用于室内无线电地图估计的协同上下文学习

# Collaborative In-Context Learning for Indoor Radio Map Estimation
Source: [https://arxiv.org/html/2610.31702](https://arxiv.org/html/2610.31702)
## Collaborative In\-Context Learning for Indoor Radio Map EstimationThanks:The work of Yuanzhe Peng \(email:y\.peng@unf\.edu\) was partially supported by UNF startup funding\. The work of Jie Xu was partially supported by the NSF under Grants 2433886, 2515982, and 2505381\.

Yuanzhe Peng1, Jie Xu2Affiliation:1University of North Florida, Jacksonville, FL, USA 2University of Florida, Gainesville, FL, USAAffiliation:

###### Abstract

Indoor radio map estimation \(RME\) is essential for wireless applications such as coverage planning, localization, and network deployment\. However, most existing methods follow a supervised learning paradigm that requires costly labeled data and task\-specific model training\. Moreover, indoor environments involve privacy\-sensitive information, such as building layouts and structures, that often cannot be centralized\. To address these challenges, we make the first attempt to formulate indoor RME as a collaborative in\-context learning \(ICL\) problem\. We proposePoly\-Ripple, a framework in which multiple clients perform ICL using large autoregressive vision models \(LAVMs\) without model updates\. Each client is prompted by its private demonstrations and predicts the radio map for the same scene query\. To capture the causal structure of wireless propagation outward from the transmitter \(Tx\), we adopt ripple\-ordered autoregressive modeling\. We further introduce importance\-aware aggregation to exploit heterogeneous contextual knowledge distributed across clients under the context length constraint\. We provide theoretical analysis and conduct extensive experiments on a public indoor RME benchmark\. The experimental results show thatPoly\-Rippleconsistently outperforms representative ICL baselines, while aligning with our theoretical analysis\.

###### Index Terms:

In\-Context Learning \(ICL\), Large Autoregressive Vision Model \(LAVM\), Radio Map Estimation \(RME\), Foundation Models, Big Data Applications in Telecommunications\.

††aftertitle:## IIntroduction

Accurate and cost\-effective indoor radio map estimation \(RME\) is essential for wireless applications such as coverage planning, localization, and network deployment\[[1](https://arxiv.org/html/2610.31702#bib.bib24),[2](https://arxiv.org/html/2610.31702#bib.bib1),[3](https://arxiv.org/html/2610.31702#bib.bib31)\]\. A radio map provides spatially resolved estimates of path loss \(PL\), which quantifies signal attenuation in decibels throughout an indoor environment for a given transmitter \(Tx\) location\. Indoor RME is particularly challenging because of complex propagation effects such as reflection and transmission\. Traditionalmodel\-drivenapproaches, including the log\-distance PL model and 3GPP channel models, offer physical interpretability but often generalize poorly across diverse building layouts\.

More recently,data\-drivenmethods have formulated RME as an image\-to\-image supervised learning task\. Prior studies mainly improve indoor RME from three aspects: \(1\) designing encoder\-decoder architectures based on convolutional neural networks \(CNNs\) or vision transformers \(ViTs\)\[[4](https://arxiv.org/html/2610.31702#bib.bib33),[5](https://arxiv.org/html/2610.31702#bib.bib27),[6](https://arxiv.org/html/2610.31702#bib.bib18),[7](https://arxiv.org/html/2610.31702#bib.bib9),[8](https://arxiv.org/html/2610.31702#bib.bib10),[9](https://arxiv.org/html/2610.31702#bib.bib21),[10](https://arxiv.org/html/2610.31702#bib.bib15),[11](https://arxiv.org/html/2610.31702#bib.bib13),[12](https://arxiv.org/html/2610.31702#bib.bib11)\]; \(2\) customizing loss functions\[[13](https://arxiv.org/html/2610.31702#bib.bib25),[14](https://arxiv.org/html/2610.31702#bib.bib20),[15](https://arxiv.org/html/2610.31702#bib.bib12),[16](https://arxiv.org/html/2610.31702#bib.bib16),[17](https://arxiv.org/html/2610.31702#bib.bib19)\]; and \(3\) applying feature fusion or data augmentation techniques\[[18](https://arxiv.org/html/2610.31702#bib.bib28),[19](https://arxiv.org/html/2610.31702#bib.bib14),[20](https://arxiv.org/html/2610.31702#bib.bib26)\]\. Despite these advances, most existing methods still follow the supervised learning paradigm and focus on improving task\-specific models\.

However, practical indoor deployment exposes fundamental limitations of supervised learning\. First, obtaining labeled PL maps requires costly ray\-tracing simulations or labor\-intensive field measurements for each new building, making task\-specific training expensive and difficult to scale\. Second, building layouts and indoor structures contain privacy\-sensitive information that often cannot be centralized across sites\. This restriction makes it difficult to train a single global model using data collected from different environments\. Therefore, although supervised learning has achieved promising performance, its practical deployment for indoor RME remains limited\.

![Refer to caption](https://arxiv.org/html/2610.31702v1/fig1.png)Fig\. 1:Illustration ofPoly\-Ripple\. Each clientm∈\[M\]m\\in\[M\]performs ICL for the same server\-issued query𝒙q\\boldsymbol\{x\}\_\{q\}using its private demonstration set𝒫m\\mathcal\{P\}\_\{m\}\. Under privacy and context length constraints, the server applies importance\-aware aggregation to combine diverse client knowledge into the final prediction𝒚^q\\hat\{\\boldsymbol\{y\}\}\_\{q\}\.Recent advances in large language models \(LLMs\), such as GPTs, have demonstrated the ability to perform new tasks using only a few input\-output demonstrations \(or prompts\), a capability known as in\-context learning \(ICL\)\[[21](https://arxiv.org/html/2610.31702#bib.bib23)\]\. ICL eliminates the need for task\-specific model updates and suggests a different approach to indoor RME\. Instead of training a model, a large autoregressive vision model \(LAVM\) can be prompted with a few relevant demonstrations to infer the radio map for a new scene query\. This property makes ICL particularly appealing for indoor RME, where labeled data are costly and rapid adaptation to unseen environments is desirable\. Meanwhile, indoor data are naturally distributed across buildings, floors, and organizations\. Each client maintains its own private layout and measurements, while sharing raw demonstrations may be infeasible because of privacy concerns\. Moreover, the limited context length of LAVMs restricts the number of demonstrations that can be processed within a single context\. These observations motivate the collaborative ICL setting shown in Fig\.[1](https://arxiv.org/html/2610.31702#S1.F1), where multiple clients contribute distributed contextual knowledge without sharing their private demonstrations\.

![Refer to caption](https://arxiv.org/html/2610.31702v1/ripple.png)Fig\. 2:Raster scan order \(left\) is commonly used in visual autoregressive modeling\[[22](https://arxiv.org/html/2610.31702#bib.bib22)\]\. In contrast, inspired by the physical law of ripple\-like outward signal propagation from a Tx in indoor radio map estimation, we adopt a ripple order \(right\) to better capture the causal structure for next\-token prediction\.Challenges\.Despite its potential, applying collaborative ICL to indoor RME is nontrivial\. First, scene inputs and radio maps are continuous visual data and must be converted into meaningful discrete tokens before autoregressive modeling\. Second, most existing visual autoregressive models adopt raster\-scan ordering, which fails to capture the causal structure of wireless propagation\. Signals propagate outward from the Tx and interact with increasingly distant obstacles, resembling ripples across still water and forming ripple\-like spatial dependencies\. Third, practical deployment is distributed and subject to privacy and context length constraints\. Each client’s demonstrations contain private building information and cannot be shared, while the maximum token limit allows only a few high\-quality demonstrations in a single context\. The server thus requires an effective aggregation mechanism to combine heterogeneous client predictions into an accurate final estimate\.

These challenges raise aquestion:*how can we design a collaborative ICL framework that captures the causal structure of RME under privacy and context length constraints?*

In this paper, we proposePoly\-Ripple, a collaborative ICL framework for indoor RME\. The term “poly” reflects the collaboration of multiple clients that generate ripple\-structured local predictions for server\-side aggregation\. On each client,Poly\-Rippleconstructs a high\-quality demonstration set that considers both Tx location relevance and spatial diversity\. It then tokenizes continuous visual inputs into meaningful discrete tokens and arranges them in an outward ripple order\. As shown in Fig\.[2](https://arxiv.org/html/2610.31702#S1.F2), this physics\-guided ordering enables the LAVM to capture the causal structure of wireless propagation for next\-token prediction\. On the server side,Poly\-Rippleapplies importance\-aware aggregation based on client\-query similarity\. This design emphasizes predictions from more relevant demonstrations while leveraging complementary information across clients\. It exploits distributed contextual knowledge without sharing private indoor information while respecting the context length imposed by the maximum token limit\.

Contributions\.Our contributions are summarized as follows\.

- •We formulate indoor RME as a collaborative ICL problem beyond the conventional supervised learning paradigm\.
- •We proposePoly\-Ripple, an ICL framework combining ripple autoregressive inference with importance\-aware aggregation of distributed contextual knowledge\.
- •We provide theoretical analysis of estimation error bounds and multi\-client collaboration gains\.
- •We conduct experiments on a public dataset, showing thatPoly\-Rippleoutperforms representative ICL baselines\. Experimental results align with our theoretical analysis\.

## IIRelated Work

Radio Map Estimation\.RME methods can be divided into model\-driven and data\-driven approaches\. Model\-driven methods use analytical or physics\-based models, including the log\-distance PL model, 3GPP channel models, and ray\-tracing simulations\[[3](https://arxiv.org/html/2610.31702#bib.bib31)\]\. Although physically interpretable, they often generalize poorly across diverse indoor layouts\. Data\-driven methods formulate RME as supervised learning from multi\-channel scene inputs\[[4](https://arxiv.org/html/2610.31702#bib.bib33),[20](https://arxiv.org/html/2610.31702#bib.bib26),[13](https://arxiv.org/html/2610.31702#bib.bib25),[18](https://arxiv.org/html/2610.31702#bib.bib28)\]\. They commonly adopt encoder\-decoder architectures based on CNNs or ViTs\[[5](https://arxiv.org/html/2610.31702#bib.bib27),[12](https://arxiv.org/html/2610.31702#bib.bib11)\]\. Despite advances in model architectures, losses, and feature fusion, they still require costly labeled data and model training\.

