Stochastic Precoding Techniques

• Stochastic precoding is a design paradigm that optimizes transmission parameters using statistical channel descriptors and probabilistic models instead of relying solely on instantaneous CSI.
• It is employed in massive MIMO, IRS-aided systems, and federated learning to enhance performance and robustness under diverse and uncertain channel conditions.
• The approach leverages stochastic optimization, chance constraints, and two-timescale architectures to balance resource allocation, mitigate CSI imperfections, and improve overall system efficiency.

Stochastic precoding denotes a family of transmitter-side design paradigms in which the precoder is optimized against randomness rather than against a single fully known channel realization. In the cited literature, that randomness arises from several distinct sources: random fading and ergodic rate objectives; randomized control-state selection and time-sharing; statistical CSI in place of instantaneous CSI; probabilistic CSI-error models; random information-bearing waveforms; and two-stage formulations in which a long-term variable is optimized over the expected value of short-term per-realization precoding outcomes (Chen et al., 2014, Liu et al., 2018, Tian et al., 2019, Wang et al., 9 Jul 2025, Hashmi et al., 20 Sep 2025). The term therefore does not identify one algorithm or one hardware architecture; it identifies a class of formulations in which precoding is coupled to probability, statistics, or stochastic optimization.

1. Definitions and conceptual scope

In FDD massive MIMO, stochastic precoding can mean a randomized control policy

$\Omega=\{\Gamma,q\},$

where $q$ is a probability vector over control states and each state contains an analog phase-shifter vector and a power-allocation vector; the transmitter time-shares among multiple analog precoders rather than fixing one analog beamformer for the entire statistical coherence interval (Tian et al., 2019). In multi-cell MIMO under generalized fading, it can mean long-term weighted sum-rate maximization under an expectation over random channel matrices when only the first and second moments are known (Wang et al., 9 Jul 2025). In two-stage IRS-aided systems, the “stochastic” aspect can reside in the outer problem,

$$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$

where $\boldsymbol{\theta}$ is a long-term variable and $\boldsymbol{W}$ is recomputed for each realized channel state $\omega$ (Hashmi et al., 20 Sep 2025).

The cited work also uses the term in robust and task-specific senses. In symbol-level precoding, stochasticity arises from Gaussian CSI errors and is handled through probabilistic constructive-interference constraints (Haqiqatnejad et al., 2019). In wireless federated learning over a MAC, precoding is matched to the signSGD objective and uses only one-bit CSIT, namely the sign of the fading coefficient, to prevent sign-flipping errors (Park et al., 2021). In sensing-oriented ISAC, the relevant randomness is the information-bearing transmit signal itself, so precoding is designed for the ergodic LMMSE induced by random Gaussian signaling rather than for a deterministic training covariance (Lu et al., 2023).

A recurring implication is that stochastic precoding should not be reduced to random beam selection or blind averaging. The cited literature treats it as a structured response to uncertainty: randomized policies when user grouping and pilot budgets must be balanced, statistical designs when only second-order information is available, and chance-constrained or robust formulations when CSI is imperfect.

2. Statistical information and ergodic formulations

A central strand of stochastic precoding replaces instantaneous CSI by long-term statistical descriptors. In multi-cell MIMO over generalized fading, the design target is the ergodic weighted sum rate

$$\underset{\mathbf{V}}{\text{maximize}}\quad \sum^L_{j=1}\sum^K_{k=1}\mathbb{E}\Big[\omega_{jk}\mathcal{R}_{jk}\Big]$$

subject to per-BS power constraints, while assuming only

$$\mathbf C_{jk}=\mathbb E\big[\mathbf H_{jk,j}\big], \qquad \mathbf D_{jk,\ell}= \mathbb E\big[\mathrm{vec}(\mathbf H_{jk,\ell})\mathrm{vec}(\mathbf H_{jk,\ell})^H\big].$$

The resulting formulation is explicitly distribution-free in the sense of not requiring a full parametric fading law, and is specialized in the paper to Rayleigh fading, Rician fading, Nakagami-$m$ fading, and other generalized fading/channel-uncertainty models for which those moments are known (Wang et al., 9 Jul 2025).

