Online Beamforming Algorithms

• Online beamforming algorithms are signal processing methods that adapt beamformers in real time using causal observations and rapid model updates.
• They integrate techniques like deep learning, meta-learning, and hybrid analog-digital design to optimize performance under dynamic channel conditions.
• These algorithms address challenges such as varying CSI, high system dimensionality, and low-latency requirements in massive MIMO and heterogeneous networks.

Online beamforming algorithms are a class of signal processing, optimization, or learning-based methods that compute or adapt transmit or receive beamformers in real time (or low-latency iterative mode) as new channel state information (CSI), network topology, or user conditions arrive. In contrast to classical offline or batch approaches, these algorithms exploit causal observations, low-complexity structures, and rapid model updates to provide robustness and near-optimal performance in dynamic, heterogeneous, or large-scale multi-antenna wireless and acoustic systems. Approaches span and integrate techniques including deep learning, meta-learning, adaptive optimization, hybrid analog-digital design, probabilistic generative modeling, bandit learning, and distributed decision-making.

1. Fundamental Problem Structure and Motivation

Online beamforming formulations are driven by real-time requirements and rapidly varying environments in both communications and sensor array processing. Core challenges include unknown or time-varying CSI, high system dimensionality (massive MIMO, cell-free networks, mmWave arrays), architectural constraints (hybrid analog/digital, RIS/IRS, unit-modulus phase shifters), and stringent latency or switching constraints. Optimization objectives typically target sum-rate maximization, SNR/SINR balancing, transmission power minimization, robust worst-case performance, or multi-task tradeoffs such as admission control and user selection.

A prototypical online beamforming problem takes the form

$$\max_{\{\mathbf{w}_k\}}\; \sum_{k=1}^K R_k(\mathbf{w}_1,\dots,\mathbf{w}_K; \mathbf{H})$$

subject to system-level constraints (power, QoS, per-antenna, hardware), where CSI $\mathbf{H}$ evolves on a fast timescale. Methods are sought that avoid global optimization per new realization, favoring incremental updates, forward-passing neural architectures, or sample-based optimization to meet latency requirements (Chen et al., 2024, Gao et al., 2020, Zou et al., 16 Dec 2025, Yuan et al., 2020).

2. Algorithmic Architectures and Online Adaptation Mechanisms

A broad taxonomy of online beamforming algorithms includes:

• Deep Learning–based Direct Inference: Lightweight feedforward neural networks are trained on realistic channel samples to approximate the mapping from CSI (or features thereof) to beamforming parameters. Real-time inference is accomplished via a millisecond-scale forward pass (Gao et al., 2020, Chen et al., 2024, Xia et al., 2020). Online update typically occurs by rapid gradient steps on a subset of the network (e.g., batch norm parameters, final linear layers) with unsupervised losses on currently observed CSI and transmission statistics.
• Meta-Learning and Fast Adaptation: To address domain and distribution shift, model-agnostic meta-learning (MAML) frameworks are used to pretrain base models that can be rapidly adapted online to new channel or network conditions with minimal fine-tuning data. Online adaptation employs buffered channel samples and gradient-based inner-loop updates, combined with meta-learning outer loops (e.g., “follow-the-leader”) to track non-stationary channel statistics (Yuan et al., 2020, Zou et al., 16 Dec 2025).
• Hybrid Offline-Online and Unsupervised Approaches: Offline training is combined with rapid online tuning, where models incorporate both empirical statistics and learned priors. Covariance prediction networks, MAML with multiple basis initializations, and embedded optimization steps (e.g., WMMSE updates) are prominent. Online adaptation is performed through fusion of network predictions with real-time sample statistics, with fine-tuning via a small number of gradient steps per coherence block (Zou et al., 16 Dec 2025, Li et al., 2024).
• Generative and Bayesian Methods under Limited Feedback: Deep generative models (e.g., conditional VAEs) are online-trained on high-resolution channel feedback and deployed to rapidly generate refined channel realizations conditional on limited feedback (such as Type I codebook PMI/CQI). This enables online sample-average stochastic optimization (e.g., for robust WMMSE) even under compressed feedback regimes (Li et al., 2024).
• Adaptive and Mixed-Timescale Design: For hybrid and passive beamforming systems (e.g., RIS/IRS, hybrid analog-digital MIMO), online adaptation may involve separated timescales, with slow-timescale (frame) optimization of analog or passive elements using deep-unfolded networks, and fast-timescale (slot) updates of digital beamformers using low-dimensional equivalent CSI estimated online. Fine-tuning of analog parameters is accomplished via a small number of gradient steps on fresh samples per frame, with digital updates every slot (Kang et al., 2022, Liu et al., 2021).
• Adaptive Filtering and Affine-Projection Methods in Acoustics: In speech enhancement, low-latency online convolutional or MVDR beamforming leverages recursive filtering algorithms derived from Kalman filters or affine-projection updates, exploiting block- or frame-wise online accumulation of speech/noise covariance statistics from recent source separation or dereverberation modules (Nugraha et al., 2022, Braun et al., 2021).

