Self-Evolving In-Context Learning for Direct Pilot-to-Beamformer Design in MU-MISO Systems

# Self-Evolving In-Context Learning for Direct Pilot-to-Beamformer Design in MU-MISO Systems
Source: [https://arxiv.org/html/2607.11970](https://arxiv.org/html/2607.11970)
Yubo Zhang, and Xiaodong Wang Y\. Zhang and X\. Wang are with the Department of Electrical Engineering, Columbia University, New York, NY 10027\.

###### Abstract

We develop an enhanced in\-context learning \(ICL\) framework to improve the performance of pilot\-based beamforming in multi\-user multiple\-input single\-output \(MU\-MISO\) systems\. The proposed scheme integrates the ICL\-Transformer backbone with the pilot encoder\-decoder network \(EDN\) and the beamformer EDN\. A crucial feature of our ICL network is that it can handle multiple channel models without retraining, enabled by the construction of model\-specific context datasets\. To improve convergence and robustness, we introduce three key innovations: \(a\) a curriculum learning \(CL\) strategy that smoothly transitions from supervised LMMSE\-labeled imitation to unsupervised sum\-rate maximization, \(b\) a self\-evolving mechanism that dynamically expands and refines the context datasets for all channel models during CL\-based training, and \(c\) a mismatch\-aware extension that incorporates several mismatches into the general ICL framework and bypasses explicit channel calibrations\. Ablation studies validate the effectiveness of the in\-context architecture and enhanced training strategies\. Simulation results over diverse communication environments show that the proposed scheme is able to rapidly adapt to both seen and unseen channel models without gradient\-based parameter updates, and can mitigate the mismatch issues via intelligent context constructions\. Furthermore, our scheme consistently outperforms the existing beamforming schemes under pilot\-based settings, including the WMMSE benchmark and the recent Transformer\-based methods\.

## IIntroduction

Next\-generation wireless systems are expected to operate with larger antenna arrays, denser user populations, and more diverse propagation environments, creating an urgent need for scalable, generalizable and low\-latency physical\-layer signal processing\[[1](https://arxiv.org/html/2607.11970#bib.bib1)\]\. Classical iterative methods such as the weighted minimum mean squared error \(WMMSE\) algorithm\[[2](https://arxiv.org/html/2607.11970#bib.bib2)\]and the branch\-reduce\-and\-bound \(BRB\) method\[[3](https://arxiv.org/html/2607.11970#bib.bib3)\]can attain near\-optimal solutions, but their computational cost is often prohibitive in large\-scale systems\. At the other extreme, low\-complexity linear schemes such as maximum\-ratio transmission \(MRT\)\[[4](https://arxiv.org/html/2607.11970#bib.bib4)\]and linear MMSE \(LMMSE\) beamforming\[[1](https://arxiv.org/html/2607.11970#bib.bib1)\]provide fast solutions at the cost of substantial performance degradation\. To bridge this complexity\-performance gap,deep learning\(DL\) approaches, especially Transformer\-based models, have recently been investigated to enable high\-quality beamforming with improved scalability and real\-time adaptability\.

### I\-APilot\-based Beamforming Learning

Deep learning for beamforming and precoding in multi\-user systems has undergone rapid development in recent years\. Early works mainly relied on lightweight architectures such as fully\-connected neural networks \(FCNNs\)\[[5](https://arxiv.org/html/2607.11970#bib.bib5),[6](https://arxiv.org/html/2607.11970#bib.bib6),[7](https://arxiv.org/html/2607.11970#bib.bib7),[8](https://arxiv.org/html/2607.11970#bib.bib8)\]and convolutional neural networks \(CNNs\)\[[9](https://arxiv.org/html/2607.11970#bib.bib9),[10](https://arxiv.org/html/2607.11970#bib.bib10),[11](https://arxiv.org/html/2607.11970#bib.bib11)\]to learn compact channel representations or to directly predict beamformers\. More recently, graph neural networks \(GNNs\)\[[12](https://arxiv.org/html/2607.11970#bib.bib12),[13](https://arxiv.org/html/2607.11970#bib.bib13)\]have been introduced to improve scalability by exploiting the interaction structures among users and antennas, while Transformer\-based schemes\[[14](https://arxiv.org/html/2607.11970#bib.bib14),[15](https://arxiv.org/html/2607.11970#bib.bib15),[16](https://arxiv.org/html/2607.11970#bib.bib16)\]further enhance the expressive power by capturing long\-range global dependencies in large\-scale optimization problems\. In parallel, several studies have begun to explore wireless foundation models\[[17](https://arxiv.org/html/2607.11970#bib.bib17),[18](https://arxiv.org/html/2607.11970#bib.bib18)\]toward broader multi\-task and multi\-scenario adaptability\.

Despite these advances, most learning\-based beamforming methods still assume perfect CSI at the base station \(BS\), whereas practical systems provide only noisy, limited\-length pilot signals\. In particular, when the pilot length is smaller than the number of users, the full channel cannot be exactly recovered even in the noiseless case\[[19](https://arxiv.org/html/2607.11970#bib.bib19),[20](https://arxiv.org/html/2607.11970#bib.bib20)\]\. This information bottleneck limits the conventional two\-stage pipeline of channel estimation followed by beamforming\[[21](https://arxiv.org/html/2607.11970#bib.bib21),[22](https://arxiv.org/html/2607.11970#bib.bib22)\]: channel estimation minimizes reconstruction error rather than the ultimate sum\-rate objective, and its errors propagate to the subsequent beamforming stage\. These limitations motivate*direct pilot\-to\-beamformer learning*, which generates beamformers directly from pilot observations without explicitly reconstructing the full channel\[[9](https://arxiv.org/html/2607.11970#bib.bib9),[23](https://arxiv.org/html/2607.11970#bib.bib23)\]\. This approach is particularly well suited to sparse mmWave and sub\-THz channels, where angular sparsity enables compressed pilots to retain beamforming\-relevant information\[[21](https://arxiv.org/html/2607.11970#bib.bib21),[24](https://arxiv.org/html/2607.11970#bib.bib24)\]\. Accordingly, encoder\-decoder network \(EDN\) architectures can extract task\-relevant features from the pilots and map them directly to beamforming solutions, an approach whose effectiveness has been demonstrated in recent studies\[[19](https://arxiv.org/html/2607.11970#bib.bib19),[6](https://arxiv.org/html/2607.11970#bib.bib6),[25](https://arxiv.org/html/2607.11970#bib.bib25)\]\.

### I\-BIn\-Context Learning

In\-context learning \(ICL\) has emerged as a promising paradigm for multi\-scenario adaptation without parameter updates\[[26](https://arxiv.org/html/2607.11970#bib.bib26),[27](https://arxiv.org/html/2607.11970#bib.bib27)\]\. Originally popularized by large language models, ICL enables a model to infer the underlying task from a few input demonstrations and adapt through forward inference alone\[[28](https://arxiv.org/html/2607.11970#bib.bib28)\]\. Theoretical studies have further shown that self\-attention can implicitly emulate gradient\-based adaptation, providing a principled interpretation of this capability\[[29](https://arxiv.org/html/2607.11970#bib.bib29),[30](https://arxiv.org/html/2607.11970#bib.bib30)\]\.

In wireless communications, ICL has been applied primarily to receiver\-side detection and estimation, where pilot\-based demonstrations enable real\-time adaptation without retraining\[[31](https://arxiv.org/html/2607.11970#bib.bib31),[32](https://arxiv.org/html/2607.11970#bib.bib32)\]\. Recent results further establish Transformer\-based ICL as an optimal in\-context estimator for certain wireless estimation problems\[[33](https://arxiv.org/html/2607.11970#bib.bib33)\]\. Of particular relevance to our work,\[[34](https://arxiv.org/html/2607.11970#bib.bib34)\]extends ICL to transmitter\-side beamforming using demonstration pairs for task adaptation\. Nevertheless, it assumes perfect CSI and requires a large corpus of pre\-computed WMMSE labels, limiting its practicality for large\-scale high\-frequency systems\.

### I\-CCurriculum Learning and Self\-Evolving Mechanism

Our training framework combines curriculum learning \(CL\) with a self\-evolving mechanism\. CL gradually increases the learning difficulty\[[35](https://arxiv.org/html/2607.11970#bib.bib35),[36](https://arxiv.org/html/2607.11970#bib.bib36)\], facilitating a stable optimization and reducing the risk of poor early\-stage convergence\. This is particularly beneficial for beamforming, whose non\-convex sum\-rate objective becomes increasingly difficult under heavier user loads and larger system scales\. Recent studies\[[37](https://arxiv.org/html/2607.11970#bib.bib37),[16](https://arxiv.org/html/2607.11970#bib.bib16)\]have also demonstrated that CL can improve the scalability and generalization of learning\-based beamforming methods\.

Complementing CL, the self\-evolving strategy progressively augments a small seed dataset with high\-quality model\-generated samples\. This idea is related to self\-training and pseudo\-labeling\[[38](https://arxiv.org/html/2607.11970#bib.bib38),[39](https://arxiv.org/html/2607.11970#bib.bib39)\], where carefully designed quality control prevents unreliable predictions from degrading training\[[40](https://arxiv.org/html/2607.11970#bib.bib40)\]\. Similar iterative learning from filtered model generations has also shown promise for large language models\[[41](https://arxiv.org/html/2607.11970#bib.bib41)\]\. In our ICL framework, this mechanism reduces the reliance on expensive pre\-computed beamforming labels while continuously enriching the context distribution\. More importantly, the resulting context dataset provides increasingly diverse and informative demonstrations, thereby improving the model’s generalization and adaptation capabilities\.

### I\-DContributions and Outline

Despite recent progress, existing pilot\-based beamforming methods remain limited in several respects\. Conventional pilot\-to\-beamformer schemes typically employ conventional neural networks tailored to fixed channel models, resulting in limited scalability and cross\-scenario generalization\[[24](https://arxiv.org/html/2607.11970#bib.bib24),[9](https://arxiv.org/html/2607.11970#bib.bib9),[23](https://arxiv.org/html/2607.11970#bib.bib23),[19](https://arxiv.org/html/2607.11970#bib.bib19)\]\. Transformer\-based schemes improve scalability, but generally remain scenario\-specific\[[14](https://arxiv.org/html/2607.11970#bib.bib14),[15](https://arxiv.org/html/2607.11970#bib.bib15)\]; extending them to heterogeneous channel environments often requires additional modules\[[17](https://arxiv.org/html/2607.11970#bib.bib17)\], architectural modifications, or costly retraining and fine\-tuning\[[16](https://arxiv.org/html/2607.11970#bib.bib16)\]\. Moreover, the most relevant ICL\-based beamforming method, ICWLM\[[34](https://arxiv.org/html/2607.11970#bib.bib34)\], assumes perfect CSI and requires a large corpus of pre\-computed WMMSE labels, limiting its applicability to practical pilot\-based high\-frequency systems\. To tackle these issues, we propose a self\-evolving ICL framework with context bootstrapping for direct pilot\-to\-beamformer design in MU\-MISO systems, enabling scalable multi\-model adaptation from limited\-length pilots with substantially fewer pre\-computed labels\. Our main contributions are summarized as follows\.

- •ICL\-Transformer architecture:We develop an enhanced ICL framework that directly generates beamformers from noisy, length\-limited pilot signals without explicit channel estimation\. The proposed architecture integrates pilot EDN and beamformer EDN with an ICL Transformer backbone, allowing the Transformer to reason based on the pilot\-beamformer demonstration pairs in a low\-dimensional feature space\.
- •Curriculum self\-evolving training with context bootstrapping:We design a CL\-based training scheme that gradually transitions from supervised imitation of inexpensive LMMSE labels to unsupervised sum\-rate maximization\. This strategy prevents the early\-stage poor convergence to local optima and enables the model to improve beyond the initial solutions without requiring near\-optimal labels\. To improve the training effect, a self\-evolving strategy that dynamically expands and refines the context datasets is adopted, where model\-generated solutions are selectively admitted into the datasets\. This mechanism reduces the dependence on pre\-computed labels while providing increasingly diverse and informative demonstrations for ICL\.
- •Multi\-model adaptation via context datasets:A single shared ICL network can adapt to multiple channel models through model\-specific context datasets, without adding additional modules or updating network parameters\. We further extend its adaptation space to model\-mismatch settings by incorporating several mismatches into the proposed ICL framework and bypassing explicit channel calibrations\.
- •Performance evaluations:Ablation studies justify the effectiveness of the the proposed ICL\-Transformer architecture and enhanced training strategies\. Extensive simulations demonstrate that the proposed scheme rapidly adapts to both seen and unseen channel models without gradient\-based parameter updates, while effectively mitigating model mismatches through context construction\. Furthermore, our scheme consistently outperforms the existing beamforming schemes under pilot\-based settings, including WMMSE benchmark and recent Transformer\-based methods\.