In\-Context Learning\.ICL enables models to perform new tasks using a few input\-output demonstrations without model updates\[[23](https://arxiv.org/html/2610.31702#bib.bib32),[24](https://arxiv.org/html/2610.31702#bib.bib6),[25](https://arxiv.org/html/2610.31702#bib.bib7),[26](https://arxiv.org/html/2610.31702#bib.bib8),[27](https://arxiv.org/html/2610.31702#bib.bib39),[28](https://arxiv.org/html/2610.31702#bib.bib30),[29](https://arxiv.org/html/2610.31702#bib.bib40)\]\. It has been extended to visual tasks, where images are tokenized using VQ\-VAE\[[30](https://arxiv.org/html/2610.31702#bib.bib34)\]or VQ\-GAN\[[31](https://arxiv.org/html/2610.31702#bib.bib35)\], followed by autoregressive next\-token prediction\[[22](https://arxiv.org/html/2610.31702#bib.bib22)\]\. Visual ICL has shown promising performance on semantic segmentation, image inpainting, and other tasks\[[32](https://arxiv.org/html/2610.31702#bib.bib36)\]\. Recent studies have explored ICL for wireless applications\[[33](https://arxiv.org/html/2610.31702#bib.bib4),[34](https://arxiv.org/html/2610.31702#bib.bib5),[35](https://arxiv.org/html/2610.31702#bib.bib17)\], but they do not address collaborative inference across distributed clients\[[36](https://arxiv.org/html/2610.31702#bib.bib3),[37](https://arxiv.org/html/2610.31702#bib.bib2)\], where private demonstrations cannot be shared and the maximum token limit constrains context length\.

## IIIProblem Formulation

Indoor RME Task\.Consider an indoor environment discretized into a two\-dimensional grid of sizeH×WH\\times W\. Let𝒩=\{i\}i=1N\\mathcal\{N\}=\\\{i\\\}\_\{i=1\}^\{N\}denote the set of spatial coordinates, whereN=H×WN=H\\times W\. Given a transmitter \(Tx\) locationiTx∈𝒩i^\{\\mathrm\{Tx\}\}\\in\\mathcal\{N\}, indoor RME aims to predict the path loss valuey^i\\hat\{y\}^\{i\}at every locationi∈𝒩i\\in\\mathcal\{N\}, forming a radio map𝒚^=\{y^1,…,y^N\}∈ℝH×W\.\\hat\{\\boldsymbol\{y\}\}=\\\{\\hat\{y\}^\{1\},\\ldots,\\hat\{y\}^\{N\}\\\}\\in\\mathbb\{R\}^\{H\\times W\}\.

The corresponding scene input is denoted by𝒙∈ℝH×W×C\\boldsymbol\{x\}\\in\\mathbb\{R\}^\{H\\times W\\times C\}, whereCCis the number of input channels\. Following the dataset used in this paper, each scene input contains three physically meaningful channels: the distance from each spatial location to the Tx, transmittance, and reflectance\. The transmittance and reflectance channels are determined by the material interfaces of the building layout and remain unchanged across different Tx locations within the same building layout\.

Conventional Supervised Formulation\.Most existing RME methods adopt centralized supervised learning\. A modelfθf\_\{\\theta\}is trained to map a scene input𝒙\\boldsymbol\{x\}to the corresponding radio map,𝒚^=fθ​\(𝒙\),\\hat\{\\boldsymbol\{y\}\}=f\_\{\\theta\}\(\\boldsymbol\{x\}\),where the parametersθ\\thetaare optimized using a task\-specific loss such as the root mean squared error\. Despite diverse designs in model architectures and loss functions, these methods generally require large labeled datasets and dedicated training for effective deployment in new environments\.

Our Collaborative ICL Formulation\.We instead formulate indoor RME as a collaborative ICL problem withMMclients and a central server, enabling large\-scale deployment without task\-specific model training\. Each client is associated with a distinct building layout and maintains private input\-output demonstrations collected from that layout\. In practice, these demonstrations cannot be centralized because they contain private information about building layouts and structures\. In addition, the LAVM has a fixed context window measured in tokens, which limits the number of demonstrations it can process\. Collaboration therefore enables distributed contextual knowledge to contribute to indoor RME without sharing private demonstrations or combining them all into one context\.

Specifically, for each clientm∈\{1,…,M\}m\\in\\\{1,\\ldots,M\\\}, let𝒩m\\mathcal\{N\}\_\{m\}denote the spatial coordinate set of its building layout\. Different Tx locations within the same layout produce different scene inputs and corresponding radio maps\. The local demonstration set of clientmmis defined as𝒫m=\{\(𝒙m,k,𝒚m,k\)\}k=1Km,\\mathcal\{P\}\_\{m\}=\\\{\(\\boldsymbol\{x\}\_\{m,k\},\\boldsymbol\{y\}\_\{m,k\}\)\\\}\_\{k=1\}^\{K\_\{m\}\},where\(𝒙m,k,𝒚m,k\)\(\\boldsymbol\{x\}\_\{m,k\},\\boldsymbol\{y\}\_\{m,k\}\)is an input\-output demonstration associated with Tx locationim,kTx∈𝒩mi\_\{m,k\}^\{\\mathrm\{Tx\}\}\\in\\mathcal\{N\}\_\{m\}\. These private demonstrations remain local and are not exchanged among clients or with the server\.

The server, representing the user side, maintains a set of query scene inputs\{𝒙q\}\\\{\\boldsymbol\{x\}\_\{q\}\\\}, where each query𝒙q\\boldsymbol\{x\}\_\{q\}corresponds to either a new building layout or a new Tx location\. The corresponding radio maps\{𝒚q\}\\\{\\boldsymbol\{y\}\_\{q\}\\\}are unavailable and must be estimated without costly task\-specific model training\. The server therefore broadcasts the same query𝒙q\\boldsymbol\{x\}\_\{q\}to all clients\. Each clientmmthen performs ICL using an LAVM with frozen parametersω\\omega, prompted by its private demonstrations𝒫m\\mathcal\{P\}\_\{m\},

𝒚^q,m=fω​\(𝒫m,𝒙q\),m∈\{1,…,M\}\.\\hat\{\\boldsymbol\{y\}\}\_\{q,m\}=f\_\{\\omega\}\(\\mathcal\{P\}\_\{m\},\\boldsymbol\{x\}\_\{q\}\),\\quad m\\in\\\{1,\\ldots,M\\\}\.

\(1\)Notably, the clients produce different predictions for the same query because they are prompted by different demonstrations\.

The objective of collaborative ICL is to aggregate these local predictions into a final radio map𝒚^q\\hat\{\\boldsymbol\{y\}\}\_\{q\}under privacy and context length constraints\. Specifically, the server computes

𝒚^q=𝒜⁡\(𝒚^q,1,…,𝒚^q,M\),\\hat\{\\boldsymbol\{y\}\}\_\{q\}=\\mathcal\{A\}\\big\(\\hat\{\\boldsymbol\{y\}\}\_\{q,1\},\\ldots,\\hat\{\\boldsymbol\{y\}\}\_\{q,M\}\\big\),

\(2\)where𝒜⁡\(⋅\)\\mathcal\{A\}\(\\cdot\)denotes an aggregation operator\. For example, weighted aggregation is given by𝒚^q=∑m=1Mαm​𝒚^q,m,\\hat\{\\boldsymbol\{y\}\}\_\{q\}=\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}\\hat\{\\boldsymbol\{y\}\}\_\{q,m\},whereαm≥0\\alpha\_\{m\}\\geq 0and∑m=1Mαm=1\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}=1\. The weights reflect the relevance of each client’s demonstrations to the query, such as the similarity between a representative client scene and𝒙q\\boldsymbol\{x\}\_\{q\}\.

![Refer to caption](https://arxiv.org/html/2610.31702v1/long.png)Fig\. 3:Each clientm∈\[M\]m\\in\[M\]locally performs ICL using an LAVM, which consists of a VQ\-GAN encoder and decoder for tokenization and reconstruction, and a transformer for next\-token prediction\. The server aggregates the ripple\-structured local predictions via𝒜⁡\(⋅\)\\mathcal\{A\}\(\\cdot\)to collaboratively produce the final radio map\.
## IVMethodology

We present thePoly\-Rippleframework for indoor RME\. On the client side, each client performs query dependent demonstration construction, visual tokenization, ripple ordered sequence generation, and autoregressive prediction using an LAVM\. On the server side, the local predictions are aggregated using importance\-aware weights, as shown in Fig\.[3](https://arxiv.org/html/2610.31702#S3.F3)\.

### IV\-AClient\-Side In\-Context Learning

Each clientmmcorresponds to a building layout and stores its demonstrations locally\. Upon receiving a query𝒙q\\boldsymbol\{x\}\_\{q\}broadcast by the server, clientmmperforms the following four steps\.

Local Demonstration Construction\.Each clientm∈\[M\]m\\in\[M\]receives the query input𝒙q\\boldsymbol\{x\}\_\{q\}with transmitter locationiqTxi\_\{q\}^\{\\mathrm\{Tx\}\}\. Let𝒟m=\{\(𝒙m,i,𝒚m,i,im,iTx\)\}i=1Nm\\mathcal\{D\}\_\{m\}=\\\{\(\\boldsymbol\{x\}\_\{m,i\},\\boldsymbol\{y\}\_\{m,i\},i\_\{m,i\}^\{\\mathrm\{Tx\}\}\)\\\}\_\{i=1\}^\{N\_\{m\}\}denote its local dataset, whereim,iTx∈ℐTxi\_\{m,i\}^\{\\mathrm\{Tx\}\}\\in\\mathcal\{I\}\_\{\\mathrm\{Tx\}\}is a candidate transmitter position\. Due to the context length constraint, clientmmselects at mostKKdemonstrations\. It first chooses the position aligned anchork1=arg⁡mini⁡‖im,iTx−iqTx‖2k\_\{1\}=\\arg\\min\_\{i\}\\\|i\_\{m,i\}^\{\\mathrm\{Tx\}\}\-i\_\{q\}^\{\\mathrm\{Tx\}\}\\\|\_\{2\}, whose transmitter geometry is closest to the query\. It then applies K\-means clustering toℐTx\\mathcal\{I\}\_\{\\mathrm\{Tx\}\}, formingKKdisjoint clusters\{𝒞k\}k=1K\\\{\\mathcal\{C\}\_\{k\}\\\}\_\{k=1\}^\{K\}with centroids\{μk\}k=1K\\\{\\mu\_\{k\}\\\}\_\{k=1\}^\{K\}, and selectsiTxk=arg⁡mini∈𝒞k⁡‖i−μk‖2i\_\{\\mathrm\{Tx\}\}^\{k\}=\\arg\\min\_\{i\\in\\mathcal\{C\}\_\{k\}\}\\\|i\-\\mu\_\{k\}\\\|\_\{2\}from each cluster\. This yields𝒦Tx=\{iTx1,…,iTxK\}\\mathcal\{K\}\_\{\\mathrm\{Tx\}\}=\\\{i\_\{\\mathrm\{Tx\}\}^\{1\},\\ldots,i\_\{\\mathrm\{Tx\}\}^\{K\}\\\}\. If the anchor is not included, the representative of the cluster containingim,k1Txi\_\{m,k\_\{1\}\}^\{\\mathrm\{Tx\}\}is replaced by the anchor\. Finally, clientmmconstructs𝒫q,m=\{\(𝒙m,k,𝒚m,k\)\}k∈𝒦Tx\\mathcal\{P\}\_\{q,m\}=\\\{\(\\boldsymbol\{x\}\_\{m,k\},\\boldsymbol\{y\}\_\{m,k\}\)\\\}\_\{k\\in\\mathcal\{K\}\_\{\\mathrm\{Tx\}\}\}\. All operations remain local\. The anchor preserves positional relevance, while the remaining representatives provide spatial diversity\.