3. Feature Engineering, Loss Functions, and Model Structures

Input feature construction is critical in online learning-based beamforming:

• Bilinear/Band-Limited Representation: Neural networks may receive as input explicit bilinear terms characterizing the channel and system geometry (e.g., vec($G \odot h_r$), real/imaginary parts, or concatenated CSI tensors) to encode interaction structure and reduce input dimensionality (Gao et al., 2020, Chen et al., 2024).
• Domain-Generalizing Modules: Feature selection or pruning layers, e.g., gradient-reversal and classifier-guided masking within convolutional backbones, are used to ensure robustness to shifts in fading law, network size, or distribution (Chen et al., 2024).
• Surrogate and Unsupervised Losses: Since ground-truth beamformers may be unavailable, many methods use unsupervised loss functions directly tied to beamforming performance metrics (e.g., $-\|\text{composite channel}\|^2$, negative sum rate, information entropy surrogates). Some frameworks employ sample-average approximations of expected utility based on model or VAE-generated channel realizations (Gao et al., 2020, Li et al., 2024, Zou et al., 16 Dec 2025).
• Meta-Objective Functions: For rapid adaptation, meta-learning schemes use per-task support and query losses, sample-weighted or soft-selected basis models, and regularization terms to maintain diversity of meta-initializations across environmental tasks (Yuan et al., 2020, Zou et al., 16 Dec 2025).

4. Complexity, Real-Time Feasibility, and Scalability

A hallmark of online beamforming algorithms is real-time operation under stringent computational and hardware constraints:

• Deep Learning and Hybrid Models: Inference via well-designed DNNs (e.g., 5-layer FCN as in RISBFNN, dimension-adaptive CNNs as in HGNet) achieves sub-millisecond execution on CPUs/GPUs, enabling adaptation at or below the wireless channel coherence time (Gao et al., 2020, Chen et al., 2024). Online fine-tuning, when limited to a small parameter subset (e.g., batch-norm scale/shift), adds only microseconds to total latency.
• Sample Complexity and Training Overhead: Offline pretraining often requires $10^5$–$10^6$ samples, but online adaptation typically converges in $1$–$20$ gradient steps per environment update. Streaming-type or meta-learning approaches maintain small task buffers, permitting rapid adaptation with $N_t \leq 100$ (Yuan et al., 2020).
• Optimization-Based and Recursive Methods: Deep-unfolded versions of conventional solvers (e.g., SSCA for IRS beamforming) collapse matrix inversion and iterative blocks into shallow networks with per-sample complexity $\mathcal O(N^{2.37})$, dramatically below the $\mathcal O(N^3)$ or higher of classic approaches (Liu et al., 2021, Kang et al., 2022).
• Scalability in Cell-Free and Massive Networks: Models with input and output modules explicitly designed for variable network size (e.g., dimension-adaptive convolution, zero padding) generalize efficiently to scenarios with changing numbers of APs/users, enabling run-times on the order of a few milliseconds per period in large-scale cell-free MIMO (Chen et al., 2024).
• Resource-Aware Speech Beamforming: Online convolutional affine-projection algorithms and covariance-accumulation from block-online BSS achieve end-to-end frame latency of $22$–$32$ ms, much less than DNN-based block-MVDR schemes (Nugraha et al., 2022, Braun et al., 2021).