## IIBackground

### II\-ADownlink Beamforming

Consider a TDD MU\-MISO downlink system, where a base station \(BS\) equipped withNNtransmit antennas servesKKsingle\-antenna users\. The downlink channel from the BS to userkkis denoted by𝒉k∈CN×1\\bm\{h\}\_\{k\}\\in\\mathbb\{C\}^\{N\\times 1\}, and the aggregated channel matrix is𝑯=\[𝒉1,…,𝒉K\]∈CN×K\\bm\{H\}=\[\\bm\{h\}\_\{1\},\\ldots,\\bm\{h\}\_\{K\}\]\\in\\mathbb\{C\}^\{N\\times K\}\. Assume that the BS applies a linear beamformer𝑾=\[𝒘1,…,𝒘K\]∈CN×K\\bm\{W\}=\[\\bm\{w\}\_\{1\},\\ldots,\\bm\{w\}\_\{K\}\]\\in\\mathbb\{C\}^\{N\\times K\}to transmit user symbols𝒂=\[a1,…,aK\]T\\bm\{a\}=\[a\_\{1\},\\ldots,a\_\{K\}\]^\{T\}, whereE​\[‖ak‖2\]=1\\mathbb\{E\}\[\\\|a\_\{k\}\\\|^\{2\}\]=1for allkk\. Then the transmitted signal is given by

𝒙=∑i=1K𝒘i​ai=𝑾​𝒂\.\\displaystyle\\bm\{x\}=\\sum\_\{i=1\}^\{K\}\\bm\{w\}\_\{i\}a\_\{i\}=\\bm\{W\}\\bm\{a\}\.\(1\)Accordingly, the received signal at userkkis

rk=𝒉kH​𝒘k​ak\+∑i≠k𝒉kH​𝒘i​ai\+zk,\\displaystyle r\_\{k\}=\\bm\{h\}\_\{k\}^\{H\}\\bm\{w\}\_\{k\}a\_\{k\}\+\\sum\_\{i\\neq k\}\\bm\{h\}\_\{k\}^\{H\}\\bm\{w\}\_\{i\}a\_\{i\}\+z\_\{k\},\(2\)wherezk∼𝒞​𝒩​\(0,σk2\)z\_\{k\}\\sim\\mathcal\{CN\}\(0,\\sigma\_\{k\}^\{2\}\)denotes the additive noise at userkk\. Hence the achievable sum\-rate is

Rsum​\(𝑯,𝑾\)=∑k=1Klog2⁡\(1\+\|𝒉kH​𝒘k\|2∑i≠k\|𝒉kH​𝒘i\|2\+σk2\)\.\\displaystyle R\_\{\\rm sum\}\(\\bm\{H\},\\bm\{W\}\)=\\sum\_\{k=1\}^\{K\}\\log\_\{2\}\\\!\\left\(1\+\\frac\{\|\\bm\{h\}\_\{k\}^\{H\}\\bm\{w\}\_\{k\}\|^\{2\}\}\{\\sum\_\{i\\neq k\}\|\\bm\{h\}\_\{k\}^\{H\}\\bm\{w\}\_\{i\}\|^\{2\}\+\\sigma\_\{k\}^\{2\}\}\\right\)\.\(3\)Given the channel state information \(CSI\)𝑯\\bm\{H\}, the optimal beamformer is then formulated as

𝑾∗​\(𝑯\)=arg⁡max‖𝑾‖F2⩽P⁡Rsum​\(𝑯,𝑾\),\\displaystyle\\bm\{W\}^\{\*\}\(\\bm\{H\}\)=\\arg\\max\_\{\\\|\\bm\{W\}\\\|\_\{F\}^\{2\}\\leqslant P\}R\_\{\\text\{sum\}\}\(\\bm\{H\},\\bm\{W\}\),\(4\)wherePPis the maximum transmit power budget at the BS\. Typically, the optimal beamformer in \([4](https://arxiv.org/html/2607.11970#S2.E4)\) is hard to compute\. In contrast, the suboptimal LMMSE beamformer𝑾m​\(𝑯\)=\[𝒘1m,…,𝒘Km\]\\bm\{W\}^\{\\text\{m\}\}\(\\bm\{H\}\)=\[\\bm\{w\}^\{\\text\{m\}\}\_\{1\},\\dots,\\bm\{w\}^\{\\text\{m\}\}\_\{K\}\]is much easier to obtain, given by

𝒘km=PK⋅\(𝑰N\+∑i=1KPK​𝒉i​𝒉iH\)−1​𝒉k‖\(𝑰N\+∑i=1KPK​𝒉i​𝒉iH\)−1​𝒉k‖2,k∈\[K\]\.\\displaystyle\\bm\{w\}^\{\\text\{m\}\}\_\{k\}=\\sqrt\{\\frac\{P\}\{K\}\}\\cdot\\frac\{\(\\bm\{I\}\_\{N\}\+\\sum\_\{i=1\}^\{K\}\\frac\{P\}\{K\}\\bm\{h\}\_\{i\}\\bm\{h\}\_\{i\}^\{H\}\)^\{\-1\}\\bm\{h\}\_\{k\}\}\{\\\|\(\\bm\{I\}\_\{N\}\+\\sum\_\{i=1\}^\{K\}\\frac\{P\}\{K\}\\bm\{h\}\_\{i\}\\bm\{h\}\_\{i\}^\{H\}\)^\{\-1\}\\bm\{h\}\_\{k\}\\\|\_\{2\}\},\\ k\\in\[K\]\.\(5\)
In practice, the CSI is not available at the BS\. Instead, the BS only has access to the received uplink pilot signal\. Specifically, let𝚿∈CK×Lp\\bm\{\\Psi\}\\in\\mathbb C^\{K\\times L\_\{p\}\}denote the pilot matrix, whereLpL\_\{p\}is the pilot length\. The received pilot signal at the BS is given by

𝒀=𝑯​𝚿\+𝑵∈CN×Lp,\\displaystyle\\bm\{Y\}=\\bm\{H\}\\bm\{\\Psi\}\+\\bm\{N\}\\in\\mathbb\{C\}^\{N\\times L\_\{p\}\},\(6\)where𝑵∈CN×Lp\\bm\{N\}\\in\\mathbb\{C\}^\{N\\times L\_\{p\}\}denotes the additive white Gaussian noise with i\.i\.d\. entriesni​j∼𝒞​𝒩​\(0,σB2\)n\_\{ij\}\\sim\\mathcal\{CN\}\(0,\\sigma\_\{B\}^\{2\}\)\. Since pilot transmission consumes valuable time\-frequency resources, the pilot length is usually limited, especially in large\-scale high\-frequency systems\. Therefore, communication systems typically operate in the underdetermined regime such thatLp<KL\_\{p\}<K, where the channel matrix cannot be uniquely recovered even in the noiseless case, since any perturbationΔ​𝑯\\Delta\\bm\{H\}satisfyingΔ​𝑯⋅𝚿=𝟎\\Delta\\bm\{H\}\\cdot\\bm\{\\Psi\}=\\bm\{0\}is unobservable from the pilot signal\.

Several learning\-based approaches to beamforming optimization\[[14](https://arxiv.org/html/2607.11970#bib.bib14),[15](https://arxiv.org/html/2607.11970#bib.bib15),[16](https://arxiv.org/html/2607.11970#bib.bib16)\]have been proposed to approximate the optimal beamformer in \([4](https://arxiv.org/html/2607.11970#S2.E4)\), assuming that perfect CSI is available at the BS\. When only the pilot signal is available, existing learning\-based methods\[[42](https://arxiv.org/html/2607.11970#bib.bib42),[43](https://arxiv.org/html/2607.11970#bib.bib43)\]typically follow a two\-stage process: first an estimate𝑯^\\hat\{\\bm\{H\}\}of the channel matrix is obtained based on the pilot signal𝒀\\bm\{Y\}, and then the beamformer is formed using the estimated CSI𝑯^\\hat\{\\bm\{H\}\}\. Moreover, recent works\[[9](https://arxiv.org/html/2607.11970#bib.bib9),[23](https://arxiv.org/html/2607.11970#bib.bib23)\]advocate*direct pilot\-to\-beamformer*learning, where the received pilot signal is mapped directly to the beamformer without explicit channel estimation\. Such an approach can potentially avoid error propagation from the imperfect CSI recovery and use the short pilot signal more efficiently\.

### II\-BIn\-Context Learning

We briefly review the traditional in\-context learning \(ICL\) mechanism and its implementation based on Transformer models\. When a single model is expected to operate across*multiple tasks or scenarios*, a central challenge lies in the rapid adaptation to the specific scenario without explicit supervision or retraining\. ICL addresses this challenge by presenting a few input\-output samples as part of the model input, which serve as the context and allow the model to infer the task/scenario information directly from the input sequence, rather than relying on specific architectural branches or explicit metadata\. Specifically, denote by𝒮\\mathcal\{S\}the set of all scenarios\. For each scenarios∈𝒮s\\in\\mathcal\{S\}, anℓ\\ell\-shot input sequence to the neural network modelℳ𝜽\\mathcal\{M\}\_\{\\bm\{\\theta\}\}is formed as

𝒵sℓ=\[\(𝒛1,f​\(𝒛1\)\),\(𝒛2,f​\(𝒛2\)\),…,\(𝒛ℓ,f​\(𝒛ℓ\)\),𝒛ℓ\+1\],\\mathcal\{Z\}\_\{s\}^\{\\ell\}=\[\(\\bm\{z\}\_\{1\},f\(\\bm\{z\}\_\{1\}\)\),\(\\bm\{z\}\_\{2\},f\(\\bm\{z\}\_\{2\}\)\),\\ldots,\(\\bm\{z\}\_\{\\ell\},f\(\\bm\{z\}\_\{\\ell\}\)\),\\bm\{z\}\_\{\\ell\+1\}\],\(7\)where𝒞s=\[\(𝒛j,f​\(𝒛j\)\)\]j=1ℓ\\mathcal\{C\}\_\{s\}=\[\(\\bm\{z\}\_\{j\},f\(\\bm\{z\}\_\{j\}\)\)\]\_\{j=1\}^\{\\ell\}denotes the input\-output samples, referred to as the*context*,𝒛ℓ\+1\\bm\{z\}\_\{\\ell\+1\}denotes the*query input*, andf​\(⋅\)f\(\\cdot\)denotes the objective function\. The model then generates the*query output*ℳ𝜽​\(𝒵sℓ\)\\mathcal\{M\}\_\{\\bm\{\\theta\}\}\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\)to predict the ground truthf​\(𝒛ℓ\+1\)f\(\\bm\{z\}\_\{\\ell\+1\}\)\.

During training, the model acquires its ICL capability by minimizing the following expected loss over all scenarioss∈𝒮s\\in\\mathcal\{S\}and over all input\-output pairs:

min𝜽​∑s∈𝒮E𝒵sℓ​\[ℒ​\(ℳ𝜽​\(𝒵sℓ\),f​\(𝒛ℓ\+1\)\)\],\\min\_\{\\bm\{\\theta\}\}\\ \\sum\_\{s\\in\\mathcal\{S\}\}\\mathbb\{E\}\_\{\\mathcal\{Z\}\_\{s\}^\{\\ell\}\}\\bigg\[\\mathcal\{L\}\\big\(\\mathcal\{M\}\_\{\\bm\{\\theta\}\}\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\),f\(\\bm\{z\}\_\{\\ell\+1\}\)\\big\)\\bigg\],\(8\)whereℒ​\(⋅,⋅\)\\mathcal\{L\}\(\\cdot,\\cdot\)is a properly chosen loss function, such as the mean squared error \(MSE\) for regression tasks\.

Transformer architectures are particularly suitable for ICL frameworks\[[29](https://arxiv.org/html/2607.11970#bib.bib29),[30](https://arxiv.org/html/2607.11970#bib.bib30)\]\. This is because ICL naturally organizes the input data as a context\-query token sequence, where each token represents either an input or an output sample\. Owing to its strong sequence\-modeling capability, Transformer can effectively process such a context\-query sequence and serves as a query\-output predictor\. In the context of wireless communication applications, recent works\[[32](https://arxiv.org/html/2607.11970#bib.bib32),[31](https://arxiv.org/html/2607.11970#bib.bib31)\]show that ICL Transformers are well suited for receiver\-side functionalities, such as symbol detection\. Furthermore,\[[34](https://arxiv.org/html/2607.11970#bib.bib34)\]demonstrates the potential of ICL Transformers for transmitter\-side optimization, based on perfect CSI and a large labeled dataset\.

## IIIPilot\-based Beamforming

### III\-AProblem Statement

We aim to train a single neural network that handles direct pilot\-to\-beamformer learning across multiple channel models\. Let𝒮\\mathcal\{S\}denote the set of all channel models, where eachs∈𝒮s\\in\\mathcal\{S\}corresponds to a channel distributionpsp\_\{s\}, i\.e\.,𝑯∼ps\\bm\{H\}\\sim p\_\{s\}\. As discussed in Sec\.[II\-A](https://arxiv.org/html/2607.11970#S2.SS1), for a given channel models∈𝒮s\\in\\mathcal\{S\}, aCSI\-basedbeamforming optimization scheme learns a mapping𝑾​\(𝑯\)\\bm\{W\}\(\\bm\{H\}\)that approaches the optimal mapping𝑾∗​\(𝑯\)\\bm\{W\}^\{\*\}\(\\bm\{H\}\)in \([4](https://arxiv.org/html/2607.11970#S2.E4)\), for𝑯∼ps\\bm\{H\}\\sim p\_\{s\}\. On the other hand, to have a single network for multiple channel models, the ICL can be employed\[[34](https://arxiv.org/html/2607.11970#bib.bib34)\]\. In particular, for each channel models∈𝒮s\\in\\mathcal\{S\}, a context dataset𝒱s\\mathcal\{V\}\_\{s\}composed of samples\(𝑯,𝑾∗​\(𝑯\)\)\(\\bm\{H\},\\bm\{W\}^\{\*\}\(\\bm\{H\}\)\)can be precomputed, where𝑯∼ps\\bm\{H\}\\sim p\_\{s\}, and𝑾∗​\(𝑯\)\\bm\{W\}^\{\*\}\(\\bm\{H\}\)is the corresponding near\-optimal WMMSE beamformer\. The network input includes a context set𝒞s\\mathcal\{C\}\_\{s\}consisting ofℓ\\ellsamples from𝒱s\\mathcal\{V\}\_\{s\}, and a new channel𝑯∼ps\\bm\{H\}\\sim p\_\{s\}, and the expected output is the corresponding beamformer𝑾∗​\(𝑯\)\\bm\{W\}^\{\*\}\(\\bm\{H\}\)\. However, such an approach requires computing a large number of near\-optimal beamformer labels, which is computationally expensive, and becomes infeasible for large\-scale MIMO systems\.

In this paper, our goal is to learn a direct mapping from the pilot signal to the beamformer across different channel models\. Note that such apilot\-basedbeamforming cannot be formulated similar to \([4](https://arxiv.org/html/2607.11970#S2.E4)\) with𝑯\\bm\{H\}simply replaced by𝒀\\bm\{Y\}, since unlike \([3](https://arxiv.org/html/2607.11970#S2.E3)\), there is no explicit expression for the sum rate based on the pilot signal𝒀\\bm\{Y\}\.

Nevertheless, a straightforward way is to simply modify the above ICL approach as follows: for each channel models∈𝒮s\\in\\mathcal\{S\}, we draw channel samples𝑯∼ps\\bm\{H\}\\sim p\_\{s\}, and form the corresponding pilot signal𝒀\\bm\{Y\}according to \([6](https://arxiv.org/html/2607.11970#S2.E6)\) and compute the WMMSE beamformer𝑾∗​\(𝑯\)\\bm\{W\}^\{\*\}\(\\bm\{H\}\)\. The context dataset𝒱s\\mathcal\{V\}\_\{s\}is then composed of samples\(𝒀,𝑾∗​\(𝑯\)\)\(\\bm\{Y\},\\bm\{W\}^\{\*\}\(\\bm\{H\}\)\)\. However, such an approach suffers from the same prohibitively high complexity for precomputing the context datasets for large\-scale systems\.

To that end, we propose anenhanced ICL schemebased on the curriculum learning \(CL\) paradigm, in which the model first learns from low\-complexity LMMSE labels to obtain a stable initialization, and then gradually shifts toward unsupervised sum\-rate maximization on query samples\. In particular, the context datasets𝒱s\\mathcal\{V\}\_\{s\},s∈𝒮s\\in\\mathcal\{S\}, are not precomputed, but expanded and refined during the training stage\. Moreover, during training only LMMSE beamformers are computed and no WMMSE beamformer is needed\.