Visual Tokenization\.After constructing the query dependent demonstration set𝒫q,m\\mathcal\{P\}\_\{q,m\}, clientmmconverts the demonstrations and query into discrete tokens for causal sequence modeling\. We use a pretrained VQ\-GAN with a fixed codebook𝒵\\mathcal\{Z\}, consisting of an encoderE⁡\(⋅\)E\(\\cdot\), a quantizerQ⁡\(⋅\)Q\(\\cdot\), and a decoderD⁡\(⋅\)D\(\\cdot\)\. For any input𝒙\\boldsymbol\{x\}, the encoder produces a latent tensor𝒛^=E⁡\(𝒙\)∈ℝh×w×b\\hat\{\\boldsymbol\{z\}\}=E\(\\boldsymbol\{x\}\)\\in\\mathbb\{R\}^\{h\\times w\\times b\}, whereh=H/fh=H/f,w=W/fw=W/f,ffis the downsampling factor, andbbis the embedding dimension\. Each latent vector𝒛^i,j\\hat\{\\boldsymbol\{z\}\}^\{i,j\}is mapped to the index of its nearest codeword in𝒵=\{𝒛v\}v=1V⊂ℝb\\mathcal\{Z\}=\\\{\\boldsymbol\{z\}\_\{v\}\\\}\_\{v=1\}^\{V\}\\subset\\mathbb\{R\}^\{b\}according toQ⁡\(𝒛^i,j\)=arg⁡minv∈\[V\]​‖𝒛^i,j−𝒛v‖\.Q\(\\hat\{\\boldsymbol\{z\}\}^\{i,j\}\)=\\arg\\min\_\{v\\in\[V\]\}\\bigl\\\|\\hat\{\\boldsymbol\{z\}\}^\{i,j\}\-\\boldsymbol\{z\}\_\{v\}\\bigr\\\|\.This produces a discrete token grid𝒛∈\{1,…,V\}h×w\\boldsymbol\{z\}\\in\\\{1,\\ldots,V\\\}^\{h\\times w\}\. Clientmmapplies the tokenizer to all inputs and outputs in𝒫q,m\\mathcal\{P\}\_\{q,m\}and to the query input𝒙q\\boldsymbol\{x\}\_\{q\}, yielding their corresponding token grids\.

Ripple Ordering\.To reflect the causal structure of wireless propagation, tokens are organized in a ripple order rather than a raster scan order\. Specifically, tokens are arranged by increasing radial distance from the transmitter, so autoregressive prediction proceeds from nearby regions to farther regions\.

For a demonstration pair\(𝒛kin,𝒛kout\)\(\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{k\},\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{k\}\)with transmitter locationikTxi^\{\\mathrm\{Tx\}\}\_\{k\}mapped to the latent token grid, the grid positions are represented asℐ~k=\{i~k,t\}t=1T\\tilde\{\\mathcal\{I\}\}\_\{k\}=\\\{\\tilde\{i\}\_\{k,t\}\\\}\_\{t=1\}^\{T\}, whereT=h⋅wT=h\\cdot w\. The radial distance of each position isrk,t=‖i~k,t−ikTx‖2\.r\_\{k,t\}=\\bigl\\\|\\tilde\{i\}\_\{k,t\}\-i\_\{k\}^\{\\mathrm\{Tx\}\}\\bigr\\\|\_\{2\}\.Given increasing radial thresholds0=r0<r1<⋯<rLmax0=r\_\{0\}<r\_\{1\}<\\cdots<r\_\{L\_\{\\max\}\}, theℓ\\ellth ripple layer is defined as

ℛkℓ=\{i~k,t∈ℐ~k\|rℓ−1<‖i~k,t−ikTx‖2≤rℓ\},ℓ∈\[Lmax\]\.\\mathcal\{R\}^\{\\ell\}\_\{k\}=\\bigl\\\{\\tilde\{i\}\_\{k,t\}\\in\\tilde\{\\mathcal\{I\}\}\_\{k\}\\;\\big\|\\;r\_\{\\ell\-1\}<\\\|\\tilde\{i\}\_\{k,t\}\-i\_\{k\}^\{\\mathrm\{Tx\}\}\\\|\_\{2\}\\leq r\_\{\\ell\}\\bigr\\\},\\ell\\in\[L\_\{\\max\}\]\.

\(3\)LetRkℓ=\|ℛkℓ\|R\_\{k\}^\{\\ell\}=\|\\mathcal\{R\}\_\{k\}^\{\\ell\}\|\. Within each ripple layer, input and output tokens at the same position are interleaved to associate environmental context with local propagation behavior\. Fori~ℓ,j∈ℛkℓ\\tilde\{i\}\_\{\\ell,j\}\\in\\mathcal\{R\}^\{\\ell\}\_\{k\}, the interleaved token pair is𝒔kℓ,j=\[𝒛kin​\(i~ℓ,j\),𝒛kout​\(i~ℓ,j\)\]\.\\boldsymbol\{s\}^\{\\ell,j\}\_\{k\}=\[\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{k\}\(\\tilde\{i\}\_\{\\ell,j\}\),\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{k\}\(\\tilde\{i\}\_\{\\ell,j\}\)\]\.The sequence for layerℓ\\ellis

𝒔kℓ=\[𝒛kin​\(i~ℓ,1\),𝒛kout​\(i~ℓ,1\),…,𝒛kin​\(i~ℓ,Rkℓ\),𝒛kout​\(i~ℓ,Rkℓ\)\]\.\\displaystyle\\boldsymbol\{s\}^\{\\ell\}\_\{k\}=\[\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{k\}\(\\tilde\{i\}\_\{\\ell,1\}\),\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{k\}\(\\tilde\{i\}\_\{\\ell,1\}\),\\ldots,\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{k\}\(\\tilde\{i\}\_\{\\ell,R^\{\\ell\}\_\{k\}\}\),\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{k\}\(\\tilde\{i\}\_\{\\ell,R^\{\\ell\}\_\{k\}\}\)\]\.\(4\)We concatenate the layers in ascending radial order and append a separator token after each layer, including empty layers, to maintain a consistent sequence structure\. The sequence for demonstrationkkis𝒔k=\[𝒔k1,SEP,𝒔k2,SEP,…,𝒔kLmax,SEP\]\.\\boldsymbol\{s\}\_\{k\}=\[\\boldsymbol\{s\}^\{1\}\_\{k\},\\mathrm\{SEP\},\\boldsymbol\{s\}^\{2\}\_\{k\},\\mathrm\{SEP\},\\ldots,\\boldsymbol\{s\}^\{L\_\{\\max\}\}\_\{k\},\\mathrm\{SEP\}\]\.Finally, clientmmconcatenates all selected demonstration sequences to construct𝒫q,m∗=\[𝒔k1,𝒔k2,…,𝒔kKm\]\.\\mathcal\{P\}^\{\*\}\_\{q,m\}=\[\\boldsymbol\{s\}\_\{k\_\{1\}\},\\boldsymbol\{s\}\_\{k\_\{2\}\},\\ldots,\\boldsymbol\{s\}\_\{k\_\{K\_\{m\}\}\}\]\.

Autoregressive Prediction\.Given the ripple ordered prompt context𝒫q,m∗\\mathcal\{P\}^\{\*\}\_\{q,m\}and query input tokens𝒛qin\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{q\}, clientmmperforms autoregressive prediction using a causal transformer with fixed parameters\. The transformer sequentially generates the client specific output tokens𝒛q,mout\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{q,m\}, conditioned on the prompt context, the query input, and previously generated output tokens\. Formally, the joint likelihood factorizes as

p\(𝒛q,mout∣𝒫q,m∗,𝒛qin\)=∏t=1Tp\(zq,m,tout\|𝒛q,m,1:\(t−1\)out,𝒫q,m∗,𝒛qin\),p\\bigl\(\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{q,m\}\\mid\\mathcal\{P\}^\{\*\}\_\{q,m\},\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{q\}\\bigr\)=\\prod\_\{t=1\}^\{T\}p\(z^\{\\mathrm\{out\}\}\_\{q,m,t\}\|\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{q,m,1\\mathrel\{\\mathop\{\\mathchar 58\\relax\}\}\(t\-1\)\},\\mathcal\{P\}^\{\*\}\_\{q,m\},\\boldsymbol\{z\}^\{\\mathrm\{in\}\}\_\{q\}\),

\(5\)whereT=h⋅wT=h\\cdot wis the number of tokens in the latent grid, andzq,m,toutz^\{\\mathrm\{out\}\}\_\{q,m,t\}is generated at thettth position in the ripple ordered sequence\. The causal attention mask prevents access to future output tokens, so the generated outputs follow the outward ripple order while conditioning on the observed query input and prompt context\. The generated output𝒛q,mout\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{q,m\}is rearranged into a discrete token grid in\{1,…,V\}h×w\\\{1,\\ldots,V\\\}^\{h\\times w\}\. A lookup operatorZ⁡\(⋅\)Z\(\\cdot\)maps each token index to its codeword in𝒵=\{𝒛v\}v=1V\\mathcal\{Z\}=\\\{\\boldsymbol\{z\}\_\{v\}\\\}\_\{v=1\}^\{V\}, and the VQ\-GAN decoder reconstructs the local radio map as

𝒚^q,m=D⁡\(Z⁡\(𝒛q,mout\)\),∀m∈\[M\]\.\\hat\{\\boldsymbol\{y\}\}\_\{q,m\}=D\\bigl\(Z\(\\boldsymbol\{z\}^\{\\mathrm\{out\}\}\_\{q,m\}\)\\bigr\),\\quad\\forall\\,m\\in\[M\]\.

\(6\)Finally, only𝒚^q,m\\hat\{\\boldsymbol\{y\}\}\_\{q,m\}is transmitted, while the private demonstrations and intermediate token sequences remain stored locally\.

