Learn to Rapidly and Robustly Optimize Hybrid Precoding

The feasible mappings that can be realized by the analog precoder $\textbf{W}_{a}$, encapsulated in the set $\mathcal{A}$, depend on the hardware architecture. We consider two different architectures for the analog precoder: unconstrained precodersZ and fully-connected phase shifter networks.

We aim at designing the hybrid precoding operation given a channel realization to maximize the achievable sum-rate of multiuser downlink MIMO communications. Defining the sum-rate as the precoding objective is a common approach [18, 41, 13], being a communication measure of the overall combined effect of the channel and the hybrid precoder. We particularly focus on two scenarios – sum-rate maximization (given accurate CSI) and robust sum-rate maximization (given noisy CSI).

Problem (7) represents constrained maximization with $B+1$ optimization matrix variables $\textbf{W}_{a},\{\textbf{W}_{d,b}\}_{b\in\mathcal{B}}$. Such problems can be tackled using the PGA algorithm combined with alternating optimization. In this iterative method, each iteration first optimizes $\textbf{W}_{a}$ while keeping $\{\textbf{W}_{d,b}\}_{b\in\mathcal{B}}$ fixed, then repeats this process for every $\textbf{W}_{d,b}$. The optimized matrices are projected to guarantee the constraints are not violated.

To formulate this operation mathematically, we define $\tilde{\textbf{H}}_{b}\triangleq\sqrt{\frac{1}{N\sigma^{2}}}\textbf{H}_{b}$. Each iteration of index $k+1$ is comprised of two alternating stages: analog PGA and digital PGA.

Analog PGA: The update of $\textbf{W}_{a}$ is carried out according to

where $\mu_{a}^{(k)}$ is the step size of the gradient step, and $\Pi_{\mathcal{A}}\{\cdot\}$ is the projection operator onto the set $\mathcal{A}$. The gradient of R with respect to $\textbf{W}_{a}$ is given by (see Appendix A)

where $\textbf{G}_{b}(\textbf{W}_{a},\textbf{W}_{d,b},\textbf{H}_{b})\triangleq(\textbf{I}_{N}+\tilde{\textbf{H}}_{b}\textbf{W}_{a}\textbf{W}_{d,b}\textbf{W}_{d,b}^{H}\textbf{W}_{a}^{H}\tilde{\textbf{H}}_{b}^{H})$.

The projection operator $\Pi_{\mathcal{A}}$ in (9) depends on the feasible analog mappings. For unconstrained architectures, clearly $\Pi_{\mathcal{A}}(\textbf{A})=\textbf{A}$. For fully connected phase shifters based hardware, the projection is given by

Digital PGA: The digital precoder is updated after the analog precoder, where the update is carried out for all frequencies in parallel. Namely, the gradient step for each $b\in\mathcal{B}$ is computed as

where $\mu_{d,b}^{(k)}$ is the step size. The gradient of R with respect to $\textbf{W}_{d,b}$ for each $b\in\mathcal{B}$ is computed as (see Appendix A)

After the gradient step is taken for all frequency bins, the solution is projected to meet the power constraint via

The overall procedure is summarized as Algorithm 1. While the initial settings of $\{\textbf{W}_{d,b}^{(0)}\}_{b\in\mathcal{B}}$ are taken to be random, the initial analog combiner $\textbf{W}_{a}^{(0)}$ is set to be the first $L$ right-singular vectors of $\frac{1}{B}\sum_{b=1}^{B}\tilde{\textbf{H}}_{b}$ (the frequency bands sub-channels average). This setting corresponds to analog beamforming towards the eigenmodes of the (frequency-average) channel, being the part of the capacity achieving precoding method for frequency flat MIMO channels [43, Ch. 10]. This principled initialization is numerically shown to notably improve the performance of hybrid precoders tuned to directly optimize the rate in (7).