Specifically, during training, for each channel models∈𝒮s\\in\\mathcal\{S\}, we dynamically update a context dataset𝒱s\\mathcal\{V\}\_\{s\}consisting of triplets\(𝑯,𝒀,𝑾^\)\(\\bm\{H\},\\bm\{Y\},\\hat\{\\bm\{W\}\}\)\. For a channel sample𝑯∼ps\\bm\{H\}\\sim p\_\{s\}, the pilot signal𝒀\\bm\{Y\}is generated according to \([6](https://arxiv.org/html/2607.11970#S2.E6)\), and the beamformer solution𝑾^\\hat\{\\bm\{W\}\}is either an LMMSE beamformer computed by \([5](https://arxiv.org/html/2607.11970#S2.E5)\), or a model\-generated beamformer\. In particular, initially, we set

𝒱\(0\)=⋃s∈𝒮𝒱s\(0\),𝒱s\(0\)=\{\(𝑯j,𝒀j,𝑾jm\)\}j=1M0,\\displaystyle\\mathcal\{V\}^\{\(0\)\}=\\bigcup\_\{s\\in\\mathcal\{S\}\}\\mathcal\{V\}\_\{s\}^\{\(0\)\},\\ \\mathcal\{V\}\_\{s\}^\{\(0\)\}=\\bigl\\\{\(\\bm\{H\}\_\{j\},\\bm\{Y\}\_\{j\},\\bm\{W\}\_\{j\}^\{\\text\{m\}\}\)\\bigr\\\}\_\{j=1\}^\{M\_\{0\}\},\(9\)where𝒀j\\bm\{Y\}\_\{j\}and𝑾jm\\bm\{W\}^\{\\text\{m\}\}\_\{j\}are the corresponding pilot signal and LMMSE beamformer based on the channel sample𝑯j∼ps\\bm\{H\}\_\{j\}\\sim p\_\{s\}\. Following the ICL formulation in \([7](https://arxiv.org/html/2607.11970#S2.E7)\) and \([8](https://arxiv.org/html/2607.11970#S2.E8)\), the context is formed by randomly selectingℓ\\ellpilot\-beamformer demonstration pairs from𝒱s\\mathcal\{V\}\_\{s\}:

𝒞s=\[\(𝒀j,𝑾^j\)\]j=1ℓ\.\\displaystyle\\mathcal\{C\}\_\{s\}=\[\(\\bm\{Y\}\_\{j\},\\hat\{\\bm\{W\}\}\_\{j\}\)\]\_\{j=1\}^\{\\ell\}\.\(10\)Given a new query input𝒀\\bm\{Y\}, the context\-query sequence is written as𝒵sℓ=\[𝒞s,𝒀\]\\mathcal\{Z\}\_\{s\}^\{\\ell\}=\[\\mathcal\{C\}\_\{s\},\\bm\{Y\}\]\. Then the context\-conditioned pilot\-to\-beamformer mapping can be represented by a neural networkℳ𝜽\\mathcal\{M\}\_\{\\bm\{\\theta\}\}:

𝑾=ℳ𝜽​\(𝒵sℓ\)\.\\displaystyle\\bm\{W\}=\\mathcal\{M\}\_\{\\bm\{\\theta\}\}\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\)\.\(11\)Then the ultimate objective is to maximize the expected sum\-rate over all channel models:

max𝜽​∑s∈𝒮E𝑯∼ps,𝑵𝒀=𝑯​𝚿\+𝑵𝒞s∈𝒱s​\(𝜽\)​\[Rsum​\(𝑯,ℳ𝜽​\(𝒵sℓ\)\)\]\.\\displaystyle\\max\_\{\\bm\{\\theta\}\}\\ \\sum\_\{s\\in\\mathcal\{S\}\}\\ \\mathbb E\_\{\\begin\{subarray\}\{c\}\\bm\{H\}\\sim p\_\{s\},\\bm\{N\}\\\\ \\bm\{Y\}=\\bm\{H\}\\bm\{\\Psi\}\+\\bm\{N\}\\\\ \\mathcal\{C\}\_\{s\}\\in\\mathcal\{V\}\_\{s\}\(\\bm\{\\theta\}\)\\end\{subarray\}\}\\left\[R\_\{\\text\{sum\}\}\\left\(\\bm\{H\},\\mathcal\{M\}\_\{\\bm\{\\theta\}\}\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\)\\right\)\\right\]\.\(12\)Note that the context dataset𝒱s\\mathcal\{V\}\_\{s\}depends on the network parameters𝜽\\bm\{\\theta\}and therefore is dynamically updated during training, as will be detailed in Sec\.[IV](https://arxiv.org/html/2607.11970#S4)\. Starting from these initial datasets, we adopt a CL\-based training scheme that combines a supervised MSE loss with an unsupervised sum\-rate loss as the objective\. At the same time, starting from𝒱s\(0\)\\mathcal\{V\}\_\{s\}^\{\(0\)\}, each context dataset𝒱s\\mathcal\{V\}\_\{s\}is expanded and updated to improve its quality\. Consequently, the training procedure produces both a trained ICL model and a context dataset𝒱s\\mathcal\{V\}\_\{s\}for each channel models∈𝒮s\\in\\mathcal\{S\}\.

*During inference*, given the received pilot signal𝒀\\bm\{Y\}and its underlying channel modelss, a context𝒞s\\mathcal\{C\}\_\{s\}is sampled from the dataset𝒱s\\mathcal\{V\}\_\{s\}, and the beamformer solution is generated by a single forward pass𝑾=ℳ𝜽∗​\(𝒵sℓ\)\\bm\{W\}=\\mathcal\{M\}\_\{\\bm\{\\theta\}^\{\*\}\}\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\), where𝒵sℓ=\[𝒞s,𝒀\]\\mathcal\{Z\}\_\{s\}^\{\\ell\}=\[\\mathcal\{C\}\_\{s\},\\bm\{Y\}\], andℳ𝜽∗\\mathcal\{M\}\_\{\\bm\{\\theta\}^\{\*\}\}is the trained model\.

### III\-BOverall Network Architecture

![Refer to caption](https://arxiv.org/html/2607.11970v1/icl_nn_architecture_4.png)Figure 1:The network architecture of the proposed ICL beamforming framework\.We design a network architecture tailored to the ICL beamformer learning problem described in Sec\.[III\-A](https://arxiv.org/html/2607.11970#S3.SS1), as illustrated in Fig\.[1](https://arxiv.org/html/2607.11970#S3.F1)\. To effectively capture the features of context samples, a pretrained pilot encoder network and a pretrained beamformer encoder network are adopted to compress the pilot signals and beamformers into the same\-size low\-dimensional tokens, respectively\. This also reduces the subsequent computation costs and enables efficient context\-query processing\. The resulting compressed token sequence is then fed into a Transformer backbone, and finally the beamformer solution is generated by a beamformer decoder network\.

### III\-CEDN Pretraining

We use a convolutional encoder network with attention pooling to compress the pilot signal, aiming to exploit its structural correlations induced by sparse channels and adaptively aggregates the informative features into a compact token\. On the other hand, a simple MLP network is adopted to compress the beamformer label since it is a dense optimization outputs with no clear local structures\.

#### III\-C1Pilot EDN

We first describe the pilot EDN architecture\. Given the received pilot signal𝒀∈CN×Lp\\bm\{Y\}\\in\\mathbb\{C\}^\{N\\times L\_\{p\}\}, we stack its real and imaginary parts to obtain𝒚=vec​\(\[ℜ⁡\(𝒀\);ℑ⁡\(𝒀\)\]\)∈R2​N​Lp\\bm\{y\}=\\text\{vec\}\(\[\\Re\(\\bm\{Y\}\);\\,\\Im\(\\bm\{Y\}\)\]\)\\in\\mathbb\{R\}^\{2NL\_\{p\}\}\. As shown in Fig\.[1](https://arxiv.org/html/2607.11970#S3.F1), the pilot encoderℰp:R2​N​Lp→Rd\\mathcal\{E\}\_\{p\}:\\mathbb\{R\}^\{2NL\_\{p\}\}\\to\\mathbb\{R\}^\{d\}maps𝒚\\bm\{y\}to a compact pilot token𝒛∈Rd\\bm\{z\}\\in\\mathbb\{R\}^\{d\}\. The encoder network consists of two 1D convolutional modules, an attention\-pooling module, and a two\-layer MLP module, where each MLP layer includes a fully\-connected \(FC\) component, a GELU activation component, and a LayerNorm component\. To pretrain the pilot encoder, we employ another three\-layer MLP network as an auxiliary channel decoder𝒟c:Rd→R2​K​N\\mathcal\{D\}\_\{c\}:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{2KN\}, which reconstructs the real\-valued channel vector𝒉\\bm\{h\}from the pilot token𝒛\\bm\{z\}\. Note that this auxiliary decoder is used only during pretraining to encourage the pilot encoder to extract channel\-informative latent features from compressed pilot signals, instead of performing explicit CSI reconstructions\.

We then pretrain the pilot EDN using the multi\-scenario labeled datasets in \([9](https://arxiv.org/html/2607.11970#S3.E9)\), which allow the encoder networks to learn shared token representations across different channel models\. Denote the trainable parameters of the pilot encoder and the auxiliary channel decoder asϕp\\bm\{\\phi\}\_\{p\}and𝝍c\\bm\{\\psi\}\_\{c\}, respectively\. The pilot EDN is trained by minimizing the following channel reconstruction loss

minϕp,𝝍c​∑s∈𝒮E\(𝑯,𝒀\)∈𝒱s\(0\)​‖𝒉−𝒟c​\(ℰp​\(𝒚\)\)‖22,\\displaystyle\\min\_\{\\bm\{\\phi\}\_\{p\},\\bm\{\\psi\}\_\{c\}\}\\ \\sum\_\{s\\in\\mathcal\{S\}\}\\mathbb\{E\}\_\{\(\\bm\{H\},\\bm\{Y\}\)\\in\\mathcal\{V\}\_\{s\}^\{\(0\)\}\}\\bigl\\\|\\bm\{h\}\-\\mathcal\{D\}\_\{c\}\\bigl\(\\mathcal\{E\}\_\{p\}\(\\bm\{y\}\)\\bigr\)\\bigr\\\|\_\{2\}^\{2\},\(13\)where𝒉=vec​\(\[ℜ⁡\(𝑯\);ℑ⁡\(𝑯\)\]\)∈R2​N​K\\bm\{h\}=\\text\{vec\}\(\[\\Re\(\\bm\{H\}\);\\Im\(\\bm\{H\}\)\]\)\\in\\mathbb\{R\}^\{2NK\}and𝒚=vec​\(\[ℜ⁡\(𝒀\);ℑ⁡\(𝒀\)\]\)∈R2​N​Lp\\bm\{y\}=\\text\{vec\}\(\[\\Re\(\\bm\{Y\}\);\\Im\(\\bm\{Y\}\)\]\)\\in\\mathbb\{R\}^\{2NL\_\{p\}\}\. After convergence, the trained channel decoder𝒟c∗\\mathcal\{D\}^\{\*\}\_\{c\}is discarded while the trained pilot encoderℰp∗\\mathcal\{E\}^\{\*\}\_\{p\}is frozen and used for pilot\-token construction in the subsequent training stages\.

#### III\-C2Beamformer EDN

We now describe the beamformer EDN architecture\. Given a beamformer matrix𝑾∈CN×K\\bm\{W\}\\in\\mathbb C^\{N\\times K\}, we stack its real and imaginary parts to obtain𝒘=vec​\(\[ℜ⁡\(𝑾\);ℑ⁡\(𝑾\)\]\)∈R2​N​Lp\\bm\{w\}=\\text\{vec\}\(\[\\Re\(\\bm\{W\}\);\\,\\Im\(\\bm\{W\}\)\]\)\\in\\mathbb\{R\}^\{2NL\_\{p\}\}\. As shown in Fig\.[1](https://arxiv.org/html/2607.11970#S3.F1), the beamformer encoderℰb:R2​N​K→Rd\\mathcal\{E\}\_\{b\}:\\mathbb R^\{2NK\}\\rightarrow\\mathbb R^\{d\}compresses𝒘\\bm\{w\}into a beamformer token, i\.e\.,𝒄=ℰb​\(𝒘\)∈Rd\\bm\{c\}=\\mathcal\{E\}\_\{b\}\(\\bm\{w\}\)\\in\\mathbb R^\{d\}\. Both encoderℰb\\mathcal\{E\}\_\{b\}and decoder𝒟b\\mathcal\{D\}\_\{b\}are implemented as three\-layer MLP networks\. The beamformer decoder𝒟b:Rd→R2​N​K\\mathcal\{D\}\_\{b\}:\\mathbb R^\{d\}\\rightarrow\\mathbb R^\{2NK\}then maps a compressed beamformer token back to the real\-valued beamformer vector\. The reconstructed vector is first normalized to satisfy the transmit power constraint‖𝒘‖2=P\\\|\\bm\{w\}\\\|^\{2\}=Pand then reshaped into a complex beamformer matrix𝑾\\bm\{W\}\.

The pretraining of beamformer EDN is similar to that of pilot EDN\. Denote the trainable parameters of the beamformer encoder and beamformer decoder asϕb\\bm\{\\phi\}\_\{b\}and𝝍b\\bm\{\\psi\}\_\{b\}, respectively\. The beamformer EDN is pretrained by minimizing the reconstruction loss

minϕb,𝝍b​∑s∈𝒮E𝑾m∈𝒱s\(0\)​‖𝒘m−𝒟b​\(ℰb​\(𝒘m\)\)‖2,\\displaystyle\\min\_\{\\bm\{\\phi\}\_\{b\},\\bm\{\\psi\}\_\{b\}\}\\ \\sum\_\{s\\in\\mathcal\{S\}\}\\mathbb\{E\}\_\{\\bm\{W\}^\{\\text\{m\}\}\\in\\mathcal\{V\}\_\{s\}^\{\(0\)\}\}\\bigl\\\|\\bm\{w\}^\{\\text\{m\}\}\-\\mathcal\{D\}\_\{b\}\\bigl\(\\mathcal\{E\}\_\{b\}\(\\bm\{w\}^\{\\text\{m\}\}\)\\bigr\)\\bigr\\\|^\{2\},\(14\)where𝒘m=vec​\(\[ℜ⁡\(𝑾m\);ℑ⁡\(𝑾m\)\]\)\\bm\{w\}^\{\\text\{m\}\}=\\text\{vec\}\(\[\\Re\(\\bm\{W\}^\{\\text\{m\}\}\);\\Im\(\\bm\{W\}^\{\\text\{m\}\}\)\]\)\. After convergence, both the trained beamformer encoderℰ~b\\tilde\{\\mathcal\{E\}\}\_\{b\}and the trained beamformer decoder𝒟~b\\tilde\{\\mathcal\{D\}\}\_\{b\}are retained as the initializations for the subsequent training stages\.

## IVEnhanced ICL for Multi\-Model Pilot\-based Beamforming

Building on the pretrained EDNs, we now present the enhanced ICL framework by first detailing the ICL Transformer architecture and then presenting our proposed curriculum self\-evolving training scheme\.