### IV\-BServer\-Side Importance\-Aware Aggregation

After all clients generate local predictions using their private prompts, the server aggregates them to obtain the final estimate\. Uniform averaging is straightforward, but it may be suboptimal because the local prompt sets can differ substantially in their relevance to the query\. We thus express the aggregation rule as

𝒚^q=𝒜⁡\(𝒚^q,1,…,𝒚^q,M\),\\hat\{\\boldsymbol\{y\}\}\_\{q\}=\\mathcal\{A\}\\big\(\\hat\{\\boldsymbol\{y\}\}\_\{q,1\},\\ldots,\\hat\{\\boldsymbol\{y\}\}\_\{q,M\}\\big\),

\(7\)where𝒜⁡\(⋅\)\\mathcal\{A\}\(\\cdot\)is a weighted aggregator that assigns each client an importance weight based on its relevance to the query\. Let𝒙m\\boldsymbol\{x\}\_\{m\}denote a representative scene input of clientmm, such as the anchor selected for the current query\. Given the query scene𝒙q\\boldsymbol\{x\}\_\{q\}, their cosine similarity isSm=⟨ϕ⁡\(𝒙m\),ϕ⁡\(𝒙q\)⟩‖ϕ⁡\(𝒙m\)‖2​‖ϕ⁡\(𝒙q\)‖2,S\_\{m\}=\\frac\{\\langle\\phi\(\\boldsymbol\{x\}\_\{m\}\),\\phi\(\\boldsymbol\{x\}\_\{q\}\)\\rangle\}\{\\\|\\phi\(\\boldsymbol\{x\}\_\{m\}\)\\\|\_\{2\}\\,\\\|\\phi\(\\boldsymbol\{x\}\_\{q\}\)\\\|\_\{2\}\},whereϕ⁡\(⋅\)\\phi\(\\cdot\)is a shared scene encoder\. This score measures the structural relevance of clientmmto the query\. The embeddingϕ⁡\(𝒙m\)\\phi\(\\boldsymbol\{x\}\_\{m\}\)or the resulting scoreSmS\_\{m\}is computed locally and transmitted to the server, while the original scene remains private\. The similarity scores are converted into aggregation weights through a temperature scaled softmax function,

αm=exp⁡\(Sm/τ\)∑m′=1Mexp⁡\(Sm′/τ\),∀m∈\[M\],\\displaystyle\\alpha\_\{m\}=\\frac\{\\exp\(S\_\{m\}/\\tau\)\}\{\\sum\_\{m^\{\\prime\}=1\}^\{M\}\\exp\(S\_\{m^\{\\prime\}\}/\\tau\)\},\\quad\\forall\\,m\\in\[M\],\(8\)whereτ\>0\\tau\>0controls the concentration of the weights\. A smallerτ\\tauemphasizes the most relevant clients, whereas a largerτ\\taudistributes the weights more evenly\.

The final radio map prediction for server query𝒙q\\boldsymbol\{x\}\_\{q\}is

𝒚^q=∑m=1Mαm​𝒚^q,m\.\\hat\{\\boldsymbol\{y\}\}\_\{q\}=\\sum\\nolimits\_\{m=1\}^\{M\}\\alpha\_\{m\}\\hat\{\\boldsymbol\{y\}\}\_\{q,m\}\.

\(9\)
This importance\-aware aggregation emphasizes predictions from relevant clients while reducing the influence of less informative ones\. Other relevance measures, including confidence based or hybrid criteria, can also be incorporated without changing the overall framework\.

## VTheoretical Analysis

### V\-AEstimation Error Analysis

We introduce a surrogate model to characterize the implicit adaptation induced by contextual conditioning in ICL\. We consider a common server\-issued query broadcast to all clients, whose private local demonstrations may produce different predictions\. Each clientm∈\[M\]m\\in\[M\]observesKKindependent and identically distributed demonstrations\{\(𝒙k,𝒚km\)\}k=1K\\\{\(\\boldsymbol\{x\}\_\{k\},\\boldsymbol\{y\}^\{m\}\_\{k\}\)\\\}\_\{k=1\}^\{K\}\. For any subset𝒮⊆\[d\]≜\{1,…,d\}\\mathcal\{S\}\\subseteq\[d\]\\triangleq\\\{1,\\ldots,d\\\}, let𝒙𝒮∈ℝ\|𝒮\|\\boldsymbol\{x\}\_\{\\mathcal\{S\}\}\\in\\mathbb\{R\}^\{\|\\mathcal\{S\}\|\}denote the restriction of the input vector𝒙\\boldsymbol\{x\}to the coordinates indexed by𝒮\\mathcal\{S\}\. We begin by making the following mild assumptions\.

###### Assumption 1\.

For every output coordinatei∈\[d\]i\\in\[d\]and every subset𝒮⊆\[d\]\\mathcal\{S\}\\subseteq\[d\], there exist coefficientswi,𝒮⋆∈ℝ\|𝒮\|w^\{\\star\}\_\{i,\\mathcal\{S\}\}\\in\\mathbb\{R\}^\{\|\\mathcal\{S\}\|\}and client\-specific fluctuation termsϵim\\epsilon^\{m\}\_\{i\}such that, for all clientsm∈\[M\]m\\in\[M\],

yim=𝒙𝒮⊤​wi,𝒮⋆\+ϵim,𝔼⁡\[ϵim\|𝒙𝒮\]=0,Var⁡\(ϵim\|𝒙𝒮\)=σi2\.y^\{m\}\_\{i\}=\\boldsymbol\{x\}\_\{\\mathcal\{S\}\}^\{\\top\}w^\{\\star\}\_\{i,\\mathcal\{S\}\}\+\\epsilon^\{m\}\_\{i\},\\ \\mathbb\{E\}\\\!\\left\[\\epsilon^\{m\}\_\{i\}\|\\boldsymbol\{x\}\_\{\\mathcal\{S\}\}\\right\]=0,\\ \\mathrm\{Var\}\\\!\\left\(\\epsilon^\{m\}\_\{i\}\|\\boldsymbol\{x\}\_\{\\mathcal\{S\}\}\\right\)=\\sigma\_\{i\}^\{2\}\.

\(10\)

###### Assumption 2\.

The inputs are standardized with zero mean and identity covariance:𝔼⁡\[𝐱\]=0,\\mathbb\{E\}\[\\boldsymbol\{x\}\]=0,Cov⁡\(𝐱\)=Id\.\\mathrm\{Cov\}\(\\boldsymbol\{x\}\)=I\_\{d\}\.

###### Assumption 3\.

There exists a physical ordering of output coordinates by increasing distance to the transmitter\. Without loss of generality, indices are ordered so thati=1i=1corresponds to the closest location andi=di=dto the farthest\. The dynamic range levels satisfyσ12≥σ22≥⋯≥σd2\.\\sigma\_\{1\}^\{2\}\\geq\\sigma\_\{2\}^\{2\}\\geq\\cdots\\geq\\sigma\_\{d\}^\{2\}\.

###### Assumption 4\.

For any clientm∈\[M\]m\\in\[M\]and any two coordinate subsets𝒮≠𝒮′\\mathcal\{S\}\\neq\\mathcal\{S\}^\{\\prime\}, the initialization error satisfies

‖δ𝒮,m‖22=‖δ𝒮′,m‖22=‖δm‖22\.\\displaystyle\\\|\\delta\_\{\\mathcal\{S\},m\}\\\|\_\{2\}^\{2\}=\\\|\\delta\_\{\\mathcal\{S\}^\{\\prime\},m\}\\\|\_\{2\}^\{2\}=\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\.\(11\)

Assumption[1](https://arxiv.org/html/2610.31702#Thmassumption1)models the shared propagation relationship throughwi,𝒮⋆w^\{\\star\}\_\{i,\\mathcal\{S\}\}, whileϵim\\epsilon\_\{i\}^\{m\}captures variations induced by different local contexts\. The coordinate\-specific varianceσi2\\sigma\_\{i\}^\{2\}represents the signal dynamic range\. Assumption[2](https://arxiv.org/html/2610.31702#Thmassumption2)standardizes the input covariance, while the proof of Lemma[1](https://arxiv.org/html/2610.31702#Thmlemma1)additionally uses the isotropic fourth\-moment identity specified in the proof section\. Assumption[3](https://arxiv.org/html/2610.31702#Thmassumption3)reflects the decreasing signal variability from the Tx outward and motivates ripple\-ordered estimation\. Finally, Assumption[4](https://arxiv.org/html/2610.31702#Thmassumption4)characterizes each client’s initialization mismatch by‖δm‖22\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}, thereby isolating the effect of the estimation order\.

###### Lemma 1\(One\-step estimation error on clientmm\)\.

Under Assumptions[1](https://arxiv.org/html/2610.31702#Thmassumption1)and[2](https://arxiv.org/html/2610.31702#Thmassumption2), for any clientm∈\[M\]m\\in\[M\], any output coordinatei∈\[d\]i\\in\[d\], and any subset𝒮⊆\[d\]\\mathcal\{S\}\\subseteq\[d\], after one \(implicit\) update step with step sizeη\\etausing allKKdemonstrations on clientmm, the expected squared estimation error satisfies

𝔼⁡\[\(yim−𝒙𝒮⊤​wi,𝒮1\)2\]\\displaystyle\\mathbb\{E\}\\Bigl\[\\bigl\(y\_\{i\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\}^\{\\top\}w^\{1\}\_\{i,\\mathcal\{S\}\}\\bigr\)^\{2\}\\Bigr\]≤\(1\+η2​\|𝒮\|K\)​σi2\+\(1−η\)2​‖δ𝒮,m‖22\\displaystyle\\leq\\Bigl\(1\+\\eta^\{2\}\\tfrac\{\|\\mathcal\{S\}\|\}\{K\}\\Bigr\)\\sigma\_\{i\}^\{2\}\+\(1\-\\eta\)^\{2\}\\\|\\delta\_\{\\mathcal\{S\},m\}\\\|\_\{2\}^\{2\}\+η2​\|𝒮\|​\(\|𝒮\|\+1\)K​‖δ𝒮,m‖22\.\\displaystyle\\quad\+\\eta^\{2\}\\tfrac\{\|\\mathcal\{S\}\|\(\|\\mathcal\{S\}\|\+1\)\}\{K\}\\,\\\|\\delta\_\{\\mathcal\{S\},m\}\\\|\_\{2\}^\{2\}\.\(12\)

### V\-BOptimal Estimation Order Analysis

We next examine how the estimation order affects the cumulative error on each client\. Letoobe a permutation of\{1,…,d\}\\\{1,\\ldots,d\\\}specifying the output order\. At steptt, coordinateo⁡\(t\)o\(t\)is estimated using the cumulative feature set𝒮⁡\(t\)≜\{o⁡\(1\),…,o⁡\(t\)\},\|𝒮⁡\(t\)\|=t\.\\mathcal\{S\}\(t\)\\triangleq\\\{o\(1\),\\ldots,o\(t\)\\\},\|\\mathcal\{S\}\(t\)\|=t\.Applying Lemma[1](https://arxiv.org/html/2610.31702#Thmlemma1)at each step and summing overt∈\[d\]t\\in\[d\]gives the following bound\.