5. Performance Benchmarks and Empirical Studies

Rigorous simulation and deployment studies have benchmarked online beamforming algorithms against offline solvers and fixed DNNs:

Method/Class	Relative Rate/SNR (vs. Opt/SDR)	Time per Instance	Adaptivity
RISBFNN DNN (Gao et al., 2020)	88–96% of SDR	0.04–0.12 ms	Real-time, no label required
Model-driven BNN (Xia et al., 2020)	98–99% of WMMSE, 98% sum-rate	0.03 ms	Continual, light-touch finetune
HGNet+OAU (Chen et al., 2024)	$>$7% sum-rate gain over WMMSE	6 ms	Millisecond online update of 3% params
Online meta-FTL (Yuan et al., 2020)	90–95% of ideal BNN SINR	$<$0.5 ms/slot	Sublinear regret in switching environments
MB-MAML + SALR (Zou et al., 16 Dec 2025)	Significant gain over SOTA, fast	$1$–$5$ gradient steps	Online robust adaptation
Online CVAE-WMMSE (Li et al., 2024)	$>0.5$ bps/Hz gain over WMMSE-coarse	Minimal (per step)	Per-UE online training, robust
FastMNMF-MVDR (speech) (Nugraha et al., 2022)	5-point WER gain over DNN-MVDR	22 ms latency	Frame-online (<16ms shift)
Deep-Unfolding Beamforming (Liu et al., 2021, Kang et al., 2022)	97%+ of full SSCA / RLS-SSCA	0.06–0.3 s per slot (NN); 3–19 s (classic)	Online analog+digital, mixed timescale

These results underscore the ability of online beamforming algorithms to maintain near-optimal or rapidly adaptive performance with order-of-magnitude reductions in computational cost and latency.

6. Specialized Online Strategies: Bandit Learning, Recommendation, and Adaptive Search

Beyond neural and optimization-based methods, specialized online beamforming strategies span bandit learning and collaborative filtering:

• Online Bandit-Based Beam Search: For blind or minimally informed initial access and direction finding, online algorithms cast the beam search as a continuous-armed multi-armed bandit, solved via UCB or Thompson-sampling methods on adaptively discretized candidate directions (Amuru, 2019). The BLB (beam-learning bandit) algorithm provides theoretical vanishing per-step regret and superior learning curves compared to fixed-grid scans.
• Collaborative Filtering for Initial Access: Recommender-system methodology (e.g., matrix completion via SVD) leverages historical group-beam interaction data to recommend optimal beams to new users with minimal online probing (Yammine et al., 2022). Using fast low-rank projections and neighbor-based scoring, such algorithms outperform standard hierarchical or greedy searches, especially in multi-BS and codebook scenarios.
• ADMM-Based Online Joint Beamforming/Admission Control: In long-term resource allocation, online alternating direction methods efficiently solve per-block beamforming, admission, and switching-cost constrained problems via decomposition into convex subproblems and sample-average approximation (Lin et al., 2019).

7. Practical Considerations, Limitations, and Application Domains

Several factors influence the deployment and effectiveness of online beamforming algorithms:

• Assumptions and Preconditioning: Many approaches rely on synchronized CSI updates, modest feedback overhead, reliable initial training under representative channel models, and sufficient computational resources for mini-batch or small gradient updates.
• Domain Mismatch and Robustness: The performance of static (fully offline) DNN beamformers can degrade under out-of-distribution (OOD) channel changes or user mobility. Online adaptation and domain generalization modules are essential to mitigate these issues (Chen et al., 2024, Zou et al., 16 Dec 2025, Yuan et al., 2020).
• Application Scope: Online algorithms span 5G/6G and cell-free MIMO, RIS/IRS reflection control, acoustic beamforming and dereverberation, mmWave/THz initial access, and dynamic admission/resource control. Architectures accommodating variable network size, distributed deployment, and real-time decision making are increasingly pivotal (Chen et al., 2024, Kang et al., 2022, Yammine et al., 2022).
• Resource and Complexity Management: Techniques such as mixed-timescale modeling, sparse low-rank output representations, and meta-initialization buffering allow real-time operation on CPU/GPU hardware even in large-scale scenarios.

In summary, online beamforming algorithms synthesize model-driven and data-driven principles, enabling robust, scalable, and computationally efficient solutions across dynamic multi-antenna communications and sensor array systems. The field integrates diverse techniques—deep learning, meta and online learning, signal processing, bandit theory, and distributed inference—to realize low-latency, high-performance beamforming under practical and non-stationary conditions.