The convergence speed of gradient-based optimizers largely depends on the step sizes, i.e., $\big{\{}\big{(}\big{\{}\mu_{d,b}^{(k)}\big{\}}_{b\in\mathcal{B}},\mu_{a}^{(k)}\big{)}\big{\}}$ in Algorithm 1. However, conventional step size optimization methods based on, e.g., line search and backtracking [44, Ch. 9], typically involve additional per-iteration processing which increases the overall complexity and run-time. Hence, a common practice is to use pre-defined hand-tuned constant step sizes, which may result in lengthy convergence.

Algorithm 1 optimizes hybrid precoders for a given channel realization. However, its convergence speed largely depends on its hyperparameters, i.e., the step sizes, which tend to be difficult to set. Here, we propose to leverage automated data-based optimizers used in deep learning to tune iteration-dependent step sizes, i.e., to learn-to-optimize hybrid precoders in a small and predefined number of iterations.

Our design follows the deep unfolding methodology [30, 31], which designs DNNs as iterative optimizers with a fixed number of iterations. In particular, we use as our optimizer the PGA method in Algorithm 1 with exactly $K$ iterations. By doing so, we guarantee an exact pre-known, and typically small, run-time and complexity in real-time. While the accuracy of first-order optimizers such as PGA is typically invariant of the setting of the hyperparameters when allowed to run until convergence (under mild conditions, e.g., that the step sizes are sufficiently small), their performance is largely affected by these values when the number of iterations is fixed. Consequently, our design treats the hyperparameters of an iterative optimizer with $K$ iterations as the parameters of a DNN with $K$ layers, and tunes them via end-to-end training, based on the available data set $\mathcal{D}$, thus converting PGA into a trainable discriminative model [45].

To formulate this, the step sizes vector of the $k$th iteration is defined as $\bm{\mu}_{k}\triangleq(\mu_{a}^{(k)},\ldots,\mu_{d,B}^{(k)})$, and the step sizes matrix defined as $\bm{\mu}\triangleq(\bm{\mu}_{0},\ldots,\bm{\mu}_{K-1})^{T}$ for $K$ iterations of Algorithm 1. The entries of this $K\times(B+1)$ matrix are trainable parameters, that are learned from data. Note that end-to-end training is feasible despite the fact that $\mathcal{D}$ does not hold the ground truth precoders. This follows since the performance of hybrid precoders can be evaluated using the differentiable measure in (6), that is used to define a loss function with which the hyperparameters $\bm{\mu}$ are trained in an unsupervised manner.

The loss function, for a given normalized channel $\tilde{\textbf{H}}=\{\tilde{\textbf{H}}_{b}\}_{b\in\mathcal{B}}$ and step sizes $\bm{\mu}$, is computed as a weighted average of the negative resulting achievable sum-rates of this channel (6) in each $\textbf{PGA}_{K}(\tilde{\textbf{H}},\bm{\mu})$ iteration. This implies that the $k$th iteration sum-rate is computed when the precoders are $\Big{(}\textbf{W}_{a}^{(k)},\{\textbf{W}_{d,b}^{(k)}\}_{b\in\mathcal{B}}\Big{)}\triangleq\textbf{PGA}_{k}(\tilde{\textbf{H}},\bm{\mu})$, i.e., the precoders obtained via Algorithm 1 after $k<K$ iterations and step sizes $\bm{\mu}$. Since each iteration is required to provide a setting of the hybrid beamformer which gradually improves along the iterative procedure, we adopt the following loss, inspired by [46]

The data set $\mathcal{D}$ includes channel realizations. Since $\tilde{\textbf{H}}_{b}\triangleq\sqrt{\frac{1}{N\sigma^{2}}}\textbf{H}_{b}$ with known $N$ and $\sigma^{2}$, we henceforth write the entries of the data set $\mathcal{D}$ as $\{\tilde{\textbf{H}}^{r}\}_{r=1}^{|\mathcal{D}|}$.