In downlink C-RAN over block-ergodic channels, the central unit may have only stochastic CSI, such as the correlation matrices in a Kronecker fading model. Under this assumption, the same precoder and compression parameters are reused across blocks in Compression-After-Precoding, whereas in Compression-Before-Precoding the same precoding matrix is reused so that the fronthaul overhead for sending precoding matrices is amortized over the whole coding block and becomes negligible (Kang et al., 2014). This places stochastic precoding inside a larger resource-allocation problem in which fronthaul compression and statistical beam design are jointly optimized.

A more classical example appears in multiuser MISO-MC-CDMA with orthogonal STFBC. There, the precoder is designed from time-domain transmit correlation and tap statistics under a Kronecker correlation model, with the explicit goal of minimizing an upper bound on the average PEP. The optimal statistical precoder aligns with the eigenvectors of the transmit correlation matrix,

$V_0=V_{R,0},$

and uses a water-filling-type power allocation over the selected eigenmodes (Chen et al., 2014). The same work derives asymptotic rules: equal power allocation is asymptotically optimal at high effective SINR, while single-beam allocation is asymptotically optimal at low SINR. This suggests that stochastic precoding can be understood not only as expectation maximization, but also as deterministic optimization over second-order channel structure.

3. Randomized policies and multiscale architectures

Randomization can enter the precoder itself. In randomized channel sparsifying hybrid precoding for FDD massive MIMO, the base station adopts a stochastic policy over several analog/digital configurations rather than a single fixed hybrid precoder. Each control state contains one analog phase-shifter vector and one power-allocation vector, and the selected analog precoder is adapted to channel statistics so that the resulting effective channel has enough spatial DoF to serve a compatible user group while still being accurately estimable under a limited pilot budget (Tian et al., 2019). The framework turns imperfect-CSI hybrid precoder design into a stochastic utility maximization problem in which user grouping emerges implicitly through the optimized time-sharing policy.

A closely related but non-randomized multiscale architecture is two-timescale hybrid precoding in massive MIMO. In that model, the RF precoder is updated on the slow timescale from channel statistics, while the baseband precoder is updated every slot from the instantaneous effective channel $q$0. The paper organizes the system into super-frames and frames, assumes real-time effective CSI every slot but only one possibly outdated full channel sample once per frame, and uses SSCA-THP to learn the long-term variables online from streaming channel samples (Liu et al., 2018). The common structure is the same as in stochastic precoding more broadly: slow variables absorb distributional information, and fast variables exploit reduced-dimensional or effective CSI.

The asynchronous SCA precoding example in wireless sensor networks makes the same separation explicit through

$q$1

where $q$2 is a slowly varying static component learned across coherence intervals and $q$3 is a small instantaneous correction constrained by $q$4 (Idrees et al., 2020). In IRS-aided sum-rate maximization, the same long-term/short-term split appears at a higher architectural level: long-term IRS control is optimized over the expectation induced by random channel states, while short-term active precoding is solved per realization, possibly only approximately (Hashmi et al., 20 Sep 2025).

These multiscale constructions share a common significance. They are not merely heuristic decompositions; they are responses to hardware, pilot, latency, and fronthaul constraints. Stochastic precoding in this sense is an architectural principle for distributing adaptation across timescales.

4. Imperfect CSI, quantization, and probabilistic robustness

A major use of stochastic precoding is robust design under imperfect CSI. In robust symbol-level precoding with Gaussian CSI errors, the hard constructive-interference constraint is replaced by a chance constraint of the form

$q$5

and three convex approximations—A1, A2, and B—are developed, with A1 and A2 always outperforming the benchmark under the paper’s comparisons and each being tighter than the other under specific robustness conditions (Haqiqatnejad et al., 2019). A related SLP formulation based on DPCIRs models the CSI error as Gaussian and imposes a probabilistic CI constraint with outage probability at most $q$6, yielding a convex stochastic robust design that is sufficient but conservative (Haqiqatnejad et al., 2018). Across these formulations, stochastic precoding means power minimization subject to reliability specified in probability rather than through worst-case sets alone.

