fading channels across the users. In the i.i.d. case, the coefficients $h_{u}(t)$ are generated from a Rayleigh distribution with unit scale parameter, and a new realization is generated on each $T_{\rm s}$ time instances. We also consider a non-i.i.d. case, where the scale parameter for different users is not identical, e.g., due to different statistics of the local environment for each user, such that for each user $u\in\mathcal{U}$, the scale parameter $\sigma_{u}$ is randomized from a uniform distribution over the range $[0.5,1.5]$.

We use 16QAM modulations, i.e., $M=16$, with average transmit power per bit $\rho$, representing the signal-to-noise ratio (SNR). For the learning-based methods, the training and test datasets are comprised of $|\mathcal{D}_{train}|=2\cdot 10^{4}$ and $|\mathcal{D}_{test}|=10^{7}$ symbols, respectively. Training data is collected by the users during phase (i) of network operation, with each user recording $N_{T}=|\mathcal{D}_{u}|=2\cdot 10^{4}/U$ 16QAM pilot symbols, which are faded according to its local fading statistics. The learning-based methods train a NN with an input layer of size 2 corresponding to the in-phase and quadrature signal components, followed by a hidden layer with 16 neurons and softmax, thus $|{\boldsymbol{\theta}}|=48$. This NN architecture is trained using the Adam optimizer [25] over $25$ epochs with batch-size of $20$, and tested for each SNR value. We compare the following symbol detection schemes:

-Non-collaborative learning (NL): Each user trains the NN solely on its local dataset. The bit error rate (BER) is averaged over all user trained NNs using the test dataset.

-Centralized learning (CL): The users transmit their datasets to the BS over a noise-free link, and the NN is trained centrally.

-FedRec: The users follow Algorithm 1 with $5$ local epochs and over $5$ rounds of aggregation to train the NN collaboratively.

-Model-based detection (MD): The model-based detector uses the decision rule (4), with an estimate of the Rayleigh scale parameter, denoted by $\hat{\sigma}$, obtained from the training data. Here, each user obtains a maximum likelihood estimate of its scale parameter denoted by $\hat{\sigma}_{u}$ utilizing its local dataset. The overall estimate is then obtained by $\hat{\sigma}^{2}=1/U\sum_{u=1}^{U}\hat{\sigma}^{2}_{u}$ inserted in (4).

-MAP bound: The MAP receiver follows the closed form (4) for the i.i.d. case. Unlike the model-based approach which estimates the Rayleigh scale parameter from the user data, we here assume exact knowledge of the scale parameter $\sigma=1$ and insert it into (4). For the non-i.i.d. case, however, derivation of the MAP receiver is cumbersome and we use a numerical approach to calculate the integrals in (3) and to evaluate the BER.

In Table 1, we evaluate the BER performance of these symbol detectors for various SNRs with $U=5$ users. Among the data-driven detectors, FedRec consistently outperforms the non-collaborative scheme, while achieving similar or improved BER compared to centralized learning for both the i.i.d. and non-i.i.d cases. It is observed that the BER is significantly improved over non-collaborative learning, most notably in high SNRs when users collaborate either through FL or by exchanging their local datasets with the BS for centralized training. For the i.i.d. case, this gain is due to the increased amount of data made available for training through collaboration. For the non-i.i.d. case, the improvement is more significant as it is not only due to increased number of data samples, but also due to the diversity captured by these samples.

For the i.i.d. case, FedRec approaches the performance of the model-based and MAP detectors. For the more realistic non-i.i.d. case, FedRec generally outperforms the model-based approach computed with the estimated scale parameter. This indicates that while the model-based approach is sensitive to uncertainty in the scale parameter, FedRec learns from data how to cope with such heterogeneous fading and performs closer to the MAP lower bound.

In Fig. 3, we compare the BER versus SNR of FedRec for different number of users $U=1,2,5$. We have also added the MAP lower bound for comparison. For both i.i.d and non-i.i.d cases, FedRec BER rapidly decreases as the number of users participating in federated training grows, approaching the MAP lower bound with merely $U=5$ users. Hence, increasing the number of users participating in federated training of FedRec not only decreases $N_{T}$, hence reducing the duration of the data collection phase (i), but also improves the BER performance.

Finally, we evaluate the overhead induced on both uplink (UL) and downlink (DL) communications in the training procedure of the collaborative data-driven schemes of FedRec and centralized learning. The number of float32 words conveyed over the UL and DL channels for FedRec  and centralized learning are summarized in Table 2. For centralized training, the overhead does not depend on the number of users, and is comprised of $2\cdot|\mathcal{D}_{train}|$ and $|{\boldsymbol{\theta}}|$ words on the UL and DL, respectively. For federated training, the overhead is comprised of $5$ parameter exchange rounds of $U|{\boldsymbol{\theta}}|$ words and $|{\boldsymbol{\theta}}|$ words on UL and DL, respectively, both are much smaller compared to having the users transmit their received pilots to the BS. Hence, FedRec notably reduces the communication load.

We proposed FedRec, a data-driven symbol detector for downlink fading channels. FedRec is comprised of a NN designed based on the MAP rule for fading channels, combined with a collaborative training mechanism, which exploits the channel diversity across multiple users through federated learning. Our numerical results demonstrate that FedRec approaches the MAP performance, while achieving improved robustness to uncertainty, and significantly reduces the communication overhead compared to centralized training.