The learn-to-optimize method seeks to tune the hyperparameters vector $\bm{\mu}$ to best fit the data set $\mathcal{D}$ in the sense of the loss measure (15). Namely, we aim at setting

We tackle (16) using deep learning optimization techniques based on, e.g., mini-batch stochastic gradient descent, to tune $\bm{\mu}$ based on the data set $\mathcal{D}$. The resulting procedure is summarized as Algorithm 2. We initialize $\bm{\mu}$ before the training process, with fixed step sizes with which PGA converges. After training, which is based on past channel realization and can thus be done offline, the learned $\bm{\mu}$ is used as hyperparameters for rapidly converting a channel realization into a hybrid precoding setting via $K$ of Algorithm 1. The resulting unfolded PGA algorithm is illustrated in Fig. 2.

The proposed learn-to-optimize method leverages data to improve the performance and convergence speed of iterative PGA-based optimization. The resulting precoder design preserves the interpretability and simplicity of classic PGA optimization, while inferring at a fixed and low delay as done by DNNs applied for such tasks. We thus benefit from the best of both worlds of model-based optimization and data-driven deep learning.

The fact that the number of iterations is fixed and limited is reflected upon the computational complexity associated with setting a hybrid precoder for a given channel realization. To quantify the complexity, we examine one iteration of Algorithm 1; its complexity is dominated by the gradient computation (whose complexity order is $\mathcal{O}\bigl{(}BM^{2}(N+L)\bigl{)}$) in Step 1 (which is of the same complexity order as the $B$ gradient computations in Step 12), and by the projection required in Step 1 (whose complexity order is $\mathcal{O}\bigl{(}BNML\bigl{)}$). For $K$ iterations of Algorithm 1 and when signaling over $B$ frequency bands, it follows that the overall complexity of the PGA algorithm with pre-defined $K$ iterations, as is done by the proposed learned optimizer, is of the order of

The fact that we set our objective to be the rate $R(\cdot)$ results in its gradient computation yielding a higher complexity per iteration compared with that used when taking the gradients of a surrogate objective, e.g., [15] (where the quadratic dependence is on the number of RF chains rather than the number of antennas), while being of a similar order of that used in [16]. However, recall that our derivation of Algorithm 2 serves as the first step towards rapid and robust optimization of the rate, for which these gradient computations are useful, as shown in the sequel. Furthermore, the proposed learn-to-optimize framework facilitates implementing the optimizer with a fixed and small number of iterations, allowing to limit the overall complexity.

Our approach follows the deep unfolding methodology [30, 32, 31]. As opposed to other forms of deep unfolded networks, which designed DNNs to imitate the operation of a model-based optimizer while modifying its operation, as in, e.g., [46], our design is geared to preserve the operation of model-based iterative optimization. We use automated training capabilities of deep learning tools to tune the hyperparameters of the optimizer. By doing so, we improve upon the conventional usage of fixed hyperparameters, as demonstrated in Section V, and avoid the excessive delay of implementing hyperparameter search in each iteration.

The design objective used in our derivation is the achievable sum-rate (6), which is approached using dedicated coding over asymptotically large blocks. While one is likely to deviate from (6) in practice, it serves as a fundamental characteristic of the overall channel which encompasses hybrid precoding, transmission, and reception, making it a relevant figure-of-merit for optimizing hybrid MIMO systems. Furthermore, as the objective in (6) is explicit and differentiable, it enables our method to learn-to-optimize in an unsupervised manner, i.e., one does not need access to ground-truth precoders as in [19], and can train using solely channel realizations as data. Yet, using (6) as our objective is expected to affect the performance of the hybrid precoders when the channel matrices provided as inputs are inaccurate estimates of the true channel, in the same manner that such CSI errors affect the performance of model-based optimizers. We discuss this extension of our method to handling CSI errors in the following section.

The formulation in (8) represents a constrained maximin optimization problem with $2B+1$ optimization (and auxiliary) matrix variables $\textbf{W}_{a},\{\textbf{W}_{d,b}\}_{b\in\mathcal{B}},\{\textbf{E}_{b}\}_{b\in\mathcal{B}}$. Such problems can be tackled by combining the conceptual mirror prox (CMP) algorithm [39] with alternating optimization and projections, resulting in the PCMP method.

PCMP is an iterative method. Each iteration is comprised of two stages: CMP and projection. We next formulate these stages in the context of (8).