### IV\-AICL Transformer Network

For each pilot\-beamformer sample\(𝒀j,𝑾^j\)\(\\bm\{Y\}\_\{j\},\\hat\{\\bm\{W\}\}\_\{j\}\)in the context dataset, the pilot encoder \(pretrained and frozen\) and beamformer encoder \(remain trainable\) produce the compressed tokens as follows

𝒛j=ℰp∗​\(𝒀j\)∈Rd,𝒄^j=ℰb​\(𝑾^j\)∈Rd\.\\displaystyle\\bm\{z\}\_\{j\}=\\mathcal\{E\}^\{\*\}\_\{p\}\(\\bm\{Y\}\_\{j\}\)\\in\\mathbb\{R\}^\{d\},\\quad\\hat\{\\bm\{c\}\}\_\{j\}=\\mathcal\{E\}\_\{b\}\(\\hat\{\\bm\{W\}\}\_\{j\}\)\\in\\mathbb\{R\}^\{d\}\.\(15\)Given a query input𝒀ℓ\+1\\bm\{Y\}\_\{\\ell\+1\}, its pilot token is denoted by𝒛ℓ\+1=ℰp∗​\(𝒀ℓ\+1\)\\bm\{z\}\_\{\\ell\+1\}=\\mathcal\{E\}^\{\*\}\_\{p\}\(\\bm\{Y\}\_\{\\ell\+1\}\)\. The ICL token sequence is then constructed as

𝒵ℓ=\[\(𝒛1,𝒄^1\),\(𝒛2,𝒄^2\),…,\(𝒛ℓ,𝒄^ℓ\),𝒛ℓ\+1\]∈Rd×Lseq,\\displaystyle\\mathcal\{Z\}^\{\\ell\}=\[\(\\bm\{z\}\_\{1\},\\hat\{\\bm\{c\}\}\_\{1\}\),\(\\bm\{z\}\_\{2\},\\hat\{\\bm\{c\}\}\_\{2\}\),\\ldots,\(\\bm\{z\}\_\{\\ell\},\\hat\{\\bm\{c\}\}\_\{\\ell\}\),\\bm\{z\}\_\{\\ell\+1\}\]\\in\\mathbb R^\{d\\times L\_\{\\text\{seq\}\}\},\(16\)where\{\(𝒛j,𝒄^j\)\}j=1ℓ\\\{\(\\bm\{z\}\_\{j\},\\hat\{\\bm\{c\}\}\_\{j\}\)\\\}\_\{j=1\}^\{\\ell\}are the context tokens and𝒛ℓ\+1\\bm\{z\}\_\{\\ell\+1\}is the query input token\. The sequence length isLseq≜2​ℓ\+1L\_\{\\text\{seq\}\}\\triangleq 2\\ell\+1\.

A Transformer network withTTlayers is employed to process the token sequence𝒵ℓ\\mathcal\{Z\}^\{\\ell\}in \([16](https://arxiv.org/html/2607.11970#S4.E16)\)\. First, each token is projected to a higher\-dimensional embedding space:

𝑿\(0\)=𝓦in​𝒵ℓ\+𝒃in​𝟏LseqT∈Rα×Lseq,\\displaystyle\\bm\{X\}^\{\(0\)\}=\\bm\{\\mathcal\{W\}\}\_\{\\mathrm\{in\}\}\\mathcal\{Z\}^\{\\ell\}\+\\bm\{b\}\_\{\\mathrm\{in\}\}\\bm\{1\}\_\{L\_\{\\text\{seq\}\}\}^\{T\}\\in\\mathbb\{R\}^\{\\alpha\\times L\_\{\\text\{seq\}\}\},\(17\)where the dimensionα\>d\\alpha\>d,𝟏Lseq\\bm\{1\}\_\{L\_\{\\text\{seq\}\}\}is an all\-one vector,𝓦in∈Rα×d\\bm\{\\mathcal\{W\}\}\_\{\\mathrm\{in\}\}\\in\\mathbb\{R\}^\{\\alpha\\times d\}and𝒃in∈Rα\\bm\{b\}\_\{\\mathrm\{in\}\}\\in\\mathbb\{R\}^\{\\alpha\}are learnable parameters\. Starting from𝑿\(0\)\\bm\{X\}^\{\(0\)\}, the hidden states are updated fort=1,…,Tt=1,\\ldots,Tas

𝑿¯\(t−1\)\\displaystyle\\bar\{\\bm\{X\}\}^\{\(t\-1\)\}=LayerNorm​\(𝑿\(t−1\)\),\\displaystyle=\\mathrm\{LayerNorm\}\\bigl\(\\bm\{X\}^\{\(t\-1\)\}\\bigr\),\(18\)𝒁\(t\)\\displaystyle\{\\bm\{Z\}\}^\{\(t\)\}=𝑿\(t−1\)\+MHSA\(t\)​\(𝑿¯\(t−1\)\),\\displaystyle=\\bm\{X\}^\{\(t\-1\)\}\+\\mathrm\{MHSA\}^\{\(t\)\}\\\!\\bigl\(\\bar\{\\bm\{X\}\}^\{\(t\-1\)\}\\bigr\),\(19\)𝒁¯\(t\)\\displaystyle\\bar\{\\bm\{Z\}\}^\{\(t\)\}=LayerNorm​\(𝒁\(t\)\),\\displaystyle=\\mathrm\{LayerNorm\}\\bigl\(\{\\bm\{Z\}\}^\{\(t\)\}\\bigr\),\(20\)𝑿\(t\)\\displaystyle\\bm\{X\}^\{\(t\)\}=𝒁\(t\)\+MLP\(t\)​\(𝒁¯\(t\)\)\.\\displaystyle=\{\\bm\{Z\}\}^\{\(t\)\}\+\\mathrm\{MLP\}^\{\(t\)\}\\\!\\bigl\(\\bar\{\\bm\{Z\}\}^\{\(t\)\}\\bigr\)\.\(21\)We next specify the multi\-head self\-attention module in \([19](https://arxiv.org/html/2607.11970#S4.E19)\) at thettht^\{\\text\{th\}\}layer\. For theethe^\{\\text\{th\}\}attention head, the query, key, and value matrices are computed as

𝑸e\(t\)\\displaystyle\\bm\{Q\}\_\{e\}^\{\(t\)\}=\(𝑿¯\(t−1\)\)T​𝑹eQ,\(t\),\\displaystyle=\(\\bar\{\\bm\{X\}\}^\{\(t\-1\)\}\)^\{T\}\\bm\{R\}\_\{e\}^\{Q,\(t\)\},𝑲e\(t\)\\displaystyle\\bm\{K\}\_\{e\}^\{\(t\)\}=\(𝑿¯\(t−1\)\)T​𝑹eK,\(t\),\\displaystyle=\(\\bar\{\\bm\{X\}\}^\{\(t\-1\)\}\)^\{T\}\\bm\{R\}\_\{e\}^\{K,\(t\)\},𝑽e\(t\)\\displaystyle\\bm\{V\}\_\{e\}^\{\(t\)\}=\(𝑿¯\(t−1\)\)T​𝑹eV,\(t\),\\displaystyle=\(\\bar\{\\bm\{X\}\}^\{\(t\-1\)\}\)^\{T\}\\bm\{R\}\_\{e\}^\{V,\(t\)\},\(22\)where𝑹eQ,\(t\),𝑹eK,\(t\),𝑹eV,\(t\)∈Rα×De\\bm\{R\}\_\{e\}^\{Q,\(t\)\},\\bm\{R\}\_\{e\}^\{K,\(t\)\},\\bm\{R\}\_\{e\}^\{V,\(t\)\}\\in\\mathbb\{R\}^\{\\alpha\\times D\_\{e\}\}denote the learnable weight matrices,De=D/ED\_\{e\}=D/Edenotes the head dimension, andEEdenotes the number of attention heads\. The attention output of theethe^\{\\text\{th\}\}head is computed as

𝑨e\(t\)=softmax​\(𝑸e\(t\)​\(𝑲e\(t\)\)TDe\)​𝑽e\(t\)∈RLseq×De\.\\displaystyle\\bm\{A\}\_\{e\}^\{\(t\)\}=\\mathrm\{softmax\}\\left\(\\frac\{\\bm\{Q\}\_\{e\}^\{\(t\)\}\(\\bm\{K\}\_\{e\}^\{\(t\)\}\)^\{T\}\}\{\\sqrt\{D\_\{e\}\}\}\\right\)\\bm\{V\}\_\{e\}^\{\(t\)\}\\ \\in\\ \\mathbb\{R\}^\{L\_\{\\text\{seq\}\}\\times D\_\{e\}\}\.\(23\)The multi\-head attention output is obtained by concatenating all heads and projecting the result back to the embedding dimension:

MHSA​\(𝑿¯\(t−1\)\)=𝑹proj\(t\)​\[𝑨1\(t\),…,𝑨E\(t\)\]T,\\displaystyle\\mathrm\{MHSA\}\(\\bar\{\\bm\{X\}\}^\{\(t\-1\)\}\)=\\bm\{R\}\_\{\\text\{proj\}\}^\{\(t\)\}\\ \[\\bm\{A\}\_\{1\}^\{\(t\)\},\\ldots,\\bm\{A\}\_\{E\}^\{\(t\)\}\]^\{T\},\(24\)where𝑹proj\(t\)∈Rα×D\\bm\{R\}\_\{\\text\{proj\}\}^\{\(t\)\}\\in\\mathbb\{R\}^\{\\alpha\\times D\}is the learnable output projection matrix\.

After the final Transformer layer, the hidden state is vectorized and mapped to the predicted query beamformer token:

𝒙\(T\)\\displaystyle\\bm\{x\}^\{\(T\)\}=vec​\(𝑿\(T\)\)∈Rα​Lseq,\\displaystyle=\\text\{vec\}\(\\bm\{X\}^\{\(T\)\}\)\\in\\mathbb\{R\}^\{\\alpha L\_\{\\text\{seq\}\}\},𝒄\\displaystyle\\bm\{c\}=𝓦o⋅𝒙\(T\)\+𝒃o∈Rd,\\displaystyle=\\bm\{\\mathcal\{W\}\}\_\{\\mathrm\{o\}\}\\cdot\\bm\{x\}^\{\(T\)\}\+\\bm\{b\}\_\{\\mathrm\{o\}\}\\in\\mathbb\{R\}^\{d\},\(25\)where𝓦o∈Rd×α​Lseq\\bm\{\\mathcal\{W\}\}\_\{\\mathrm\{o\}\}\\in\\mathbb\{R\}^\{d\\times\\alpha L\_\{\\text\{seq\}\}\}and𝒃o∈Rd\\bm\{b\}\_\{\\text\{o\}\}\\in\\mathbb\{R\}^\{d\}are learnable parameters\. The predicted token𝒄\\bm\{c\}is then decoded into a real\-valued beamformer vector:

𝒘=𝒟b​\(𝒄\)∈R2​N​K\.\\displaystyle\\bm\{w\}=\\mathcal\{D\}\_\{b\}\(\\bm\{c\}\)\\in\\mathbb\{R\}^\{2NK\}\.\(26\)Finally,𝒘\\bm\{w\}is first normalized by𝒘=P⋅𝒘‖𝒘‖\\bm\{w\}=\\frac\{\\sqrt\{P\}\\cdot\\bm\{w\}\}\{\\\|\\bm\{w\}\\\|\}, and then reshaped into a complex beamformer matrix:

𝑾\\displaystyle\\bm\{W\}=vec−1\(𝒘\[1:NK\]\)\+j⋅vec−1\(𝒘\[NK\+1:2NK\]\)\.\\displaystyle=\\text\{vec\}^\{\-1\}\(\\bm\{w\}\[1:NK\]\)\+j\\cdot\\text\{vec\}^\{\-1\}\(\\bm\{w\}\[NK\+1:2NK\]\)\.\(27\)
Importantly, the Transformer blocks do not share parameters across layers\. We first collect the learnable parameters in thettht^\{\\text\{th\}\}Transformer layer as

𝜽t=\{\\displaystyle\\bm\{\\theta\}\_\{t\}=\\Bigl\\\{\{𝑹eQ,\(t\),𝑹eK,\(t\),𝑹eV,\(t\)\}e=1E,𝑹proj\(t\),\\displaystyle\\\{\\bm\{R\}\_\{e\}^\{Q,\(t\)\},\\bm\{R\}\_\{e\}^\{K,\(t\)\},\\bm\{R\}\_\{e\}^\{V,\(t\)\}\\\}\_\{e=1\}^\{E\},\\bm\{R\}\_\{\\text\{proj\}\}^\{\(t\)\},\(28\)𝜷MLP\(t\),𝜷LN,1\(t\),𝜷LN,2\(t\)\},\\displaystyle\\bm\{\\beta\}\_\{\\rm MLP\}^\{\(t\)\},\\bm\{\\beta\}\_\{\{\\rm LN\},1\}^\{\(t\)\},\\bm\{\\beta\}\_\{\{\\rm LN\},2\}^\{\(t\)\}\\Bigr\\\},where𝜷MLP\(t\)\\bm\{\\beta\}\_\{\\rm MLP\}^\{\(t\)\}denotes the parameters of the MLP network in \([21](https://arxiv.org/html/2607.11970#S4.E21)\),𝜷LN,1\(t\)\\bm\{\\beta\}\_\{\{\\rm LN\},1\}^\{\(t\)\}and𝜷LN,2\(t\)\\bm\{\\beta\}\_\{\{\\rm LN\},2\}^\{\(t\)\}denote the learnable affine parameters of the two LayerNorm modules in \([18](https://arxiv.org/html/2607.11970#S4.E18)\) and \([20](https://arxiv.org/html/2607.11970#S4.E20)\)\. The complete set of learnable parameters in the ICL Transformer is then given by

𝜽=\{\\displaystyle\\bm\{\\theta\}=\\Bigl\\\{𝓦in,𝒃in,𝓦o,𝒃o\}∪⋃t=1T𝜽t\.\\displaystyle\\bm\{\\mathcal\{W\}\}\_\{\\rm in\},\\bm\{b\}\_\{\\rm in\},\\bm\{\\mathcal\{W\}\}\_\{\\rm o\},\\bm\{b\}\_\{\\rm o\}\\Bigr\\\}\\cup\\bigcup\_\{t=1\}^\{T\}\\bm\{\\theta\}\_\{t\}\.\(29\)
For the beamformer encoder and decoder networks, the learnable parameters are given as follows

ϕb=⋃i=13\{𝝃FC\(i\)\},𝝍b=⋃i=13\{𝜻FC\(i\)\},\\displaystyle\\bm\{\\phi\}\_\{b\}=\\bigcup\_\{i=1\}^\{3\}\\Big\\\{\\bm\{\\xi\}^\{\(i\)\}\_\{\\text\{FC\}\}\\Big\\\},\\ \\bm\{\\psi\}\_\{b\}=\\bigcup\_\{i=1\}^\{3\}\\Big\\\{\\bm\{\\zeta\}^\{\(i\)\}\_\{\\text\{FC\}\}\\Big\\\},\(30\)where𝝃FC\(i\)\\bm\{\\xi\}^\{\(i\)\}\_\{\\text\{FC\}\}denotes the parameters of theithi^\{\\text\{th\}\}linear layer of the beamformer encoder, and𝜻FC\(i\)\\bm\{\\zeta\}^\{\(i\)\}\_\{\\text\{FC\}\}denotes the parameters of theithi^\{\\text\{th\}\}linear layer of the beamformer decoder\.