###### Theorem 1\(Total estimation error bound on a single clientmm\)\.

Under Assumptions[1](https://arxiv.org/html/2610.31702#Thmassumption1)–[4](https://arxiv.org/html/2610.31702#Thmassumption4), for any clientm∈\[M\]m\\in\[M\]and any estimation orderoo, the total expected prediction error accumulated over allddsequential estimation steps satisfies

∑t=1d𝔼⁡\[\(yo⁡\(t\)m−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)1\)2\]\\displaystyle\\sum\\nolimits\_\{t=1\}^\{d\}\\mathbb\{E\}\\Bigl\[\\bigl\(y^\{m\}\_\{o\(t\)\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w^\{1\}\_\{\\mathcal\{S\}\(t\)\}\\bigr\)^\{2\}\\Bigr\]≤∑i=1dσi2\+\(1−η\)2​‖δm‖22​d\\displaystyle\\leq\\sum\\nolimits\_\{i=1\}^\{d\}\\sigma\_\{i\}^\{2\}\+\(1\-\\eta\)^\{2\}\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\\,d\+η2​‖δm‖22K​d​\(d\+1\)​\(d\+2\)3\+η2K​∑t=1dt​σo⁡\(t\)2\.\\displaystyle\\quad\+\\eta^\{2\}\\frac\{\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\}\{K\}\\,\\frac\{d\(d\+1\)\(d\+2\)\}\{3\}\+\\frac\{\\eta^\{2\}\}\{K\}\\sum\\nolimits\_\{t=1\}^\{d\}t\\,\\sigma\_\{o\(t\)\}^\{2\}\.\(13\)

###### Proposition 1\(Optimal estimation order\)\.

Under Assumption[3](https://arxiv.org/html/2610.31702#Thmassumption3), for any permutationooof\{1,…,d\}\\\{1,\\ldots,d\\\}, the order\-dependent termT⁡\(o\)≜∑t=1dt​σo⁡\(t\)2T\(o\)\\triangleq\\sum\_\{t=1\}^\{d\}t\\,\\sigma\_\{o\(t\)\}^\{2\}satisfies∑t=1dt​σo⁡\(t\)2≥∑t=1dt​σ\(d\+1−t\)2,\\sum\_\{t=1\}^\{d\}t\\,\\sigma\_\{o\(t\)\}^\{2\}\\geq\\sum\_\{t=1\}^\{d\}t\\,\\sigma\_\{\(d\+1\-t\)\}^\{2\},with equality if and only if the permutationooorders the sequence\{σi2\}i=1d\\\{\\sigma\_\{i\}^\{2\}\\\}\_\{i=1\}^\{d\}in nonincreasing order\.

###### Corollary 1\(Closed\-form estimation error on a single clientmm\)\.

Under Assumptions[1](https://arxiv.org/html/2610.31702#Thmassumption1)–[4](https://arxiv.org/html/2610.31702#Thmassumption4), by choosing the optimal estimation order in Proposition[1](https://arxiv.org/html/2610.31702#Thmproposition1), the total expected prediction error accumulated over allddsequential estimation steps satisfies

∑t=1d\\displaystyle\\sum\\nolimits\_\{t=1\}^\{d\}𝔼⁡\[\(ytm−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)1\)2\]≤∑i=1d\(1\+η2​iK\)​σi2\\displaystyle\\mathbb\{E\}\\Bigl\[\(y^\{m\}\_\{t\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w^\{1\}\_\{\\mathcal\{S\}\(t\)\}\)^\{2\}\\Bigr\]\\leq\\sum\\nolimits\_\{i=1\}^\{d\}\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\)\\sigma\_\{i\}^\{2\}\+d⁡\(\(1−η\)2\+η2K​\(d\+1\)​\(d\+2\)3\)​‖δm‖22\.\\displaystyle\\quad\+d\\Bigl\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\}\{K\}\\,\\frac\{\(d\+1\)\(d\+2\)\}\{3\}\\Bigr\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\.\(14\)

### V\-CAggregation Analysis

We next analyze the estimation error obtained by aggregating the predictions ofMMclients\.

###### Assumption 5\.

For any stept∈\[d\]t\\in\[d\]and any two distinct clientsm≠m′m\\neq m^\{\\prime\}, the client prediction errors w\.r\.t\. the ground truth are uncorrelated:𝔼⁡\[\(yo⁡\(t\)m−𝐱𝒮⁡\(t\)⊤​w𝒮⁡\(t\)\)​\(yo⁡\(t\)m′−𝐱𝒮⁡\(t\)⊤​w𝒮⁡\(t\)\)\]=0\.\\mathbb\{E\}\[\(y^\{m\}\_\{o\(t\)\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\)\(y^\{m^\{\\prime\}\}\_\{o\(t\)\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\)\]=0\.

Assumption[5](https://arxiv.org/html/2610.31702#Thmassumption5)models clients whose prediction errors contain no shared systematic component\. This condition is reasonable when clients use disjoint local demonstrations and their remaining prediction fluctuations are independent across clients\.

###### Theorem 2\(Closed\-form aggregated estimation error via uniform aggregation\)\.

Under Assumptions[1](https://arxiv.org/html/2610.31702#Thmassumption1)–[5](https://arxiv.org/html/2610.31702#Thmassumption5), let each client adopt the optimal ripple estimation order characterized in Proposition[1](https://arxiv.org/html/2610.31702#Thmproposition1)\. Then, under uniform averaging acrossMMclients, the total prediction error satisfies

∑t=1d𝔼⁡\[\(1M​∑m=1Mytm−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)\)2\]\\displaystyle\\sum\\nolimits\_\{t=1\}^\{d\}\\mathbb\{E\}\\Big\[\\Big\(\\frac\{1\}\{M\}\\sum\\nolimits\_\{m=1\}^\{M\}y^\{m\}\_\{t\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\\Big\)^\{2\}\\Big\]≤1M2∑m=1M\[∑i=1d\(1\+η2​iK\)σi2\\displaystyle\\leq\\frac\{1\}\{M^\{2\}\}\\sum\\nolimits\_\{m=1\}^\{M\}\\Big\[\\sum\\nolimits\_\{i=1\}^\{d\}\\Bigl\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\\Bigr\)\\sigma\_\{i\}^\{2\}\+d\(\(1−η\)2\+η2K\(d\+1\)​\(d\+2\)3\)∥δm∥22\]\.\\displaystyle\\qquad\+d\\Bigl\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\}\{K\}\\,\\frac\{\(d\+1\)\(d\+2\)\}\{3\}\\Bigr\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\\Big\]\.\(15\)

###### Theorem 3\(Closed\-form aggregated estimation error via weighted aggregation\)\.

Under Assumptions[1](https://arxiv.org/html/2610.31702#Thmassumption1)–[5](https://arxiv.org/html/2610.31702#Thmassumption5), let each client adopt the optimal ripple estimation order characterized in Proposition[1](https://arxiv.org/html/2610.31702#Thmproposition1)\. For any nonnegative aggregation weights\{αm\}m=1M\\\{\\alpha\_\{m\}\\\}\_\{m=1\}^\{M\}satisfying∑m=1Mαm=1\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}=1, define the aggregated prediction asyt𝒜≜∑m=1Mαm​ytm\.y^\{\\mathcal\{A\}\}\_\{t\}\\triangleq\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}\\,y^\{m\}\_\{t\}\.Then the total prediction error after weighted aggregation satisfies

∑t=1d𝔼⁡\[\(yt𝒜−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)\)2\]\\displaystyle\\sum\\nolimits\_\{t=1\}^\{d\}\\mathbb\{E\}\\Big\[\\big\(y^\{\\mathcal\{A\}\}\_\{t\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\\big\)^\{2\}\\Big\]≤∑m=1Mαm2\[∑i=1d\(1\+η2​iK\)σi2\\displaystyle\\leq\\sum\\nolimits\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}\\Big\[\\sum\\nolimits\_\{i=1\}^\{d\}\\Bigl\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\\Bigr\)\\sigma\_\{i\}^\{2\}\+d\(\(1−η\)2\+η2K\(d\+1\)​\(d\+2\)3\)∥δm∥22\]\.\\displaystyle\\qquad\+d\\Bigl\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\}\{K\}\\,\\frac\{\(d\+1\)\(d\+2\)\}\{3\}\\Bigr\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\\Big\]\.\(16\)

Theorem[3](https://arxiv.org/html/2610.31702#Thmtheorem3)motivates nonuniformimportance\-aware aggregationinPoly\-Ripple\. Ideally, clients whose local demonstrations better match the server\-issued query should receive larger weights, as they are expected to have smaller mismatch\-induced errors\. We next translate the bound into an importance\-aware design principle\. Specifically, the right\-hand side of \([16](https://arxiv.org/html/2610.31702#S5.E16)\) is quadratic in the aggregation weights\. We define the client\-dependent estimation error proxy

rm2≜∑i=1d\(1\+η2​iK\)​σi2\+d⁡\(\(1−η\)2\+η2K​\(d\+1\)​\(d\+2\)3\)​‖δm‖22,r\_\{m\}^\{2\}\\triangleq\\sum\\nolimits\_\{i=1\}^\{d\}\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\)\\sigma\_\{i\}^\{2\}\+d\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\}\{K\}\\,\\frac\{\(d\+1\)\(d\+2\)\}\{3\}\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\},

\(17\)and consider the following optimization problem:

𝐏𝟏min\{αm\}m=1M\\displaystyle\\mathbf\{P1\}\\qquad\\min\_\{\\\{\\alpha\_\{m\}\\\}\_\{m=1\}^\{M\}\}\\ \\∑m=1Mαm2​rm2\\displaystyle\\sum\\nolimits\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}r\_\{m\}^\{2\}\(18\)s\.t\.αm≥0,∀m∈\[M\],\\displaystyle\\alpha\_\{m\}\\geq 0,\\quad\\forall\\,m\\in\[M\],∑m=1Mαm=1\.\\displaystyle\\sum\\nolimits\_\{m=1\}^\{M\}\\alpha\_\{m\}=1\.
ProblemP1is important for two reasons\. First, it provides an explicit and principled objective that is implied by our theoretical bound, and therefore clarifies what a better aggregation rule should optimize\. Second, its closed\-form solution reveals an inverse\-risk allocation rule, which directly inspires our importance\-aware weighting strategy inPoly\-Ripple\.

###### Proposition 2\.

Assumerm2\>0r\_\{m\}^\{2\}\>0for allm∈\[M\]m\\in\[M\]\. The solution to the optimization problem in \([18](https://arxiv.org/html/2610.31702#S5.E18)\) follows an inverse\-risk allocation rule and can be expressed as

αm⋆∝1rm2⋅κ,∀m∈\[M\],\\displaystyle\\alpha\_\{m\}^\{\\star\}\\ \\propto\\ \\frac\{1\}\{r\_\{m\}^\{2\}\}\\cdot\\kappa,\\quad\\forall\\,m\\in\[M\],\(19\)where the \(client\-independent\) factorκ\\kappais defined byκ≜\(∑m′=1M1rm′2\)−1,\\kappa\\triangleq\(\\sum\\nolimits\_\{m^\{\\prime\}=1\}^\{M\}\\frac\{1\}\{r\_\{m^\{\\prime\}\}^\{2\}\}\)^\{\-1\},so that∑m=1Mαm⋆=1\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{\\star\}=1\.