CMP: The objective in (8) is maximized according to the analog and digital precoding matrices, and minimized according to the error matrices. CMP aims at iteratively refining the optimization variables via gradient steps, which at iteration index $k+1$ take the form

The computation in (18) cannot be directly implemented in general since the gradients are taken with respect to the updated optimization variables (i.e., the iteration index $k+1$ appears on both sides of the update equation). However, it can be approached via additional iterative updates with index $i$ of the form [39, Eq. (6)]

In (19), $\mu_{a(i)}^{(k)},\mu_{d,b(i)}^{(k)},\mu_{e,b(i)}^{(k)}$ are the step sizes. The optimization variables in (19) are initialized to

The computation of the gradients in (19) with respect to the analog and digital precoders can use the gradient formulations derived for PGA in (10) and (13), respectively. The gradient of R with respect to $\textbf{E}_{b}$ for each $b\in\mathcal{B}$ is computed as (see Appendix B)

While the above CMP procedure introduces an additional iterative procedure which has to be carried out at each iteration of index $k$, it is typically sufficient to only carry out two iterations of (19) [47], i.e., repeat (19) for $i=1,\ldots,i_{\max}$ with $i_{\max}=2$.

Projection: After the CMP is conducted for two iterations, the resulting matrices, denoted $\Big{(}\hat{\textbf{W}}_{a,(i_{\max})},$ $\{\hat{\textbf{W}}_{d,b,(i_{\max})}\}_{b\in\mathcal{B}},\{\hat{\textbf{E}}_{b,(i_{\max})}\}_{b\in\mathcal{B}}\Big{)}$, are projected to meet the constraints. Thus, the update rule at iteration $k+1$ is

where $\varepsilon$ is the error bound on the entries of the sub-channel matrix, $\{\textbf{H}_{b}\}_{b\in\mathcal{B}}\in\mathbb{C}^{N\times{M}}$.

The overall procedure is summarized as Algorithm 3. The matrices $\textbf{W}_{a}^{(0)},\{\textbf{W}_{d,b}^{(0)}\}_{b\in\mathcal{B}}$ are initialized in the same manner as in Algorithm 1, while $\{\textbf{E}_{b}^{(0)}\}_{b\in\mathcal{B}}$ is generated randomly while normalizing to guarantee that $\|\textbf{E}_{b}\|_{F}<\epsilon$ for each $b\in\mathcal{B}$. Notice that Algorithm 3 describes a gradient-based optimizer, similar to Algorithm 1. Therefore, its convergence speed depends as well on the step sizes $\mu_{a(i)}^{(k)},\{\mu_{d,b(i)}^{(k)}\}_{b\in\mathcal{B}},\{\mu_{e,b(i)}^{(k)}\}_{b\in\mathcal{B}}$, which are hard to select manually. In Subsection IV-B we will describe the learn-to-rapidly-optimize method, extending the rationale used with full CSI in Algorithm 2, for accelerating Algorithm 3 convergence via data-aided hyperparameters tuning.

Algorithm 3 optimizes hybrid precoders for a given noisy channel realization. Nevertheless, it needs to be executed in real-time, whenever there is a change in the channel, and thus the need to obtain reliable hybrid precoders rapidly, i.e., within a fixed and small number of iterations, still applies. To accomplish this, we use the deep unfolding methodology following the approach as in Subsection III-B, leveraging its ability to tune hyperparameters in optimization involving projected gradients of the rate function, where the main differences are in the number of learned parameters and in the algorithm structure.