### IV\-BCurriculum Learning with Context Bootstrapping

As discussed in Sec\.[III\-A](https://arxiv.org/html/2607.11970#S3.SS1), curriculum learning \(CL\) plays a central role in the proposed enhanced ICL framework\. It addresses two challenges in ICL beamforming: highly non\-convex optimization and efficient construction of informative context datasets\. First, CL provides a smooth transition from supervised imitation to unsupervised sum\-rate maximization\. This transition anchors the model in a well\-initialized region in the early stage and mitigates poor convergence to local optima in the sum\-rate maximization\. Second, the self\-evolving context dataset reduces the need for a large precomputed labeled dataset\. Instead of relying on fixed beamformer labels, the context dataset is adaptively expanded and refined during training\. As high\-quality model\-generated samples are gradually incorporated, the context distribution becomes broader and more informative, enabling the ICL model to condition on more diverse demonstrations and generalize better across the problem family\.

Based on the architecture in Fig\.[1](https://arxiv.org/html/2607.11970#S3.F1), the complete training procedure consists of three stages: EDN pretraining, supervised warm\-start, and CL with evolving context\. The EDN pretraining has been introduced in Sec\.[III\-C](https://arxiv.org/html/2607.11970#S3.SS3)\. In the following, with the pretrained pilot encoder fixed, we focus on the joint training of the ICL Transformer and the beamformer EDN through the latter two stages\.

#### IV\-B1Supervised Warm\-Start

The*supervised warm\-start*training stage jointly trains the ICL Transformer, the beamformer encoder and decoder, by using the LMMSE\-labeled context dataset𝒱\(0\)\\mathcal\{V\}^\{\(0\)\}in \([9](https://arxiv.org/html/2607.11970#S3.E9)\)\. This stage provides a stable initialization before introducing the non\-convex sum\-rate objective\.

Each training batch contains the same number of samples from all channel modelss∈𝒮s\\in\\mathcal\{S\}\. For each training sample from modelss, we randomly selectℓ\\ellpilot\-beamformer pairs from𝒱s\(0\)\\mathcal\{V\}\_\{s\}^\{\(0\)\}to form the context𝒞s=\[\(𝒀j,𝑾jm\)\]j=1ℓ\\mathcal\{C\}\_\{s\}=\[\(\\bm\{Y\}\_\{j\},\\bm\{W\}^\{\\text\{m\}\}\_\{j\}\)\]\_\{j=1\}^\{\\ell\}, and select another\(𝒀ℓ\+1,𝑾ℓ\+1m\)∈𝒱s\(0\)\(\\bm\{Y\}\_\{\\ell\+1\},\\bm\{W\}^\{\\text\{m\}\}\_\{\\ell\+1\}\)\\in\\mathcal\{V\}\_\{s\}^\{\(0\)\}as the query pair111The query pair should be excluded from the context pairs for all training samples to prevent data\-leakage\.\. Following \([15](https://arxiv.org/html/2607.11970#S4.E15)\) and \([16](https://arxiv.org/html/2607.11970#S4.E16)\), the corresponding compressed token sequence is constructed as𝒵sℓ\\mathcal\{Z\}\_\{s\}^\{\\ell\}\. The token sequence is then processed by the ICL Transformer, and the predicted beamformer token is decoded by the beamformer decoder:

𝒄=𝒯b​\(𝒵sℓ\),𝒘=𝒟b​\(𝒄\),\\displaystyle\\bm\{c\}=\\mathcal\{T\}\_\{b\}\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\),\\ \\bm\{w\}=\\mathcal\{D\}\_\{b\}\(\\bm\{c\}\),\(31\)where𝒯b\\mathcal\{T\}\_\{b\}denotes the ICL Transformer with learnable parameters𝜽\\bm\{\\theta\}in \([29](https://arxiv.org/html/2607.11970#S4.E29)\), and the corresponding reshaped and power\-normalized beamformer matrix is denoted by𝑾​\(𝒀ℓ\+1,𝒞s\)\\bm\{W\}\(\\bm\{Y\}\_\{\\ell\+1\},\\mathcal\{C\}\_\{s\}\)\. After randomly initializing𝒯b\\mathcal\{T\}\_\{b\}and initializing\(ℰb,𝒟b\)\(\\mathcal\{E\}\_\{b\},\\mathcal\{D\}\_\{b\}\)by the pretrained models\(ℰ~b,𝒟~b\)\(\\tilde\{\\mathcal\{E\}\}\_\{b\},\\tilde\{\\mathcal\{D\}\}\_\{b\}\)obtained in Sec\.[III\-C2](https://arxiv.org/html/2607.11970#S3.SS3.SSS2), the following supervised training across all channel models is performed:

min𝜽,ϕb,𝝍b​∑s∈𝒮E\{\(𝒀j,𝑾jm\)\}j=1ℓ\+1∈𝒱s\(0\)​‖𝑾​\(𝒀ℓ\+1,𝒞s\)−𝑾ℓ\+1m‖F2\.\\displaystyle\\min\_\{\\bm\{\\theta\},\\bm\{\\phi\}\_\{b\},\\bm\{\\psi\}\_\{b\}\}\\ \\sum\_\{s\\in\\mathcal\{S\}\}\\mathbb\{E\}\_\{\\\{\(\\bm\{Y\}\_\{j\},\\bm\{W\}^\{\\text\{m\}\}\_\{j\}\)\\\}\_\{j=1\}^\{\\ell\+1\}\\in\\mathcal\{V\}\_\{s\}^\{\(0\)\}\}\\bigl\\\|\\bm\{W\}\(\\bm\{Y\}\_\{\\ell\+1\},\\mathcal\{C\}\_\{s\}\)\-\\bm\{W\}^\{\\text\{m\}\}\_\{\\ell\+1\}\\bigr\\\|\_\{F\}^\{2\}\.\(32\)By imitating the LMMSE beamformers, a reliable initialization for the subsequent CL\-based training is provided\.

#### IV\-B2CL with Context Bootstrapping

After the supervised warm\-start, the learning model gradually transitions to the sum\-rate maximization\. Let𝒱s\\mathcal\{V\}\_\{s\}denote the evolving context dataset under channel modelss, initialized as𝒱s\(0\)\\mathcal\{V\}\_\{s\}^\{\(0\)\}and bootstrapped during training\. Now consider each element\(𝑯,𝒀,𝑾^\)∈𝒱s\(\\bm\{H\},\\bm\{Y\},\\hat\{\\bm\{W\}\}\)\\in\\mathcal\{V\}\_\{s\}\. Initially,𝑾^\\hat\{\\bm\{W\}\}is the LMMSE solution𝑾m​\(𝑯\)\\bm\{W\}^\{\\text\{m\}\}\(\\bm\{H\}\)in𝒱s\(0\)\\mathcal\{V\}\_\{s\}^\{\(0\)\}; later in the dataset𝒱s\\mathcal\{V\}\_\{s\}during bootstrapping, it may also be an accepted model\-generated beamformer solution\.

For notational clarity, define the model output under a query input𝒀\\bm\{Y\}and a context𝒞s∈𝒱s\\mathcal\{C\}\_\{s\}\\in\\mathcal\{V\}\_\{s\}as follows

𝑾​\(𝒀,𝒞s\)=𝒟b​\(𝒯b​\(𝒵sℓ​\(𝒞s,𝒀\)\)\)\.\\displaystyle\\bm\{W\}\(\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)=\\mathcal\{D\}\_\{b\}\\left\(\\mathcal\{T\}\_\{b\}\\left\(\\mathcal\{Z\}\_\{s\}^\{\\ell\}\(\\mathcal\{C\}\_\{s\},\\bm\{Y\}\)\\right\)\\right\)\.\(33\)We define the supervised imitation loss as follows:

Lsup​\(𝒀,𝑾^,𝒞s\)=‖𝑾​\(𝒀,𝒞s\)−𝑾^‖F2,\\displaystyle L^\{\\text\{sup\}\}\(\\bm\{Y\},\\hat\{\\bm\{W\}\},\\mathcal\{C\}\_\{s\}\)=\\bigl\\\|\\bm\{W\}\(\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)\-\\hat\{\\bm\{W\}\}\\bigr\\\|\_\{F\}^\{2\},\(34\)and define the sum\-rate loss as:

Lrate​\(𝑯,𝒀,𝒞s\)=−Rsum​\(𝑯,𝑾​\(𝒀,𝒞s\)\)\.\\displaystyle L^\{\\text\{rate\}\}\(\\bm\{H\},\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)=\-R\_\{\\text\{sum\}\}\(\\bm\{H\},\\bm\{W\}\(\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)\)\.\(35\)
At a certain epoch, letru∈\[0,1\]r\_\{u\}\\in\[0,1\]denote the proportion of unsupervised samples within one training batch, which*increases in a stepwise manner*and ends up withru=1r\_\{u\}=1during CL\-based training\. The training objective under channel modelsscan then be written in the following expected loss form:

min𝜽,ϕb,𝝍b\[\\displaystyle\\min\_\{\\bm\{\\theta\},\\bm\{\\phi\}\_\{b\},\\bm\{\\psi\}\_\{b\}\}\\ \\Big\[\(1−ru\)​E\(𝑯,𝒀,𝑾^\)∈𝒱s𝒞s∈𝒱s​\[Lsup​\(𝒀,𝑾^,𝒞s\)\]\\displaystyle\(1\-r\_\{u\}\)\\,\\mathbb E\_\{\\begin\{subarray\}\{c\}\(\\bm\{H\},\\bm\{Y\},\\hat\{\\bm\{W\}\}\)\\in\\mathcal\{V\}\_\{s\}\\\\ \\mathcal\{C\}\_\{s\}\\in\\mathcal\{V\}\_\{s\}\\end\{subarray\}\}\\left\[L^\{\\text\{sup\}\}\(\\bm\{Y\},\\hat\{\\bm\{W\}\},\\mathcal\{C\}\_\{s\}\)\\right\]\+γ⋅ru\(ηdE\(𝑯,𝒀,𝑾^\)∈𝒱s𝒞s∈𝒱s\[Lrate\(𝑯,𝒀,𝒞s\)\]\\displaystyle\+\\gamma\\cdot r\_\{u\}\\Big\(\\eta\_\{d\}\\,\\mathbb E\_\{\\begin\{subarray\}\{c\}\(\\bm\{H\},\\bm\{Y\},\\hat\{\\bm\{W\}\}\)\\in\\mathcal\{V\}\_\{s\}\\\\ \\mathcal\{C\}\_\{s\}\\in\\mathcal\{V\}\_\{s\}\\end\{subarray\}\}\\left\[L^\{\\text\{rate\}\}\(\\bm\{H\},\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)\\right\]\+\(1−ηd\)E𝑯∼ps,𝑵𝒀=𝑯​𝚿\+𝑵𝒞s∈𝒱s\[Lrate\(𝑯,𝒀,𝒞s\)\]\)\],\\displaystyle\+\(1\-\\eta\_\{d\}\)\\,\\mathbb E\_\{\\begin\{subarray\}\{c\}\\bm\{H\}\\sim p\_\{s\},\\bm\{N\}\\\\ \\bm\{Y\}=\\bm\{H\}\\bm\{\\Psi\}\+\\bm\{N\}\\\\ \\mathcal\{C\}\_\{s\}\\in\\mathcal\{V\}\_\{s\}\\end\{subarray\}\}\\left\[L^\{\\text\{rate\}\}\(\\bm\{H\},\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)\\right\]\\Big\)\\Big\],\(36\)whereγ\\gammais a scaling factor that balances the supervised and sum\-rate losses\. To strike a balance between exploitation, i\.e\., refining existing samples in the dataset, and exploration, i\.e\., improving on newly generated samples, the unsupervised samples are further divided into those drawn from𝒱s\\mathcal\{V\}\_\{s\}, which corresponds to the second term in \([IV\-B2](https://arxiv.org/html/2607.11970#S4.Ex4)\), and those generated from the simulator, which corresponds to the third term in \([IV\-B2](https://arxiv.org/html/2607.11970#S4.Ex4)\)\.ηd∈\[0,1\]\\eta\_\{d\}\\in\[0,1\]controls the ratio between these two components\.

During CL\-based training, the model continuously generates candidate beamformer solutions for newly sampled pilot signals\. Specifically, under channel modelss, we consider a fresh sample\(𝑯~,𝒀~,𝑾~m\)\(\\tilde\{\\bm\{H\}\},\\tilde\{\\bm\{Y\}\},\\tilde\{\\bm\{W\}\}^\{\\text\{m\}\}\)generated by the simulator, where𝑾~m\\tilde\{\\bm\{W\}\}^\{\\text\{m\}\}is the LMMSE beamformer of𝑯~\\tilde\{\\bm\{H\}\}, together with a randomly selected context𝒞s\\mathcal\{C\}\_\{s\}from𝒱s\\mathcal\{V\}\_\{s\}\. The model produces𝑾~​\(𝒀~,𝒞s\)\\tilde\{\\bm\{W\}\}\(\\tilde\{\\bm\{Y\}\},\\mathcal\{C\}\_\{s\}\)based on \([33](https://arxiv.org/html/2607.11970#S4.E33)\), with achieved sum rate

R~=Rsum​\(𝑯~,𝑾~​\(𝒀~,𝒞s\)\)\.\\displaystyle\\tilde\{R\}=R\_\{\\mathrm\{sum\}\}\(\\tilde\{\\bm\{H\}\},\\tilde\{\\bm\{W\}\}\(\\tilde\{\\bm\{Y\}\},\\mathcal\{C\}\_\{s\}\)\)\.\(37\)The candidate sample is admitted into the evolving context dataset only if it outperforms a prescribed fraction of the LMMSE performance on the same channel:

R~⩾α⋅Rsum​\(𝑯~,𝑾~m\),\\displaystyle\\tilde\{R\}\\geqslant\\alpha\\cdot R\_\{\\mathrm\{sum\}\}\(\\tilde\{\\bm\{H\}\},\\tilde\{\\bm\{W\}\}^\{\\text\{m\}\}\),\(38\)whereα\\alphaincreases fromαs\\alpha\_\{s\}toαe\\alpha\_\{e\}during training\. This instance\-wise admission rule prevents low\-quality model\-generated solutions from polluting the context dataset\. The threshold is intentionally relaxed in the early stage to encourage context diversity and becomes progressively stricter to improve the quality of retained demonstrations\. Once accepted, the sample\(𝑯~,𝒀~,𝑾~​\(𝒀~,𝒞s\)\)\(\\tilde\{\\bm\{H\}\},\\tilde\{\\bm\{Y\}\},\\tilde\{\\bm\{W\}\}\(\\tilde\{\\bm\{Y\}\},\\mathcal\{C\}\_\{s\}\)\)is added to𝒱s\\mathcal\{V\}\_\{s\}, and the generated beamformer𝑾~​\(𝒀~,𝒞s\)\\tilde\{\\bm\{W\}\}\(\\tilde\{\\bm\{Y\}\},\\mathcal\{C\}\_\{s\}\)is used in the later context constructions\.