## VIExperiments

We conduct extensive experiments to answer the following three questions\. The corresponding results and analyses are presented in Sections[VI\-B](https://arxiv.org/html/2610.31702#S6.SS2),[VI\-C](https://arxiv.org/html/2610.31702#S6.SS3), and[VI\-D](https://arxiv.org/html/2610.31702#S6.SS4), respectively\.

Q1:How doesPoly\-Ripplecompare with ICL baselines in terms of quantitative and qualitative performance?

Q2:How do the key parameters affect its performance?

Q3:Do the empirical results support the theoretical analysis?

### VI\-AExperimental Setup

Dataset\.We adopt a publicly available indoor radio map dataset\[[38](https://arxiv.org/html/2610.31702#bib.bib29)\]\. Each sample is associated with a building layout, a frequency band, an antenna radiation pattern, and a Tx location\. The dataset contains 25 building layouts, three frequency bands, and five antenna radiation patterns\. For each combination of building layout, frequency band, and antenna radiation pattern, radio maps are provided for 50 to 80 Tx locations, resulting in a total of 27,750 samples\. Each input consists of three channels representing reflectance \(dB\), transmittance \(dB\), and distance to the Tx \(m\), while the output is the corresponding radio map\.

Baselines\.\(1\) Local ICL\.Each client independently predicts the query issued by the server using its private demonstrations and ripple\-ordered autoregressive modeling\. The resulting local predictions are evaluated without server\-side aggregation\.\(2\) Collaborative ICL with Uniform Aggregation\.Multiple clients independently predict the same query using their private demonstrations\. The server then uniformly averages their local predictions by assigning each client a weight of1/M1/M\.

Implementation and Metrics\.We implement our framework usingPyTorch 2\.6and conduct all experiments on NVIDIA B200 GPUs\. We deploy2020clients, denoted by C1 to C20, and assign each client a unique building layout\. The remaining five building layouts, denoted by C21 to C25, are maintained by the server as unseen target layouts representing the user side\. We use a VQ\-GAN\[[39](https://arxiv.org/html/2610.31702#bib.bib37)\]for tokenization and a LLaMA model\[[40](https://arxiv.org/html/2610.31702#bib.bib38)\]for autoregressive prediction\. With a downsampling factor off=16f=16, the VQ\-GAN tokenizes each resized256×256256\\times 256image into256256discrete tokens using a codebook containing8,1928\{,\}192entries, each with dimension6464\. For autoregressive modeling, we adopt the LLaMA\-7B checkpoint released with LVM\[[32](https://arxiv.org/html/2610.31702#bib.bib36)\], which was pretrained on5050diverse datasets with420420billion tokens and demonstrates strong generalization to unseen tasks\. We use root mean squared error \(RMSE\) as the evaluation metric, where↓\\downarrowindicates that a lower value is better\. We report mean RMSE and standard deviation over five independent runs\.

### VI\-BResults and Analysis

To answerQ1, we evaluatePoly\-RipplewithM=20M=20clients\. Each client maintains private demonstrations from one building layout and constructs a local prompt containingK=3K=3paired input\-output demonstrations\. Queries are sampled from five unseen layouts, and broadcast to all clients for local ICL\.

![Refer to caption](https://arxiv.org/html/2610.31702v1/prompts.png)Fig\. 4:Local prompt construction withK=3K=3on different clients\.Diversity in local prompt construction\.For each query, clientmmconstructs its local prompt using the two\-step strategy described in Section[IV\-A](https://arxiv.org/html/2610.31702#S4.SS1)\. The first demonstration is selected as an anchor whose Tx location is closest to the query Tx location in normalized coordinates\. The remainingK−1K\-1demonstrations are selected to provide spatially diverse Tx locations within the same client layout\. As shown in Fig\.[4](https://arxiv.org/html/2610.31702#S6.F4), the first column contains the anchor demonstrations, while the remaining columns show the diverse demonstrations selected by different clients\. Fig\.[8](https://arxiv.org/html/2610.31702#S6.F8)further visualizes the selected Tx locations for a representative client\. The anchor provides a propagation context that is geometrically aligned with the query, while the remaining demonstrations introduce complementary propagation patterns from other Tx locations\. This design balances query relevance and contextual diversity under the limited context length of the LAVM\.

![Refer to caption](https://arxiv.org/html/2610.31702v1/all_score.png)Fig\. 5:A query layout and client layouts with varying structural relevance\.Heterogeneous relevance across clients\.Although all clients receive the same query, their private demonstrations are collected from different building layouts\. As illustrated in Fig\.[5](https://arxiv.org/html/2610.31702#S6.F5), some client layouts are structurally more similar to the query layout than others\. Consequently, their demonstrations provide different levels of contextual relevance and lead to heterogeneous local prediction quality\. A client with demonstrations that better match the query environment is more likely to provide useful information about its propagation patterns\. In contrast, uniform aggregation ignores this heterogeneity and assigns equal importance to all local predictions\. This observation is consistent with the client\-dependent mismatch characterized in our theoretical analysis\.

Fig\. 6:Aggregation weights under different temperature valuesτ\\tau\.Importance\-aware aggregation of distributed contextual knowledge\.To account for heterogeneous client relevance, we assign aggregation weights based on the similarity between each client layout and the query\. Fig\.[6](https://arxiv.org/html/2610.31702#S6.F6)shows the resulting weight distributions under two temperature values\. A smallerτ\\tauproduces a more concentrated distribution and gives greater influence to a few highly relevant clients\. A largerτ\\tauproduces a smoother distribution and allows more clients to contribute to the final prediction\. Thus,τ\\taucontrols the tradeoff between emphasizing the most relevant clients and preserving complementary information from multiple clients\. Importantly, collaboration enables the server to exploit demonstrations distributed across clients without placing them all into a single context\. This effectively increases the contextual information available for ICL while respecting the LAVM context length constraint and keeping private demonstrations locally stored\.

Fig\. 7:Selected Tx locations\.Fig\. 8:RMSE comparison\.
Quantitative performance comparison\.Fig\.[8](https://arxiv.org/html/2610.31702#S6.F8)compares local ICL, collaborative ICL with uniform aggregation, andPoly\-Rippleon the five unseen layouts, C21 to C25\.Poly\-Rippleachieves the lowest RMSE on all five layouts\. Collaborative ICL with uniform aggregation generally outperforms local ICL, showing that combining predictions derived from distributed demonstrations can reduce the error caused by each client’s limited context\. This result supports the intuition that additional relevant demonstrations improve ICL performance even when the context length prevents them from being processed together\. It is also consistent with the aggregation gain characterized by our theoretical analysis\. The further improvement ofPoly\-Rippleover collaborative ICL with uniform aggregation shows that performance depends not only on the number of clients but also on their relevance to the query\. Uniform aggregation gives less relevant clients the same influence as better aligned clients\. In contrast,Poly\-Rippleuses importance\-aware aggregation to emphasize predictions supported by more relevant demonstrations while retaining complementary information from other clients, thereby achieving consistently lower estimation error\.

Qualitative performance\.Fig\.[9](https://arxiv.org/html/2610.31702#S6.F9)presents two example queries and compares uniform aggregation with the importance\-aware aggregation ofPoly\-Ripple\. The predictions fromPoly\-Ripplemore closely match the target radio maps and better preserve the dominant attenuation patterns extending outward from the Tx\. This improvement is consistent with ripple\-ordered autoregressive modeling, which generates radio map tokens according to their spatial propagation dependencies from the Tx\. Uniform aggregation produces larger deviations in several regions because predictions from less relevant clients receive the same weights as those from better aligned clients\. By assigning greater importance to more relevant contextual information,Poly\-Ripplebetter reconstructs both the global propagation structure and local path loss variations\.

![Refer to caption](https://arxiv.org/html/2610.31702v1/two_exp.png)Fig\. 9:Qualitative comparison between uniform aggregation \(avg\) and the importance\-aware \(αm\\alpha\_\{m\}\) aggregation used byPoly\-Ripple\.
### VI\-CImpact of Key Parameters

To answerQ2, we study the effects of the number of participating clients and the aggregation temperatureτ\\tau\.

Impact of the number of clients\.Fig\.[11](https://arxiv.org/html/2610.31702#S6.F11)reports the average RMSE as the number of clients increases from55to2020\. The RMSE generally decreases as more clients participate, indicating that additional clients provide complementary contextual information for the same query\. Since the context length constraint of LAVM prevents all demonstrations from being processed within a single context, collaboration allows the server to exploit a larger set of distributed demonstrations through importance\-aware aggregation\. The improvement is more pronounced when the number of clients is small, while the marginal gain gradually decreases as more clients are included\. This trend suggests that the most relevant clients provide the largest benefit, whereas additional clients may contribute redundant or less relevant information\. This trend also supports our theoretical result that collaboration improves estimation accuracy by combining complementary client predictions\.

Impact of the aggregation temperatureτ\\tau\.Fig\.[11](https://arxiv.org/html/2610.31702#S6.F11)compares the RMSE under different values ofτ\\tau\. The results show thatτ\\tauaffects performance, but the variation remains relatively limited over a moderate range\. This indicates that the main gain ofPoly\-Ripplecomes from importance\-aware aggregation rather than careful tuning ofτ\\tau\. In our weighting mechanism,τ\\taucontrols the concentration of the client weights\. A smallerτ\\tauemphasizes the most relevant clients, whereas a largerτ\\tauallows more clients to contribute\. Moderate values therefore provide a practical balance between client relevance and complementary information\. Thus,Poly\-Rippledoes not require extensive tuning ofτ\\tauwhen applied to RME in a new building layout\.

Fig\. 10:Impact of number of clients\.Fig\. 11:Impact of temperatureτ\\tau\.

### VI\-DDiscussion of Theory vs\. Practice

To answerQ3, we examine whether the empirical results are consistent with the theoretical analysis in Section[V](https://arxiv.org/html/2610.31702#S5)\.

Ripple ordering and causal structure\.Theorem[1](https://arxiv.org/html/2610.31702#Thmtheorem1)shows the cumulative estimation error contains the order\-dependent term∑t=1dt​σo⁡\(t\)2\\sum\_\{t=1\}^\{d\}t\\sigma\_\{o\(t\)\}^\{2\}\. Since coordinates predicted later receive larger coefficients, coordinates with larger uncertainty should be placed earlier in the autoregressive sequence\. Proposition[1](https://arxiv.org/html/2610.31702#Thmproposition1)further shows that the optimal order arranges coordinates in nonincreasing order of their signal dynamic ranges\. Under Assumption[3](https://arxiv.org/html/2610.31702#Thmassumption3), this ordering corresponds to predicting spatial locations outward from the Tx, which matches the ripple order used byPoly\-Ripple\. The qualitative results in Fig\.[9](https://arxiv.org/html/2610.31702#S6.F9)show that the predicted maps preserve the dominant outward attenuation patterns, supporting the physical motivation and theoretical advantage of ripple ordering\.