We use the PCMP method as the optimizer with exactly $K$ iterations and $i_{\max}=2$ internal iterations. Let us define the step sizes vector for the analog update of the $k$th iteration as $\bm{\mu}_{a}^{k}\triangleq(\mu_{a(1)}^{(k)},\mu_{a(2)}^{(k)})^{T}$; step sizes vectors for the digital updates of the $k$th iteration as $\{\bm{\mu}_{d,b}^{k}\}_{b\in\mathcal{B}}\triangleq\{(\mu_{d,b(1)}^{(k)},\mu_{d,b(2)}^{(k)})^{T}\}_{b\in\mathcal{B}}$; and the step sizes vector for the error updates as $\{\bm{\mu}_{e,b}^{k}\}_{b\in\mathcal{B}}\triangleq\{(\mu_{e,b(1)}^{(k)},\mu_{e,b(2)}^{(k)})^{T}\}_{b\in\mathcal{B}}$. Accordingly, we define the $k$th iteration $2\times 2B+1$ step sizes matrix as $\bm{\mu}^{k}\triangleq(\bm{\mu}_{a}^{k},\bm{\mu}_{d,1}^{k},\ldots,\bm{\mu}_{d,B}^{k},\bm{\mu}_{e,1}^{k},\ldots,\bm{\mu}_{e,B}^{k})$, obtaining the $K\times 2\times 2B+1$ step sizes tensor $\bm{\mu}\triangleq(\bm{\mu}^{0},\ldots,\bm{\mu}^{K-1})$ for $K$ iterations of Algorithm 3.

The unfolded architecture converts the PCMP method with $K$ iteration into a discriminative algorithm whose trainable parameters are the entries of $\bm{\mu}$. Similarly to the approach used in Subsection III-B, we train the unfolded PCMP in an unsupervised manner, i.e., the data set only includes channel realizations, while the level of tolerable error $\varepsilon$ is given. The loss function for a given normalized channel $\tilde{\textbf{H}}=\{\tilde{\textbf{H}}_{b}\}_{b\in\mathcal{B}}$ and step sizes $\bm{\mu}$ is computed as the maximal negative resulting achievable sum-rate of this channel, (6) when the maximum is taken over all $\|\textbf{E}_{b}\|_{F}<\varepsilon$, and the precoders are $\Big{(}\textbf{W}_{a}^{(K)},\{\textbf{W}_{d,b}^{(K)}\}_{b\in\mathcal{B}}\Big{)}\triangleq\textbf{PCMP}_{K}(\tilde{\textbf{H}},\bm{\mu})$, i.e., the precoders obtained via Algorithm 3 with $K$ iterations and step sizes $\bm{\mu}$. The resulting loss is

The maximization term in (23) notably complicates its usage as a training objective for learning the PCMP hyperparameters. To overcome this, we generate a finite set of $n_{e}$ error patterns (satisfying $\|\textbf{E}_{b}\|_{F}<\varepsilon$) via random generation, to which we add the zero error, i.e., $\mathcal{E}\triangleq\bigl{\{}\{\textbf{E}_{b}^{t}\}_{b\in\mathcal{B}}\bigl{\}}_{t=1}^{n_{e}}\cup\{{\bm{0}}\}$, and seek the setting which minimizes the maximal loss among these error patterns. Namely, to maintain feasible training, the objective used for training the unfolded algorithm evaluates the loss based on its output after $K$ iterations and is given by

The robust learn-to-optimize method, summarized as Algorithm 4, uses data to tune $\bm{\mu}$. We initialize $\bm{\mu}$ before the training process, with fixed step sizes with which PCMP converges (though not necessarily rapidly). After training, the learned matrix $\bm{\mu}$ is used as hyperparameters for rapidly converting a noisy channel realization into a hybrid precoding setting via $K$ iterations of Algorithm 3.

The proposed robust learn-to-optimize method in Algorithm 4 extends the method in Algorithm 2 to cope with noisy CSI. In particular, for the special case of $\varepsilon=0$ and $i_{\max}=1$, it holds that the PCMP algorithm (Algorithm 3) reduces to the PGA method (Algorithm 1). Accordingly, the data-aided optimized Algorithm 4 specializes Algorithm 2. Consequently, our gradual design, which started with designing learned PGA optimization applied to the rate function in Section III, allowed us to extend its derivation to realize robust learned PCMP algorithm for maximin rate optimization. In addition, this implementation shows that the general learn-to-optimize method can be adapted for speeding the convergence of a broad range iterative model-based hyperparameters-depending algorithms, and particularly those utilizing gradient and projection steps.