In addition toadding qualified new samples, each context dataset also keepsrevisiting and refiningits existing samples to improve the context quality\. Specifically, given a sampled triplet\(𝑯,𝒀,𝑾^\)∈𝒱s\(\\bm\{H\},\\bm\{Y\},\\hat\{\\bm\{W\}\}\)\\in\\mathcal\{V\}\_\{s\}, if the model generates a beamformer solution𝑾​\(𝒀,𝒞s\)\\bm\{W\}\(\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)during training, such that

Rsum​\(𝑯,𝑾​\(𝒀,𝒞s\)\)\>Rsum​\(𝑯,𝑾^\),\\displaystyle R\_\{\\mathrm\{sum\}\}\(\\bm\{H\},\\bm\{W\}\(\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)\)\>R\_\{\\mathrm\{sum\}\}\(\\bm\{H\},\\hat\{\\bm\{W\}\}\),\(39\)\(𝑯,𝒀,𝑾^\)\(\\bm\{H\},\\bm\{Y\},\\hat\{\\bm\{W\}\}\)will be replaced by\(𝑯,𝒀,𝑾​\(𝒀,𝒞s\)\)\(\\bm\{H\},\\bm\{Y\},\\bm\{W\}\(\\bm\{Y\},\\mathcal\{C\}\_\{s\}\)\)in𝒱s\\mathcal\{V\}\_\{s\}\.

To conclude, unlike prior ICL beamforming scheme\[[34](https://arxiv.org/html/2607.11970#bib.bib34)\]that requires a large\-scale WMMSE\-labeled dataset for context construction, the enhanced ICL scheme only requires precomputing a small seed dataset with inexpensive LMMSE labels, which is enabled by the context bootstrapping mechanism during CL\-based training\.

Algorithm 1Enhanced ICL Pilot\-based Beamforming Scheme under Multiple Channel Models\.1:Parameters: epoch number of EDN pretraining stage

E0E\_\{0\}, epoch number of warm\-start stage

E1E\_\{1\}, epoch number of CL stage

E2E\_\{2\}, sample ratios

\(ru,ηd\)\(r\_\{u\},\\eta\_\{d\}\), scaling factor

γ\\gamma, context length

ℓ\\ell
2:Training:

3:EDN pretraining:

4:Build

𝒱\(0\)=⋃s∈𝒮𝒱s\(0\)\\mathcal\{V\}^\{\(0\)\}=\\bigcup\_\{s\\in\\mathcal\{S\}\}\\mathcal\{V\}\_\{s\}^\{\(0\)\}with LMMSE labels

5:forepoch

e=1,…,E0e=1,\\ldots,E\_\{0\}do

6:Construct a training batch containing the samples from all channel models

7:Train

\(ℰp,𝒟c\)\(\\mathcal\{E\}\_\{p\},\\mathcal\{D\}\_\{c\}\)on

𝒱\(0\)\\mathcal\{V\}^\{\(0\)\}via \([13](https://arxiv.org/html/2607.11970#S3.E13)\)

8:Train

\(ℰb,𝒟b\)\(\\mathcal\{E\}\_\{b\},\\mathcal\{D\}\_\{b\}\)on

𝒱\(0\)\\mathcal\{V\}^\{\(0\)\}via \([14](https://arxiv.org/html/2607.11970#S3.E14)\)

9:end for

10:Discard

𝒟c∗\\mathcal\{D\}^\{\*\}\_\{c\}, freeze

ℰp∗\\mathcal\{E\}^\{\*\}\_\{p\}, retain

ℰ~b\\tilde\{\\mathcal\{E\}\}\_\{b\}and

𝒟~b\\tilde\{\\mathcal\{D\}\}\_\{b\}
11:Supervised warm\-start:

12:forepoch

e=1,…,E1e=1,\\ldots,E\_\{1\}do

13:Construct a training batch containing the samples from all channel models

14:Draw

ℓ\\ellcontext pairs and a query pair from

𝒱s\(0\)\\mathcal\{V\}^\{\(0\)\}\_\{s\}, compress them via

ℰp∗\\mathcal\{E\}^\{\*\}\_\{p\}and

ℰb\\mathcal\{E\}\_\{b\}
15:Construct the input token sequence

𝒵sℓ\\mathcal\{Z\}\_\{s\}^\{\\ell\}
16:Perform the supervised training based on \([32](https://arxiv.org/html/2607.11970#S4.E32)\)

17:end for

18:CL with context bootstrapping:

19:Initialize

𝒱s=𝒱s\(0\)\\mathcal\{V\}\_\{s\}=\\mathcal\{V\}^\{\(0\)\}\_\{s\}for all models

s∈𝒮s\\in\\mathcal\{S\}
20:forepoch

e=E1\+1,…,E1\+E2e=E\_\{1\}\+1,\\ldots,E\_\{1\}\+E\_\{2\}do

21:Update

rur\_\{u\}according to the curriculum

22:Construct a training batch containing the samples from all channel models

23:Based on

\(ru,ηd\)\(r\_\{u\},\\eta\_\{d\}\), select samples from

𝒱s\\mathcal\{V\}\_\{s\}and generate fresh samples from the simulator

24:Perform the CL\-based training using \([IV\-B2](https://arxiv.org/html/2607.11970#S4.Ex4)\)

25:Add each fresh sample into

𝒱s\\mathcal\{V\}\_\{s\}if it satisfies \([38](https://arxiv.org/html/2607.11970#S4.E38)\)

26:Update each existing sample in

𝒱s\\mathcal\{V\}\_\{s\}if it satisfies \([39](https://arxiv.org/html/2607.11970#S4.E39)\)

27:end for

28:Obtain the trained ICL model

\(ℰp∗,ℰb∗,𝒯b∗,𝒟b∗\)\(\\mathcal\{E\}^\{\*\}\_\{p\},\\mathcal\{E\}^\{\*\}\_\{b\},\\mathcal\{T\}^\{\*\}\_\{b\},\\mathcal\{D\}^\{\*\}\_\{b\}\)and the bootstrapped datasets

\{𝒱s\}s∈𝒮\\\{\\mathcal\{V\}\_\{s\}\\\}\_\{s\\in\\mathcal\{S\}\}for all channel models

29:Inference:

30:Receive the new pilot signal

𝒀\\bm\{Y\}
31:Given channel model

ss, randomly sample

𝒞s\\mathcal\{C\}\_\{s\}from

𝒱s\\mathcal\{V\}\_\{s\}
32:Construct token sequence

𝒵sℓ\\mathcal\{Z\}\_\{s\}^\{\\ell\}and generate the solution

## VExtension to Model\-mismatch Settings

The previous sections consider an aligned setting, where the pilot signal and the beamforming objective are associated with the same channel realization\. We now extend the proposed ICL framework to more practical model\-mismatch settings, in which the received pilot signal is generated from a channel state or propagation condition that is not fully aligned with the target channel used for beamformer evaluation\. We first introduce several representative sources of mismatch, and then develop a mismatch\-aware ICL scheme that incorporates these mismatches into the general framework\.

### V\-AModel Mismatches

#### V\-A1Uplink\-Downlink Mismatch

In FDD systems, the uplink and downlink channels share similar topology\-related parameters, but operate at different carrier frequencies\[[45](https://arxiv.org/html/2607.11970#bib.bib45)\]\. Therefore, frequency\-dependent parameters may differ between the uplink and downlink\. Accordingly, we model them as two correlated sparse geometric channels:

𝒉kul\\displaystyle\\bm\{h\}\_\{k\}^\{\\mathrm\{ul\}\}=NL​∑p=1Lαk,pul​𝒂ful​\(ϕk,p\),\\displaystyle=\\sqrt\{\\frac\{N\}\{L\}\}\\sum\_\{p=1\}^\{L\}\\alpha\_\{k,p\}^\{\\mathrm\{ul\}\}\\bm\{a\}\_\{f\_\{\\mathrm\{ul\}\}\}\(\\phi\_\{k,p\}\),𝒉kdl\\displaystyle\\bm\{h\}\_\{k\}^\{\\mathrm\{dl\}\}=NL​∑p=1Lαk,pdl​𝒂fdl​\(ϕk,p\),\\displaystyle=\\sqrt\{\\frac\{N\}\{L\}\}\\sum\_\{p=1\}^\{L\}\\alpha\_\{k,p\}^\{\\mathrm\{dl\}\}\\bm\{a\}\_\{f\_\{\\mathrm\{dl\}\}\}\(\\phi\_\{k,p\}\),\(40\)wherefulf\_\{\\mathrm\{ul\}\}andfdlf\_\{\\mathrm\{dl\}\}denote the uplink and downlink carrier frequencies, respectively, and the steering vector is given by

𝒂f​\(ϕ\)=1N​\[1,ej​2​π​f​d0​sin⁡ϕ/c,…,ej​2​π​f​d0​\(N−1\)​sin⁡ϕ/c\]T,\\displaystyle\\bm\{a\}\_\{f\}\(\\phi\)=\\frac\{1\}\{\\sqrt\{N\}\}\\left\[1,e^\{\\text\{j\}2\\pi fd\_\{0\}\\sin\\phi/c\},\\ldots,e^\{\\text\{j\}2\\pi fd\_\{0\}\(N\-1\)\\sin\\phi/c\}\\right\]^\{T\},\(41\)whereffis the carrier frequency,ccis the speed of light, andd0d\_\{0\}is the antenna spacing\. The path gains in \([V\-A1](https://arxiv.org/html/2607.11970#S5.Ex6)\) are typically modeled as the following correlated perturbation model\[Zhong2020PartialReciprocity\]:

αk,ℓdl=ρud​αk,ℓul\+1−ρud2​ϵk,ℓud,\\displaystyle\\alpha\_\{k,\\ell\}^\{\\mathrm\{dl\}\}=\\rho\_\{\\mathrm\{ud\}\}\\alpha\_\{k,\\ell\}^\{\\mathrm\{ul\}\}\+\\sqrt\{1\-\\rho\_\{\\mathrm\{ud\}\}^\{2\}\}\\epsilon\_\{k,\\ell\}^\{\\mathrm\{ud\}\},\(42\)whereρud∈\[0,1\]\\rho\_\{\\mathrm\{ud\}\}\\in\[0,1\]controls the uplink\-downlink correlation, andϵk,ℓud∼𝒞​𝒩​\(0,1\)\\epsilon\_\{k,\\ell\}^\{\\mathrm\{ud\}\}\\sim\\mathcal\{CN\}\(0,1\)is an independent perturbation\. In this setting, the pilot signal𝒀¯\\bar\{\\bm\{Y\}\}is computed by \([6](https://arxiv.org/html/2607.11970#S2.E6)\) based on the uplink channel𝑯ul=\[𝒉1ul,…,𝒉Kul\]\\bm\{H\}^\{\\text\{ul\}\}=\[\\bm\{h\}\_\{1\}^\{\\text\{ul\}\},\\dots,\\bm\{h\}\_\{K\}^\{\\text\{ul\}\}\]\.

#### V\-A2Channel\-Aging Mismatch

We next consider the channel\-aging mismatch\. If the processing delay from pilot reception to beamformer generation exceeds the channel coherence time, the received pilot signal is generated from an outdated channel, whereas the beamformer is applied to an updated channel\. Specifically, let𝒉¯k\\bar\{\\bm\{h\}\}\_\{k\}and𝒉k\\bm\{h\}\_\{k\}denote the outdated and updated channels of userkk, respectively\. Since the path topology and angular support usually vary slowly over a short delay, we keep the same AoDs and model the temporal variation through Doppler\-induced path\-gain evolution:

𝒉¯k\\displaystyle\\bar\{\\bm\{h\}\}\_\{k\}=NL​∑ℓ=1Lαk,ℓ​𝒂​\(ϕk,ℓ\),\\displaystyle=\\sqrt\{\\frac\{N\}\{L\}\}\\sum\_\{\\ell=1\}^\{L\}\\alpha\_\{k,\\ell\}\\bm\{a\}\(\\phi\_\{k,\\ell\}\),𝒉k\\displaystyle\\bm\{h\}\_\{k\}=NL​∑ℓ=1L\(ρ​αk,ℓ​ej​2​π​νk,ℓ​Δ\+1−ρ2​ξk,ℓ\)​𝒂​\(ϕk,ℓ\),\\displaystyle=\\sqrt\{\\frac\{N\}\{L\}\}\\sum\_\{\\ell=1\}^\{L\}\\left\(\\rho\\alpha\_\{k,\\ell\}e^\{\\text\{j\}2\\pi\\nu\_\{k,\\ell\}\\Delta\}\+\\sqrt\{1\-\\rho^\{2\}\}\\xi\_\{k,\\ell\}\\right\)\\bm\{a\}\(\\phi\_\{k,\\ell\}\),\(43\)whereαk,ℓ∼𝒞​𝒩​\(0,1\)\\alpha\_\{k,\\ell\}\\sim\\mathcal\{CN\}\(0,1\),νk,ℓ\\nu\_\{k,\\ell\}is the Doppler shift,Δ\\Deltais the processing delay of the BS, andξk,ℓ∼𝒞​𝒩​\(0,1\)\\xi\_\{k,\\ell\}\\sim\\mathcal\{CN\}\(0,1\)captures residual fading uncertainty\. In this setting, the pilot signal𝒀¯\\bar\{\\bm\{Y\}\}is computed by \([6](https://arxiv.org/html/2607.11970#S2.E6)\) based on the outdated CSI𝑯¯=\[𝒉¯1,…,𝒉¯K\]\\bar\{\\bm\{H\}\}=\[\\bar\{\\bm\{h\}\}\_\{1\},\\dots,\\bar\{\\bm\{h\}\}\_\{K\}\]\.