Other works address uncertainty distributed across nodes or hardware. In network MIMO with hierarchical CSIT, each transmitter has its own imperfect channel estimate and the exact problem is formulated as a team decision problem. The proposed hierarchical WMMSE-type heuristic exploits the ordering of CSI quality across transmitters so that lower-quality nodes make conservative decisions while higher-quality nodes refine the undecided precoder blocks using better CSI (Kerret et al., 2014). In quantized-CSI Tomlinson–Harashima precoding, the transmitter performs QR decomposition on the quantized channel, the residual interference due to CSI quantization is analyzed under RVQ, and upper bounds are derived for the average sum rate and mean rate loss (Sun et al., 2014). In coarsely quantized MU-MIMO with low-resolution DACs, a Gauss-Markov uncertainty model together with a bounded uncertainty set is converted through binary reformulation, the S-procedure, and semidefinite relaxation into the RSDR precoder (Chu et al., 2019).

Limited CSI need not imply weaker task performance if the task structure changes. In wireless federated learning with signSGD, the proposed sign-alignment precoder

$q$7

uses only one-bit CSIT and aligns channel polarity so that fading cannot flip the transmitted gradient sign; under a Gaussian prior on local gradients, BayAirComp performs posterior-mean aggregation (Park et al., 2021). In centralized JT-CoMP with partial feedback and limited backhaul, unknown CSI entries are not set to zero; instead, pathloss and shadow fading are used to model the statistical interference produced by unknown links, and SSOCP or WMMSE is then applied using active links only (Lakshmana et al., 2015). These examples show that “more CSI” is not automatically the relevant criterion: what matters is whether the available information is aligned with the communication or learning objective.

5. Optimization methods

The algorithmic core of stochastic precoding is dominated by surrogate-based stochastic optimization. In RCSHP, the nonconvex expected-utility problem is solved by a stochastic successive convex approximation method that samples channel and estimation-noise realizations, builds a strongly concave surrogate, solves a convex problem in the time-sharing vector and independent convex quadratic problems in the individual control states, and updates by relaxation. Under the stated conditions on $q$8 and $q$9, any limiting point is proved to be a KKT point almost surely (Tian et al., 2019). SSCA-THP uses the same logic in an online setting, where each new channel sample produces quadratic surrogates for the objective and constraints, and subsequential limits are stationary points almost surely (Liu et al., 2018).

The asynchronous SCA framework generalizes this surrogate view to delayed computations. It maintains a gradient-tracking recursion and allows the surrogate minimizer $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$0 to be computed from stale information with delay $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$1. The non-asymptotic analysis yields the condition $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$2 and the convergence-rate theorem stating that if

$$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$3

then the method achieves an $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$4-stationary point with iteration complexity $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$5 (Idrees et al., 2020). This is important because many real-time precoding subproblems cannot be solved within a single coherence interval.

A second major line uses deterministic lower bounds or upper bounds for stochastic objectives. For generalized fading channels, naive matrix FP inside the expectation fails because the auxiliary variables become realization-dependent. The proposed remedy is to move the maximization over auxiliary variables outside the expectation, which yields a computable lower bound on the expected rate and closed-form alternating updates for $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$6, $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$7, and $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$8; for large-scale MIMO, an inverse-free majorization step removes the large matrix inverse (Wang et al., 9 Jul 2025). In stochastic-CSI C-RAN, SSUM constructs sample-wise lower bounds of ergodic rates and linearized fronthaul constraints, then solves convex inner problems over covariance matrices and recovers precoders by rank reduction (Kang et al., 2014).