In terms of complexity, the unfolded PCMP with $K$ iterations and $i_{\max}=2$ shares the same complexity order as that the PGA method with full CSI. In particular, Step 3 of Algorithm 3 repeats the gradient computations of Algorithm 1 $i_{\max}$ times per iteration and includes additional gradients with respect to $\textbf{E}_{b}$ (which is of the same complexity as taking the gradients with respect to $\textbf{W}_{d,b}$). Thus, the complexity order of the unfolded PCMP algorithm is

which, for $i_{\max}=2$, yields a similar complexity as that of the unfolded PGA in (17).

The ability to rapidly optimize in a robust manner via the unfolded PCMP results in hybrid precoders that are less sensitive to channel estimation errors. The PCMP algorithm is coping with mismatched channels by considering a wide range of channel errors when setting the hybrid precoders. While the PGA algorithm is carried out each time the channel changes, the PCMP algorithm can be carried out less frequently since it is more robust and stable. Accordingly, the PCMP can be also used to reduce the rate in which the optimization procedure is carried out, and not only for coping with mismatched channels.

The error bound value, $\varepsilon$, plays a key role in the robust algorithm. It affects the robustness, reliability, and stability of the solution, and therefore, it needs to be chosen based on some prior knowledge, or on application needs. Another approach is to use channel estimation methods prior to the precoders design we presented in this work, and by this overcome the mismatched channel problem. An additional important parameter is the setting of the expected error patterns evaluated during training, i.e., $\mathcal{E}$. While in our evaluation we generated this set in a random fashion and included in it the zero-error pattern to guarantee suitability for error-free case, one can consider alternative methods for preparing this set such that the resulting optimized hyperparameters would be most suitable for the desired robust optimization metric. Since this setting is carried out offline and thus complexity is not a key issue here, one can possibly explore the usage of sequential sampling and Bayesian optimization techniques [48] for setting $\mathcal{E}$. We leave the study of extensions of our method to future work.

For full CSI, the optimizer for designing the hybrid precoder is PGA, and its data-aided learn-to-optimize version is detailed Subsection III-B. The implementation of this method can be carried out for both unconstrained analog combiners (for which the only constraint is the power constraint), as well as hardware-limited designs, where a common constraint is that of phase shifter networks. We thus divide our evaluation into learned PGA with unconstrained analog combiners and PGA with phase shifters.

The numerical studies reported in Subsection V-A empirically validate the ability of the proposed methodology to facilitate rapid optimization under different settings. We next demonstrate its ability to simultaneously enable rapid and robust hybrid precoding. Hence, in the following, we consider the unfolding of the PCMP algorithm for precoding under noisy channel estimation. Having demonstrated the ability of our approach to deal with constrained precoders, we focus here on unconstrained analog combiners. According, we implement of the learn-to-optimize methodology, i.e. Algorithm 4, for Algorithm 3, when the projection operator applied on the analog precoder in Step 3 is set to be $\Pi_{\mathcal{A}}(\textbf{A})=\textbf{A}$. The set of considered error patterns $\mathcal{E}$ is comprised of $n_{e}=20$ patterns randomly generated with Frobenius norm values in the range $(0,\epsilon)$, and we evaluate the maximin rate over $\mathcal{E}$.

We examine two different settings of the number frequency bands $B$, users $N$, RF chains $L$, transmit antennas $M$, and the data source: $B=8,N=6,L=10,M=12$, with Rayleigh channels; and $B=16,N=4,L=6,M=12$ with QuaDRiGa channel model, the simulations results for each setting are demonstrated in Fig. 8, and Fig. 9, respectively. Again, we evaluate both the convergence rate and the minimal rate (within the tolerable error regime) at the end of the optimization procedure versus SNR, and compare our results with the conventional PCMP optimizer. In the simulations, three error bound values are considered, $\varepsilon=0.005,0.05,0.5$. For each setting, we applied Algorithm 4 to learn to set hybrid precoders with $K=5$ iterations based on $|\mathcal{D}|=1000$ channels, where we used $40-50$ epochs with batch size $|\mathcal{D}_{q}|=100$, and used Adam for the update in Step 4. We compared the standard PCMP, with constant hyperparameters, to the optimized PCMP with exactly $K=5$ iterations.