#### V\-A3Pilot\-Contamination Mismatch

We further consider the mismatch induced by pilot contamination\[[46](https://arxiv.org/html/2607.11970#bib.bib46)\]\. When pilot resources are limited, users in neighboring cells may adopt non\-orthogonal pilot sequences\. As a result, the pilot signal received at the target BS is contaminated by inter\-cell interference\. Specifically, let𝑯t\\bm\{H\}^\{\\text\{t\}\}denote the target\-cell channel, and let𝑯ic\\bm\{H\}\_\{i\}^\{\\rm c\}denote the contaminating channel from theithi^\{\\text\{th\}\}neighboring cell, wherei∈ℬci\\in\\mathcal\{B\}\_\{\\rm c\}andℬc\\mathcal\{B\}\_\{\\rm c\}is the set of interfering cells\. Then the pilot signal can be modeled as

𝒀¯=𝑯t​𝚿\+∑i∈ℬcηi​𝑯ic​𝚽i\+𝑵,\\displaystyle\\bar\{\\bm\{Y\}\}=\\bm\{H\}^\{\\text\{t\}\}\\bm\{\\Psi\}\+\\sum\_\{i\\in\\mathcal\{B\}\_\{\\rm c\}\}\\sqrt\{\\eta\_\{i\}\}\\bm\{H\}^\{\\rm c\}\_\{i\}\\bm\{\\Phi\}\_\{i\}\+\\bm\{N\},\(44\)where𝚿\\bm\{\\Psi\}is the pilot matrix used by the target users,𝚽i\\bm\{\\Phi\}\_\{i\}is the pilot matrix used by theithi^\{\\text\{th\}\}cell, andηi\\eta\_\{i\}controls the relative contamination strength\.

### V\-BMismatch\-Aware ICL Scheme

We now propose a mismatch\-aware ICL scheme to handle the mismatch cases introduced in Sec\.[V\-A](https://arxiv.org/html/2607.11970#S5.SS1)\. Consider a set of channel models𝒮\\mathcal\{S\}without mismatches\. For eachs∈𝒮s\\in\\mathcal\{S\}, lets\(UD\)s^\{\(\\text\{UD\}\)\},s\(CA\)s^\{\(\\text\{CA\}\)\}, ands\(PC\)s^\{\(\\text\{PC\}\)\}denote the observed modelsssinduced by uplink\-downlink mismatch, channel\-aging mismatch, and pilot contamination mismatch, respectively\. Taking the uplink\-downlink mismatch as an example, the initial context dataset is constructed as

𝒱s,UD\(0\)=\{\(𝑯j,𝒀¯j,𝑾jm\)\}j=1M0,\\displaystyle\\mathcal\{V\}\_\{s,\\text\{UD\}\}^\{\(0\)\}=\\left\\\{\(\\bm\{H\}\_\{j\},\\bar\{\\bm\{Y\}\}\_\{j\},\\bm\{W\}\_\{j\}^\{\\text\{m\}\}\)\\right\\\}\_\{j=1\}^\{M\_\{0\}\},\(45\)where𝑯j∼ps\\bm\{H\}\_\{j\}\\sim p\_\{s\}is the target channel,𝒀¯j\\bar\{\\bm\{Y\}\}\_\{j\}is the pilot signal generated under the mismatched models\(UD\)s^\{\(\\text\{UD\}\)\}, and𝑾jm\\bm\{W\}\_\{j\}^\{\\text\{m\}\}is the LMMSE beamformer computed from𝑯j\\bm\{H\}\_\{j\}\.

Letℐ=\{UD,CA,PC\}\\mathcal\{I\}=\\\{\\text\{UD\},\\text\{CA\},\\text\{PC\}\\\}denote the set of mismatch types\. For eachs∈𝒮s\\in\\mathcal\{S\}and eachi∈ℐi\\in\\mathcal\{I\}, denote byqs,iq\_\{s,i\}the distribution of the mismatched pilot signal𝒀¯\\bar\{\\bm\{Y\}\}\. Then the ultimate objective is to maximize the expected sum rate over all channel modelss∈𝒮s\\in\\mathcal\{S\}and all mismatch typesi∈ℐi\\in\\mathcal\{I\}:

max𝜽​∑s∈𝒮,i∈ℐE𝑯∼ps,𝒀¯∼qs,i𝒞s,i∈𝒱s,i​\(𝜽\)​\[Rsum​\(𝑯,ℳ𝜽​\(𝒵s,iℓ\)\)\],\\displaystyle\\max\_\{\\bm\{\\theta\}\}\\ \\sum\_\{\\begin\{subarray\}\{c\}s\\in\\mathcal\{S\},\\\\ i\\in\\mathcal\{I\}\\end\{subarray\}\}\\ \\mathbb E\_\{\\begin\{subarray\}\{c\}\\bm\{H\}\\sim p\_\{s\},\\ \\bar\{\\bm\{Y\}\}\\sim q\_\{s,i\}\\\\ \\mathcal\{C\}\_\{s,i\}\\in\\mathcal\{V\}\_\{s,i\}\(\\bm\{\\theta\}\)\\end\{subarray\}\}\\left\[R\_\{\\text\{sum\}\}\\left\(\\bm\{H\},\\mathcal\{M\}\_\{\\bm\{\\theta\}\}\(\\mathcal\{Z\}\_\{s,i\}^\{\\ell\}\)\\right\)\\right\],\(46\)where𝒞s,i=\[\(𝒀¯j,𝑾^j\)\]j=1ℓ\\mathcal\{C\}\_\{s,i\}=\[\(\\bar\{\\bm\{Y\}\}\_\{j\},\\hat\{\\bm\{W\}\}\_\{j\}\)\]\_\{j=1\}^\{\\ell\}is sampled from the current context dataset𝒱s,i\\mathcal\{V\}\_\{s,i\},𝒵s,iℓ=\[𝒞s,i,𝒀¯\]\\mathcal\{Z\}\_\{s,i\}^\{\\ell\}=\[\\mathcal\{C\}\_\{s,i\},\\bar\{\\bm\{Y\}\}\]is the input sequence, andℳ𝜽\\mathcal\{M\}\_\{\\bm\{\\theta\}\}is the learnable neural network\. The other training strategies are similar to those described in Sec\.[IV](https://arxiv.org/html/2607.11970#S4)\.

Equivalently, the no\-mismatch ICL scheme learns to adapt to the channel models in𝒮\\mathcal\{S\}, whereas the mismatch\-aware ICL scheme extends the adaptation space to𝒮×ℐ\\mathcal\{S\}\\times\\mathcal\{I\}\. By embedding the observation\-target mismatch into the construction of demonstration pairs, the proposed scheme avoids explicit intermediate channel calibration and enables end\-to\-end mismatch\-aware beamforming\.

## VIPerformance Evaluation

### VI\-AExperiment Setup

#### VI\-A1Network Parameters

We consider a MU\-MISO downlink system where anNN\-antenna BS servesKKsingle\-antenna users under the total transmit powerPPand common noise varianceσ12=⋯=σK2=σ2\\sigma^\{2\}\_\{1\}=\\dots=\\sigma^\{2\}\_\{K\}=\\sigma^\{2\}\. Unless otherwise specified, we setK=N=32K=N=32,P=1P=1,SNR=1σ2\\text\{SNR\}=\\frac\{1\}\{\\sigma^\{2\}\}, the pilot lengthLp=20L\_\{p\}=20, the number of demonstration pairsℓ=5\\ell=5, and input sequence lengthLseq=2​ℓ\+1=11L\_\{\\text\{seq\}\}=2\\ell\+1=11\.

The parameters of Algorithm[1](https://arxiv.org/html/2607.11970#alg1)are set as follows\. The three training stages, namely EDN pretraining, supervised warm\-start, and CL with context bootstrapping, are run forE0=100E\_\{0\}=100,E1=50E\_\{1\}=50, andE2=700E\_\{2\}=700epochs, respectively\. The initial LMMSE\-labeled context dataset containsM0=1024M\_\{0\}=1024samples per channel model, and the final bootstrapped context dataset containsM=5×104M=5\\times 10^\{4\}samples per channel model\. In \([IV\-B2](https://arxiv.org/html/2607.11970#S4.Ex4)\), the scaling factor is set toγ=0\.1\\gamma=0\.1, the unsupervised ratiorur\_\{u\}is smoothly increased from0to11with step size0\.10\.1, and the ratioηd\\eta\_\{d\}is set to0\.30\.3\. The training batch size isBt=256B\_\{t\}=256, and the test set size isBi=800B\_\{i\}=800, containing samples from all\|𝒮\|=8\|\\mathcal\{S\}\|=8channel models\. During training, the channel SNR is sampled from the set𝒦t=\[5,10,15,20\]​dB\\mathcal\{K\}\_\{t\}=\[5,10,15,20\]\\ \\text\{dB\}\. The learning rate is initialized asη=10−3\\eta=10^\{\-3\}and decayed toη=10−4\\eta=10^\{\-4\}following a cosine decay schedule\. The admission thresholds in \([38](https://arxiv.org/html/2607.11970#S4.E38)\) are set toαs=0\.3\\alpha\_\{s\}=0\.3andαe=0\.7\\alpha\_\{e\}=0\.7222Note that the RHS in \([38](https://arxiv.org/html/2607.11970#S4.E38)\) is computed based on perfect CSI, but the LHS assumes that CSI is unavailable\. Hence it is reasonable to setαs<αe<1\\alpha\_\{s\}<\\alpha\_\{e\}<1\.\.

Moreover, Table\.[1](https://arxiv.org/html/2607.11970#S6.T1)summarizes the parameter sizes of the main components in the ICL network shown in Fig\.[1](https://arxiv.org/html/2607.11970#S3.F1)\. The pilot encoder maps the pilot signal from dimension2​N​Lp=12802NL\_\{p\}=1280tod=64d=64, using convolutional kernel size33and hidden dimensionDh=256D\_\{h\}=256\. The beamformer encoder maps the beamformer from dimension2​N​K=20482NK=2048tod=64d=64with hidden dimensionDh=256D\_\{h\}=256, while beamformer decoder maps the beamformer from dimensiond=64d=64to2​N​K=20482NK=2048with hidden dimensionDh=256D\_\{h\}=256\. The ICL Transformer first embeds each token fromd=64d=64toα=128\\alpha=128, processes the sequence throughT=6T=6Transformer blocks, and finally projects the vectorized hidden state from dimensionα​Lseq=1408\\alpha L\_\{\\rm seq\}=1408tod=64d=64\. Each Transformer block contains two LayerNorm modules, an MHSA module withE=8E=8attention heads and head dimensionDe=64D\_\{e\}=64, and an MLP module with layer sizes128×256128\\times 256,256×256256\\times 256, and256×128256\\times 128\. Overall, the ICL\-based neural network contains approximately4\.03×1064\.03\\times 10^\{6\}parameters\.

Table 1:Parameter sizes of the main components in the ICL network\.
#### VI\-A2Multiple Sparse Channel Models

We consider multiple cluster\-based sparse channel models\. Each channel model is specified by the triplet\(Lc,Lr,σϕ\)\(L\_\{c\},L\_\{r\},\\sigma\_\{\\phi\}\), whereLcL\_\{c\}denotes the number of scattering clusters,LrL\_\{r\}denotes the number of rays per cluster, andσϕ\\sigma\_\{\\phi\}denotes the intra\-cluster angular spread\. For each userkk, the channel is generated as

𝒉k=NLc​Lr​∑c=1Lc∑r=1Lrαk,c,r​𝒂​\(ϕk,c,r\),\\bm\{h\}\_\{k\}=\\sqrt\{\\frac\{N\}\{L\_\{c\}L\_\{r\}\}\}\\sum\_\{c=1\}^\{L\_\{c\}\}\\sum\_\{r=1\}^\{L\_\{r\}\}\\alpha\_\{k,c,r\}\\bm\{a\}\(\\phi\_\{k,c,r\}\),whereαk,c,r∼𝒞​𝒩​\(0,1\)\\alpha\_\{k,c,r\}\\sim\\mathcal\{CN\}\(0,1\)\. For a half\-wavelength\-spaced ULA, the steering vector is

𝒂​\(ϕ\)=1N​\[1,ej​π​sin⁡ϕ,…,ej​π​\(N−1\)​sin⁡ϕ\]T\.\\bm\{a\}\(\\phi\)=\\frac\{1\}\{\\sqrt\{N\}\}\\left\[1,e^\{j\\pi\\sin\\phi\},\\ldots,e^\{j\\pi\(N\-1\)\\sin\\phi\}\\right\]^\{T\}\.The ray angle is generated by

ϕk,c,r=ϕ¯k,c\+Δ​ϕk,c,r,k∈\[K\],c∈\[Lc\],r∈\[Lr\],\\displaystyle\\phi\_\{k,c,r\}=\\bar\{\\phi\}\_\{k,c\}\+\\Delta\\phi\_\{k,c,r\},\\ k\\in\[K\],\\ c\\in\[L\_\{c\}\],\\ r\\in\[L\_\{r\}\],\(47\)whereϕ¯k,c∼Uniform​\(−π/2,π/2\)\\bar\{\\phi\}\_\{k,c\}\\sim\\text\{Uniform\}\(\-\\pi/2,\\pi/2\)andΔ​ϕk,c,r∼𝒩​\(0,σϕ2\)\\Delta\\phi\_\{k,c,r\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\phi\}^\{2\}\)\. Therefore, different channel models are represented by different triplets\(Lc,Lr,σϕ\)\(L\_\{c\},L\_\{r\},\\sigma\_\{\\phi\}\), and the corresponding configurations are listed in Table[2](https://arxiv.org/html/2607.11970#S6.T2)\.

Table 2:Multiple Channel Model Configurations

### VI\-BAblation Study

We now present ablation studies on several key designs of the proposed enhanced ICL\-based beamforming scheme, including the in\-context architecture, curriculum learning \(CL\), and the context dataset bootstrapping\. The following two non\-learning baselines are used throughout simulations:

- •WMMSE\-LS: the BS first obtains the LS channel estimate from the received pilot signal, and then applies the iterative WMMSE algorithm in\[[2](https://arxiv.org/html/2607.11970#bib.bib2)\]based on the estimated channel\.
- •LMMSE\-LS: the BS first obtains the LS channel estimate from the received pilot signal, and then computes the LMMSE beamformer by \([5](https://arxiv.org/html/2607.11970#S2.E5)\) based on the estimated channel\.

For ease of the result presentation, all ICL\-based schemes in this subsection are trained over the multiple channel models listed in Table[2](https://arxiv.org/html/2607.11970#S6.T2), while the testing results are reported only on the selected “Dense Urban” channel model\.

We first evaluate the effect of thecontext lengthℓ\\ellon the proposed ICL beamforming scheme\. Fig\.[2](https://arxiv.org/html/2607.11970#S6.F2)plots the average testing sum rate versusℓ\\ell\. It can be observed that the proposed scheme significantly outperforms both WMMSE\-LS and LMMSE\-LS baselines once demonstration context is provided\. Asℓ\\ellincreases from0to a moderate value, the sum rate improves rapidly and then becomes nearly saturated aroundℓ=6\\ell=6to1212, indicating that a small number of demonstration pairs already provides effective anchors for query beamforming inference\. Whenℓ\\ellfurther increases, the performance slightly decreases, possibly because redundant or less relevant demonstrations dilute the attention to the most informative context samples\. Based on this result, we setℓ=5\\ell=5for the subsequent simulations, which achieves near\-peak performance with a small context length\. More importantly, the severe performance degradation atℓ=0\\ell=0, where the model only observes the query pilot token without any demonstration pair, confirms the necessity of context information and justifies the effectiveness of the ICL mechanism for pilot\-based beamforming\.