Gradient-based and gradient-free stochastic approximation also appear when analytic derivatives are unavailable or when the waveform itself is random. For random ISAC signaling, DIP is learned by stochastic gradient projection and MB-SGP on the ELMMSE objective (Lu et al., 2023). For two-stage IRS design with black-box effective channels and inexact WMMSE inner solutions, iZoSGA uses a two-point zeroth-order stochastic quasigradient and proves convergence to a neighborhood of stationarity governed by the average inexactness of the precoding oracle (Hashmi et al., 20 Sep 2025).

6. Applications, trade-offs, and reported behavior

The application range of stochastic precoding is unusually broad. In massive MIMO hybrid precoding, RCSHP is reported to provide better utility than ACS and THP baselines, robustness in both sparse and rich-scattering channels, improvement with fewer pilots, and performance approaching perfect-CSI performance when pilot budget is sufficient; the reported SSCA-RCSHP convergence is “about 50 iterations” in the stated setting (Tian et al., 2019). In generalized fading multi-cell MIMO, the proposed stochastic precoder is reported to outperform WMMSE by about 30% in sum rate in the reported settings, while the inverse-free Algorithm 2 attains the same achieved rate as Algorithm 1 with lower runtime (Wang et al., 9 Jul 2025).

Task-specific formulations produce equally specific gains. In wireless federated learning, BayAirComp with sign-alignment precoding outperforms OBDA by about 3.0% on MNIST and 3.7% on CIFAR-10 for heterogeneous data, and the paper explicitly states that one-bit precoding with BayAirComp aggregation can provide a better learning performance than the existing precoding method even using perfect CSI with AirComp aggregation (Park et al., 2021). In non-stationary channels, the HOGMT-based joint spatio-temporal precoder is reported to achieve $$\max_{\boldsymbol{\theta}\in\Theta} f(\boldsymbol{\theta}) \triangleq \mathbb{E}\left\{ \max_{\boldsymbol{W}\in\mathcal{W}} F(\boldsymbol{W},\boldsymbol{H}(\boldsymbol{\theta},\omega)) \right\},$$9 BER improvement (at 20dB) over existing methods, to require only about 0.5 dB additional SNR to match ideal BER for 16-QAM in the stated setting, and to reach BER around $\boldsymbol{\theta}$0 at 20 dB for 64-QAM with $\boldsymbol{\theta}$1 (Zou et al., 2022).

Robustness introduces characteristic conservatism trade-offs. In stochastic robust SLP with DPCIRs, the stochastic design is reported to achieve about 9 dBW lower transmit power than the worst-case spherical robust design in the reported setting, whereas the worst-case design achieves lower average SER with about 7 dB gain relative to the stochastic method (Haqiqatnejad et al., 2018). In the broader robust SLP comparison, A1 and A2 are more feasible than the benchmark B, and the paper states that robustness increases computational complexity by an order of the number of users in the large system limit relative to the non-robust design (Haqiqatnejad et al., 2019). In quantized-CSI THP, nonlinear precoding is reported to outperform ZF for both perfect CSI and quantized CSI, yet to suffer more from imperfect CSI; with fixed feedback bits, the high-SNR regime becomes interference-limited (Sun et al., 2014).

System-level operating points also shift under stochastic design. In C-RAN over ergodic fading, CAP is advantageous when fronthaul is plentiful and interference mitigation dominates, whereas CBP can be preferable under low fronthaul capacity, large coherence time, high transmit power, or stochastic CSI, because the overhead of sending precoding matrices is reduced or amortized (Kang et al., 2014). In ISAC, DDP is reported to improve sensing by about 1.5 dB relative to the deterministic-covariance baseline “DetOpt,” while DIP improves communication rate by several bps/Hz at the same sensing level (Lu et al., 2023).

Taken together, these results suggest a stable taxonomy. Stochastic precoding encompasses randomized policies, ergodic designs from moments or covariances, robust chance-constrained designs under statistically known CSI errors, two-stage recourse formulations, and stochastic optimization methods that learn long-term precoding variables from streaming samples. Its unifying premise is that precoding should be matched to the actual uncertainty structure of the system—fading law, moment information, pilot budget, feedback limitation, waveform randomness, or multi-timescale control—rather than to an idealized deterministic CSI model.