Error reduction through multi\-client aggregation\.Theorem[2](https://arxiv.org/html/2610.31702#Thmtheorem2)shows that uniform averaging reduces the aggregated estimation error under the cross\-client uncorrelated error condition in Assumption[5](https://arxiv.org/html/2610.31702#Thmassumption5)\. When clients have comparable error levels, the theoretical bound yields an aggregation gain that scales with1/M1/M\. This result is consistent with Fig\.[8](https://arxiv.org/html/2610.31702#S6.F8), where collaborative ICL with uniform aggregation generally outperforms local ICL\. Fig\.[11](https://arxiv.org/html/2610.31702#S6.F11)provides further evidence, as the average RMSE decreases when more clients participate\. The smaller marginal gain at largerMMis also consistent with practice because additional clients may provide increasingly redundant or less relevant contextual information\.

Client\-dependent risks and importance\-aware weighting\.Theorem[3](https://arxiv.org/html/2610.31702#Thmtheorem3)characterizes the weighted aggregation error through∑m=1Mαm2​rm2\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}r\_\{m\}^\{2\}, whererm2r\_\{m\}^\{2\}denotes a client\-dependent estimation risk proxy\. Proposition[2](https://arxiv.org/html/2610.31702#Thmproposition2)further shows that clients with lower risks should receive larger aggregation weights\. Becauserm2r\_\{m\}^\{2\}, particularly the mismatch term‖δm‖22\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}, is not directly observable at the server,Poly\-Rippleuses layout similarity as a practical proxy for client\-query relevance\. Fig\.[5](https://arxiv.org/html/2610.31702#S6.F5)illustrates the heterogeneous relevance of different client layouts, while Fig\.[6](https://arxiv.org/html/2610.31702#S6.F6)shows the resulting nonuniform weights\. The lower RMSE ofPoly\-Ripplecompared with uniform aggregation in Fig\.[8](https://arxiv.org/html/2610.31702#S6.F8)supports the theoretical benefit of assigning larger weights to more relevant clients\. The qualitative results in Fig\.[9](https://arxiv.org/html/2610.31702#S6.F9)further show that importance\-aware aggregation produces radio maps that more closely match the targets\.

Theory\-guided temperature design\.Proposition[2](https://arxiv.org/html/2610.31702#Thmproposition2)shows that clients with lower estimation risks should receive larger aggregation weights\. This result motivates our similarity\-based weighting design, where layout similarity serves as a practical proxy for the unobservable client risk\. The temperatureτ\\tauis introduced to control the concentration of these relevance\-based weights\. Fig\.[11](https://arxiv.org/html/2610.31702#S6.F11)shows that the RMSE varies withτ\\taubut remains relatively stable over a moderate range\. This result indicates thatτ\\tauprimarily adjusts the weight concentration, while the main performance gains arise from multi\-client collaboration and nonuniform weighting according to client relevance\.

## VIIConclusion

In conclusion, we formulate indoor RME as a collaborative ICL problem and proposePoly\-Ripplefor estimating radio maps without task\-specific model updates\. Each client locally performs ICL using its private demonstrations, while ripple\-ordered autoregressive modeling captures the causal structure of wireless propagation\. The server then applies importance\-aware aggregation to combine complementary client predictions under privacy and context length constraints\. Our theoretical analysis characterizes the effects of ripple ordering, multi\-client collaboration, and nonuniform aggregation on estimation error\. Experiments on a public RME benchmark show thatPoly\-Rippleconsistently outperforms representative ICL baselines, and the empirical results support our theoretical analysis\.

## Proofs of Theoretical Results

### VII\-AProof of Lemma[1](https://arxiv.org/html/2610.31702#Thmlemma1)

###### Proof\.

Fix a clientm∈\[M\]m\\in\[M\], an output coordinatei∈\[d\]i\\in\[d\], and a feature subset𝒮⊆\[d\]\\mathcal\{S\}\\subseteq\[d\]\. ClientmmobservesKKi\.i\.d\. demonstrations satisfyingyi,km=x𝒮k⊤​w𝒮⋆\+ϵi,kmy\_\{i,k\}^\{m\}=x\_\{\\mathcal\{S\}\}^\{k\\top\}w\_\{\\mathcal\{S\}\}^\{\\star\}\+\\epsilon\_\{i,k\}^\{m\}\. LetX𝒮∈ℝK×\|𝒮\|X\_\{\\mathcal\{S\}\}\\in\\mathbb\{R\}^\{K\\times\|\\mathcal\{S\}\|\}be the stacked input matrix, and letYi,m,εi,m∈ℝKY\_\{i,m\},\\varepsilon\_\{i,m\}\\in\\mathbb\{R\}^\{K\}be the stacked outputs and noise, respectively\. Hence,Yi,m=X𝒮​w𝒮⋆\+εi,mY\_\{i,m\}=X\_\{\\mathcal\{S\}\}w\_\{\\mathcal\{S\}\}^\{\\star\}\+\\varepsilon\_\{i,m\}\. Starting fromw𝒮,m0w\_\{\\mathcal\{S\},m\}^\{0\}, consider one gradient step onLm​\(w\)=12​K​‖Yi,m−X𝒮​w‖22L\_\{m\}\(w\)=\\frac\{1\}\{2K\}\\\|Y\_\{i,m\}\-X\_\{\\mathcal\{S\}\}w\\\|\_\{2\}^\{2\}\. DefineG=1K​X𝒮⊤​X𝒮G=\\frac\{1\}\{K\}X\_\{\\mathcal\{S\}\}^\{\\top\}X\_\{\\mathcal\{S\}\}andδ𝒮,m=w𝒮,m0−w𝒮⋆\\delta\_\{\\mathcal\{S\},m\}=w\_\{\\mathcal\{S\},m\}^\{0\}\-w\_\{\\mathcal\{S\}\}^\{\\star\}\. Then,em≜w𝒮,m1−w𝒮⋆=\(I\|𝒮\|−η​G\)​δ𝒮,m\+ηK​X𝒮⊤​εi,m\.e\_\{m\}\\triangleq w\_\{\\mathcal\{S\},m\}^\{1\}\-w\_\{\\mathcal\{S\}\}^\{\\star\}=\(I\_\{\|\\mathcal\{S\}\|\}\-\\eta G\)\\delta\_\{\\mathcal\{S\},m\}\+\\frac\{\\eta\}\{K\}X\_\{\\mathcal\{S\}\}^\{\\top\}\\varepsilon\_\{i,m\}\.Next, consider an independent test sample satisfyingyim=x𝒮⊤​w𝒮⋆\+ϵimy\_\{i\}^\{m\}=x\_\{\\mathcal\{S\}\}^\{\\top\}w\_\{\\mathcal\{S\}\}^\{\\star\}\+\\epsilon\_\{i\}^\{m\}\. Its prediction residual isrm=ϵim−x𝒮⊤​emr\_\{m\}=\\epsilon\_\{i\}^\{m\}\-x\_\{\\mathcal\{S\}\}^\{\\top\}e\_\{m\}\. By the independence and standardized\-input assumptions,𝔼⁡\[rm2\]=σi2\+𝔼​‖em‖22\.\\mathbb\{E\}\[r\_\{m\}^\{2\}\]=\\sigma\_\{i\}^\{2\}\+\\mathbb\{E\}\\\|e\_\{m\}\\\|\_\{2\}^\{2\}\.Writeem=am\+bme\_\{m\}=a\_\{m\}\+b\_\{m\}, wheream=\(I\|𝒮\|−η​G\)​δ𝒮,ma\_\{m\}=\(I\_\{\|\\mathcal\{S\}\|\}\-\\eta G\)\\delta\_\{\\mathcal\{S\},m\}andbm=ηK​X𝒮⊤​εi,mb\_\{m\}=\\frac\{\\eta\}\{K\}X\_\{\\mathcal\{S\}\}^\{\\top\}\\varepsilon\_\{i,m\}\. Conditioning onX𝒮X\_\{\\mathcal\{S\}\}gives𝔼⁡\[am⊤​bm\]=0\\mathbb\{E\}\[a\_\{m\}^\{\\top\}b\_\{m\}\]=0and𝔼​‖bm‖22=η2​σi2K2​𝔼​tr⁡\(X𝒮​X𝒮⊤\)=η2​σi2​\|𝒮\|K\.\\mathbb\{E\}\\\|b\_\{m\}\\\|\_\{2\}^\{2\}=\\frac\{\\eta^\{2\}\\sigma\_\{i\}^\{2\}\}\{K^\{2\}\}\\mathbb\{E\}\\operatorname\{tr\}\(X\_\{\\mathcal\{S\}\}X\_\{\\mathcal\{S\}\}^\{\\top\}\)=\\frac\{\\eta^\{2\}\\sigma\_\{i\}^\{2\}\|\\mathcal\{S\}\|\}\{K\}\.Moreover, writingG=I\|𝒮\|\+ΔG=I\_\{\|\\mathcal\{S\}\|\}\+\\Deltawith𝔼⁡\[Δ\]=0\\mathbb\{E\}\[\\Delta\]=0yieldsam=\(1−η\)​δ𝒮,m−η​Δ​δ𝒮,ma\_\{m\}=\(1\-\\eta\)\\delta\_\{\\mathcal\{S\},m\}\-\\eta\\Delta\\delta\_\{\\mathcal\{S\},m\}\. Therefore,𝔼​‖am‖22≤\(\(1−η\)2\+η2​𝔼​‖Δ‖F2\)​‖δ𝒮,m‖22\.\\mathbb\{E\}\\\|a\_\{m\}\\\|\_\{2\}^\{2\}\\leq\\left\(\(1\-\\eta\)^\{2\}\+\\eta^\{2\}\\mathbb\{E\}\\\|\\Delta\\\|\_\{F\}^\{2\}\\right\)\\\|\\delta\_\{\\mathcal\{S\},m\}\\\|\_\{2\}^\{2\}\.For isotropic Gaussian inputs,𝔼​‖Δ‖F2=\|𝒮\|\(\|𝒮\|\+1\)/K\\mathbb\{E\}\\\|\\Delta\\\|\_\{F\}^\{2\}=\|\\mathcal\{S\}\|\(\|\\mathcal\{S\}\|\+1\)/K\. Combining the above results gives𝔼⁡\[\(yim−x𝒮⊤​w𝒮,m1\)2\]≤\(1\+η2​\|𝒮\|K\)​σi2\+\(\(1−η\)2\+η2​\|𝒮\|​\(\|𝒮\|\+1\)K\)​‖δ𝒮,m‖22,\\mathbb\{E\}\\\!\[\(y\_\{i\}^\{m\}\-x\_\{\\mathcal\{S\}\}^\{\\top\}w\_\{\\mathcal\{S\},m\}^\{1\}\)^\{2\}\]\\leq\(1\+\\frac\{\\eta^\{2\}\|\\mathcal\{S\}\|\}\{K\}\)\\sigma\_\{i\}^\{2\}\+\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\|\\mathcal\{S\}\|\(\|\\mathcal\{S\}\|\+1\)\}\{K\}\)\\\|\\delta\_\{\\mathcal\{S\},m\}\\\|\_\{2\}^\{2\},which proves the result\.□\\square