![Refer to caption](https://arxiv.org/html/2607.11970v1/ablation_context_len.png)Figure 2:Performance under different context lengthsℓ\\ell\.We then demonstrate the performance of theproposed CL schemeby comparing it with the following learning\-based variants: \(a\) an*unsupervised baseline*, which trains the model solely with the sum\-rate objective and removes the supervised warm\-start stage in Sec\.[IV\-B1](https://arxiv.org/html/2607.11970#S4.SS2.SSS1); \(b\) a*supervised baseline*, which trains the model only by imitating LMMSE beamformers and removes the CL stage in Sec\.[IV\-B2](https://arxiv.org/html/2607.11970#S4.SS2.SSS2); and \(c\)*hard\-switch CL baseline*, which rapidly changesru=0r\_\{u\}=0toru=1r\_\{u\}=1in \([IV\-B2](https://arxiv.org/html/2607.11970#S4.Ex4)\), while the proposed scheme gradually increasesrur\_\{u\}from0to11with step size0\.10\.1\. All schemes are evaluated overSNR∈\{5,10,15,20\}\\text\{SNR\}\\in\\\{5,10,15,20\\\}dB\. Fig\.[3](https://arxiv.org/html/2607.11970#S6.F3)plots the average testing sum rate versus SNR\. It is seen that the proposed CL scheme consistently outperforms all three learning\-based variants, as well as the WMMSE\-LS and LMMSE\-LS baselines\. The poor performance of the unsupervised baseline shows that directly optimizing the sum\-rate objective from scratch is ineffective, confirming the necessity of supervised warm\-start\. In contrast, the supervised baseline is limited by LMMSE\-label imitation and cannot achieve further task\-oriented improvement\. Although hard\-switch CL improves over these two extremes, its clear gap from the proposed scheme shows that a smooth transition from label imitation to sum\-rate maximization is crucial for stable and effective training\.

![Refer to caption](https://arxiv.org/html/2607.11970v1/ablation_CL.png)Figure 3:Comparison between the proposed CL scheme and other training methods\.Finally, we examine the effectiveness of the proposedcontext bootstrappingmechanism, a core component of the enhanced ICL scheme\. Two variants are considered for comparison: \(a\) a*No bootstrapping baseline*, which uses the same CL strategy but keeps each context dataset fixed as the initial LMMSE\-labeled seed dataset throughout training; and \(b\) a*No update baseline*, which admits newly generated samples into the context dataset but does not refine existing samples\. All schemes are evaluated overSNR∈\{5,10,15,20\}\\text\{SNR\}\\in\\\{5,10,15,20\\\}dB\. Fig\.[4](https://arxiv.org/html/2607.11970#S6.F4)plots the average testing sum rate versus SNR\. The “No bootstrapping” baseline lags behind the proposed scheme, indicating that the initial LMMSE\-labeled dataset alone cannot provide sufficiently informative contexts\. The “No update” baseline improves over “No bootstrapping” baseline by expanding the context dataset, but its remaining gap to the proposed scheme shows that updating existing samples is also important\. Therefore, the proposed bootstrapping mechanism benefits from both admitting high\-quality fresh samples and continuously refining existing context samples\.

![Refer to caption](https://arxiv.org/html/2607.11970v1/ablation_bootstrap.png)Figure 4:Comparison between the proposed context\-bootstrapping scheme and other ICL schemes\.
### VI\-COverall Performance

The overall performance of the proposed enhanced ICL beamforming scheme is now evaluated\. We first examine itsconvergence performanceacross multiple channel models\. For each channel model, a fixed testing set is generated before training and used throughout the entire training process, thereby enabling a consistent on\-the\-fly evaluation of the model performance\. Fig\.[5](https://arxiv.org/html/2607.11970#S6.F5)reports the test sum rate curves for eight representative channel models in Table[2](https://arxiv.org/html/2607.11970#S6.T2)\. As observed, the proposed scheme exhibits stable convergence behavior across all considered channel models and eventually outperforms both WMMSE\-LS and LMMSE\-LS baselines, demonstrating its excellent multi\-model adaptability\. Although a temporary degradation occurs when the training stage switches, the sum rate increases rapidly and then gradually improves as the context dataset is refined during training\. Among all channel models, the relative performance gains in the “LoS\-dominant” and “Near\-LoS” channel models are smaller, mainly because that a dominated propagation component can increase the spatial correlation among user channels and reduce their separability, making interference suppression and pilot\-to\-beamformer learning more challenging\. Nevertheless, the proposed scheme still achieves competitive or superior performance in these challenging scenarios, showing its robustness across heterogeneous channel environments\.

![Refer to caption](https://arxiv.org/html/2607.11970v1/multimodel_otf_curves_7_10.png)Figure 5:On\-the\-fly testing performance of the enhanced ICL beamforming scheme under multiple channel models\.We then show that the proposed ICL scheme is capable to perform a fastonline adaptation to an unseenchannel model, i\.e\.,s′∉𝒮s^\{\\prime\}\\notin\\mathcal\{S\}and the corresponding channel distribution𝑯∼ps′\\bm\{H\}\\sim p\_\{s^\{\\prime\}\}is unknown\. Instead of fine\-tuning the trained ICL model, the BS constructs a small channel\-specific context dataset at inference time: it collects pilot signals\{𝒀j′\}j=1M0\\\{\\bm\{Y\}^\{\\prime\}\_\{j\}\\\}\_\{j=1\}^\{M\_\{0\}\}, obtains LS channel estimates𝑯j′=𝒀j′​𝚿†\\bm\{H\}^\{\\prime\}\_\{j\}=\\bm\{Y\}^\{\\prime\}\_\{j\}\\bm\{\\Psi\}^\{\\dagger\}, and computes the corresponding LMMSE beamformers𝑾j′⁣m​\(𝑯j′\)\\bm\{W\}^\{\\prime\\text\{m\}\}\_\{j\}\(\\bm\{H\}^\{\\prime\}\_\{j\}\)\. The resulting context dataset is then given by𝒱s′=\{\(𝑯j′,𝒀j′,𝑾j′⁣m\)\}j=1M0\\mathcal\{V\}\_\{s^\{\\prime\}\}=\\\{\(\\bm\{H\}^\{\\prime\}\_\{j\},\\bm\{Y\}^\{\\prime\}\_\{j\},\\bm\{W\}^\{\\prime\\text\{m\}\}\_\{j\}\)\\\}\_\{j=1\}^\{M\_\{0\}\}\. Conditioned on𝒱s′\\mathcal\{V\}\_\{s^\{\\prime\}\}, the trained ICL model generates beamforming solutions for new channel model without online backpropagation or parameter updates\. Specifically, Table[3](https://arxiv.org/html/2607.11970#S6.T3)reports the test sum\-rate performance under three additional channel models, beyond the eight basic channel models summarized in Table[2](https://arxiv.org/html/2607.11970#S6.T2)\. For each model, we consider two experimental settings\. In the*Seen*setting, the target channel model is included in the training model set𝒮\\mathcal\{S\}; in the*Unseen*setting, it is excluded from𝒮\\mathcal\{S\}\. It shows that the proposed ICL scheme retains strong performance even when the target channel model is unseen during training\. In all three models, the*Unseen*setting outperforms both LMMSE\-LS and WMMSE\-LS\. Compared with the corresponding*Seen*setting, the sum\-rate degradation remains moderate, despite the complete exclusion of the target channel distribution from offline training\. This confirms the adaptation capability of the proposed ICL mechanism to previously unseen channel models\.

Table 3:Testing sum\-rate performance when channel models are seen or unseen during training\.Next, we evaluate the proposed scheme under the presence of model mismatches\. The ICL network is trained over all channel models in Table[2](https://arxiv.org/html/2607.11970#S6.T2)and three mismatch types introduced in Sec\.[V](https://arxiv.org/html/2607.11970#S5), while the testing results are reported on the “Dense Urban” channel model with each type of mismatch\. Three ICL schemes are provided for comparison: \(a\) the proposed*mismatch\-aware ICL*scheme, where pilot signals are generated from the mismatched channels but beamformers are evaluated on the original channels; \(b\)*no\-mismatch benchmark*that serves as an upper bound, where pilot signal and the beamformer evaluation are both associated with the original channels; and \(c\) the*mismatch\-unaware ICL*scheme, where pilot signals and the beamformer evaluation are both associated with the mismatched channels\. Fig\.[6](https://arxiv.org/html/2607.11970#S6.F6)plots the average testing sum rate versus SNR under different mismatch types and ICL schemes\. It shows that all mismatch\-unaware ICL baselines suffer significant performance degradations, due to deviated solutions in the context\. In contrast, the proposed mismatch\-aware ICL schemes substantially improve the sum rate and remain close to the common upper bound \(no\-mismatch benchmark\)\. This demonstrates that the proposed ICL scheme can effectively absorb the corresponding mismatches into the context constructions, without explicit channel calibration operations\.

![Refer to caption](https://arxiv.org/html/2607.11970v1/model_mismatch_test_2.png)Figure 6:Performance of the mismatch\-aware ICL schemes under different mismatch types\. “UD”, “CA” and “PC” represent the uplink\-downlink mismatch, channel\-aging mismatch, and pilot\-contamination mismatch, respectively\. “Aware” and “Unaware” represent the corresponding mismatch\-aware and mismatch\-unaware ICL schemes\.We finally compare the proposed scheme with existing learning\-based beamforming schemes\. In addition to the WMMSE\-LS and LMMSE\-LS baselines introduced above, the following methods are considered:

- •ICWLM: a Transformer\-based supervised ICL model for multi\-task physical\-layer optimization, originally using CSI\-beamformer demonstration pairs\[[34](https://arxiv.org/html/2607.11970#bib.bib34)\];
- •SALLO\-M: a semi\-amortized L2O\-based Transformer optimizer for beamforming based on perfect CSI\[[16](https://arxiv.org/html/2607.11970#bib.bib16)\];
- •SA\-EDN: a semi\-amortized encoder\-decoder network for beamforming based on estimated CSI\[[6](https://arxiv.org/html/2607.11970#bib.bib6)\];
- •DCF\-DNN: an end\-to\-end learning framework that maps received pilots to distributed feedback and subsequently to beamformer\[[9](https://arxiv.org/html/2607.11970#bib.bib9)\];
- •CAP\-DNN: an end\-to\-end learning framework with channel\-adaptive pilot generation and direct beamformer prediction\[[19](https://arxiv.org/html/2607.11970#bib.bib19)\];
- •PLFP\-Net: a WMMSE\-guided framework that jointly learns pilot transmission, limited feedback, and beamforming\[[21](https://arxiv.org/html/2607.11970#bib.bib21)\]\.

For a fair comparison, all methods are evaluated under the same system dimensionsN=K=32N=K=32, transmit SNR=20=20dB, power constraintP=1P=1, while pilot lengthLpL\_\{p\}is increased from55to2525with step size55\. No mismatches are considered in this experiment\. Note that perfect CSI is not available — whenever CSI is required by a baseline, it is replaced by the LS estimate𝑯^=𝒀​𝚿†\\hat\{\\bm\{H\}\}=\\bm\{Y\}\\bm\{\\Psi\}^\{\\dagger\}, obtained from available pilot signal𝒀\\bm\{Y\}\. Accordingly, for the “SALLO\-M” baseline, the auxiliary variable and beamformer are initialized by𝑯^\\hat\{\\bm\{H\}\}and its LMMSE beamformer𝑾m​\(𝑯^\)\\bm\{W\}^\{\\text\{m\}\}\(\\hat\{\\bm\{H\}\}\), respectively\. The “SA\-EDN” baseline likewise uses𝑯^\\hat\{\\bm\{H\}\}as its input\. For “ICWLM” baseline, we maintain its supervised training procedure and replace each CSI𝑯\\bm\{H\}by the corresponding pilot signal𝒀\\bm\{Y\}in the demonstration pairs, as introduced in Sec\.[III\-A](https://arxiv.org/html/2607.11970#S3.SS1), to adapt to the pilot\-based setting\. For “DCF\-DNN” and “PLFP\-Net” baselines, their native pilot\-feedback\-beamformer pipelines are retained under the same pilot settings\. The “CAP\-DNN” baseline preserves its channel\-adaptive pilot design, but the required CSI is restricted to the LS estimate𝑯^\\hat\{\\bm\{H\}\}\. Importantly, our proposed scheme and “ICWLM” baseline are able to be trained over the same set of channel models listed in Table[2](https://arxiv.org/html/2607.11970#S6.T2)and tested on the “Dense Urban” channel model, while the remaining baselines lack the multi\-model adaptability and hence are trained and tested only on the “Dense Urban” channel model\. All learning\-based methods use the same testing channel samples\. Fig\.[7](https://arxiv.org/html/2607.11970#S6.F7)plots the average testing sum rate versus the pilot lengthLpL\_\{p\}\. It is seen that all schemes benefit from increasingLpL\_\{p\}, and the proposed scheme consistently achieves the highest sum rate over the other schemes\. It also shows that all Transformer\-based schemes outperform the other DNN\-based schemes, suggesting a superior learning capability of the Transformer model\. Among those Transformer\-based schemes, the advantage of the proposed scheme over “SALLO\-M” and “ICWLM” baselines indicates that both the in\-context structure and enhanced training strategies contribute to the final performance\.

![Refer to caption](https://arxiv.org/html/2607.11970v1/baseline_comp_7_10.png)Figure 7:Comparison between the proposed scheme and baselines under varying pilot lengthsLpL\_\{p\}\.

## VIIConclusion

We proposed a self\-evolving ICL framework for direct pilot\-to\-beamformer design in MU\-MISO systems\. By integrating pilot and beamformer EDNs with a shared ICL Transformer, the proposed framework directly generates beamformers from noisy, limited\-length pilots without explicit channel estimation\. Its CL\-based training strategy together with the self\-evolving context bootstrapping mechanism improve the convergence behavior and substantially reduce the reliance on near\-optimal labeled data\. Through intelligent model\-specific context constructions, a single shared network can adapt to diverse channel models and mismatches, without parameter updates or explicit channel calibrations\. Extensive simulations demonstrate high\-quality adaptations to both seen and unseen channel models, robust performance under model mismatches, and consistent gains over existing pilot\-based beamforming methods\. Future work will extend the proposed framework to broader physical\-layer optimization tasks, and evaluate its performance using measurement\-based channels and over\-the\-air experiments\.

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