### VII\-BProof of Theorem[1](https://arxiv.org/html/2610.31702#Thmtheorem1)

###### Proof\.

Fix a clientm∈\[M\]m\\in\[M\]\. At stept∈\[d\]t\\in\[d\], apply Lemma[1](https://arxiv.org/html/2610.31702#Thmlemma1)to coordinatei=o⁡\(t\)i=o\(t\)and subset𝒮=𝒮⁡\(t\)\\mathcal\{S\}=\\mathcal\{S\}\(t\)\. Since\|𝒮⁡\(t\)\|=t\|\\mathcal\{S\}\(t\)\|=t, Assumption[4](https://arxiv.org/html/2610.31702#Thmassumption4)gives𝔼⁡\[\(yo⁡\(t\)m−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\),m1\)2\]≤\(1\+η2​tK\)​σo⁡\(t\)2\+\(\(1−η\)2\+η2​t​\(t\+1\)K\)​‖δm‖22\.\\mathbb\{E\}\\\!\[\(y\_\{o\(t\)\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\),m\}^\{1\}\)^\{2\}\]\\leq\(1\+\\frac\{\\eta^\{2\}t\}\{K\}\)\\sigma\_\{o\(t\)\}^\{2\}\+\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}t\(t\+1\)\}\{K\}\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\.Next, summing overt=1,…,dt=1,\\ldots,dyields∑t=1d𝔼⁡\[\(yo⁡\(t\)m−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\),m1\)2\]≤∑t=1dσo⁡\(t\)2\+η2K​∑t=1dt​σo⁡\(t\)2\+d​\(1−η\)2​‖δm‖22\+η2​‖δm‖22K​∑t=1dt⁡\(t\+1\)\.\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\\\!\[\(y\_\{o\(t\)\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\),m\}^\{1\}\)^\{2\}\]\\leq\\sum\_\{t=1\}^\{d\}\\sigma\_\{o\(t\)\}^\{2\}\+\\frac\{\\eta^\{2\}\}\{K\}\\sum\_\{t=1\}^\{d\}t\\sigma\_\{o\(t\)\}^\{2\}\+d\(1\-\\eta\)^\{2\}\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\+\\frac\{\\eta^\{2\}\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\}\{K\}\\sum\_\{t=1\}^\{d\}t\(t\+1\)\.Sinceoois a permutation,∑t=1dσo⁡\(t\)2=∑i=1dσi2\\sum\_\{t=1\}^\{d\}\\sigma\_\{o\(t\)\}^\{2\}=\\sum\_\{i=1\}^\{d\}\\sigma\_\{i\}^\{2\}\. Moreover,∑t=1dt⁡\(t\+1\)=d⁡\(d\+1\)​\(d\+2\)/3\\sum\_\{t=1\}^\{d\}t\(t\+1\)=d\(d\+1\)\(d\+2\)/3\. Substituting these identities into the preceding inequality gives∑t=1d𝔼⁡\[\(yo⁡\(t\)m−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\),m1\)2\]≤∑i=1dσi2\+d​\(1−η\)2​‖δm‖22\+η2​‖δm‖22K​d​\(d\+1\)​\(d\+2\)3\+η2K​∑t=1dt​σo⁡\(t\)2\.\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\\\!\[\(y\_\{o\(t\)\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\),m\}^\{1\}\)^\{2\}\]\\leq\\sum\_\{i=1\}^\{d\}\\sigma\_\{i\}^\{2\}\+d\(1\-\\eta\)^\{2\}\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\+\\frac\{\\eta^\{2\}\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\}\{K\}\\frac\{d\(d\+1\)\(d\+2\)\}\{3\}\+\\frac\{\\eta^\{2\}\}\{K\}\\sum\_\{t=1\}^\{d\}t\\sigma\_\{o\(t\)\}^\{2\}\.□\\square

### VII\-CProof of Theorem[2](https://arxiv.org/html/2610.31702#Thmtheorem2)

###### Proof\.

Since every client follows the ripple ordero⁡\(t\)=to\(t\)=t, define the residual of clientmmat stepttasrtm≜ytm−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)r\_\{t\}^\{m\}\\triangleq y\_\{t\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\. Accordingly, the residual of the aggregated prediction isr¯t=1M​∑m=1Mrtm\\bar\{r\}\_\{t\}=\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}r\_\{t\}^\{m\}\. By Assumption[5](https://arxiv.org/html/2610.31702#Thmassumption5), the residuals of distinct clients are uncorrelated\. Therefore, all cross terms vanish, and𝔼⁡\[r¯t2\]=1M2​∑m=1M𝔼⁡\[\(rtm\)2\]\.\\mathbb\{E\}\[\\bar\{r\}\_\{t\}^\{2\}\]=\\frac\{1\}\{M^\{2\}\}\\sum\_\{m=1\}^\{M\}\\mathbb\{E\}\[\(r\_\{t\}^\{m\}\)^\{2\}\]\.Next, summing over allddsteps gives∑t=1d𝔼⁡\[r¯t2\]=1M2​∑m=1M∑t=1d𝔼⁡\[\(rtm\)2\]\.\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\[\\bar\{r\}\_\{t\}^\{2\}\]=\\frac\{1\}\{M^\{2\}\}\\sum\_\{m=1\}^\{M\}\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\[\(r\_\{t\}^\{m\}\)^\{2\}\]\.For each clientmm, Corollary[1](https://arxiv.org/html/2610.31702#Thmcorollary1)yields∑t=1d𝔼⁡\[\(rtm\)2\]≤∑i=1d\(1\+η2​iK\)​σi2\+d⁡\(\(1−η\)2\+η2​\(d\+1\)​\(d\+2\)3​K\)​‖δm‖22\.\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\[\(r\_\{t\}^\{m\}\)^\{2\}\]\\leq\\sum\_\{i=1\}^\{d\}\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\)\\sigma\_\{i\}^\{2\}\+d\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\(d\+1\)\(d\+2\)\}\{3K\}\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\.Finally, substituting this bound into the preceding equality gives∑t=1d𝔼⁡\[\(1M​∑m=1Mytm−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)\)2\]≤1M2​∑m=1M\[∑i=1d\(1\+η2​iK\)​σi2\+d⁡\(\(1−η\)2\+η2​\(d\+1\)​\(d\+2\)3​K\)​‖δm‖22\],\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\\\!\[\(\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}y\_\{t\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\)^\{2\}\]\\leq\\frac\{1\}\{M^\{2\}\}\\sum\_\{m=1\}^\{M\}\[\\sum\_\{i=1\}^\{d\}\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\)\\sigma\_\{i\}^\{2\}\+d\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\(d\+1\)\(d\+2\)\}\{3K\}\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\],which is exactly \([15](https://arxiv.org/html/2610.31702#S5.E15)\)\.□\\square

### VII\-DProof of Theorem[3](https://arxiv.org/html/2610.31702#Thmtheorem3)

###### Proof\.

Since all clients follow the ripple ordero⁡\(t\)=to\(t\)=t, define the client residual at stepttasrtm≜ytm−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)r\_\{t\}^\{m\}\\triangleq y\_\{t\}^\{m\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\. Under weighted aggregation,y^t𝒜=∑m=1Mαm​ytm\\hat\{y\}\_\{t\}^\{\\mathcal\{A\}\}=\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}y\_\{t\}^\{m\}, and hence the aggregated residual isr~t=∑m=1Mαm​rtm\\tilde\{r\}\_\{t\}=\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}r\_\{t\}^\{m\}\. By Assumption[5](https://arxiv.org/html/2610.31702#Thmassumption5), the residuals of any two distinct clients are uncorrelated\. Therefore, the cross terms vanish, and𝔼⁡\[r~t2\]=∑m=1Mαm2​𝔼​\[\(rtm\)2\]\.\\mathbb\{E\}\[\\tilde\{r\}\_\{t\}^\{2\}\]=\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}\\mathbb\{E\}\[\(r\_\{t\}^\{m\}\)^\{2\}\]\.Next, summing overt=1,…,dt=1,\\ldots,dand exchanging the two summations gives∑t=1d𝔼⁡\[r~t2\]=∑m=1Mαm2​∑t=1d𝔼⁡\[\(rtm\)2\]\.\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\[\\tilde\{r\}\_\{t\}^\{2\}\]=\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\[\(r\_\{t\}^\{m\}\)^\{2\}\]\.For each clientmm, Corollary[1](https://arxiv.org/html/2610.31702#Thmcorollary1)implies∑t=1d𝔼⁡\[\(rtm\)2\]≤∑i=1d\(1\+η2​iK\)​σi2\+d⁡\(\(1−η\)2\+η2​\(d\+1\)​\(d\+2\)3​K\)​‖δm‖22\.\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\[\(r\_\{t\}^\{m\}\)^\{2\}\]\\leq\\sum\_\{i=1\}^\{d\}\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\)\\sigma\_\{i\}^\{2\}\+d\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\(d\+1\)\(d\+2\)\}\{3K\}\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\.Finally, substituting this bound into the preceding equality yields∑t=1d𝔼⁡\[\(y^t𝒜−𝒙𝒮⁡\(t\)⊤​w𝒮⁡\(t\)\)2\]≤∑m=1Mαm2​\[∑i=1d\(1\+η2​iK\)​σi2\+d⁡\(\(1−η\)2\+η2​\(d\+1\)​\(d\+2\)3​K\)​‖δm‖22\],\\sum\_\{t=1\}^\{d\}\\mathbb\{E\}\\\!\[\(\\hat\{y\}\_\{t\}^\{\\mathcal\{A\}\}\-\\boldsymbol\{x\}\_\{\\mathcal\{S\}\(t\)\}^\{\\top\}w\_\{\\mathcal\{S\}\(t\)\}\)^\{2\}\]\\leq\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}\[\\sum\_\{i=1\}^\{d\}\(1\+\\frac\{\\eta^\{2\}i\}\{K\}\)\\sigma\_\{i\}^\{2\}\+d\(\(1\-\\eta\)^\{2\}\+\\frac\{\\eta^\{2\}\(d\+1\)\(d\+2\)\}\{3K\}\)\\\|\\delta\_\{m\}\\\|\_\{2\}^\{2\}\],which is exactly \([16](https://arxiv.org/html/2610.31702#S5.E16)\)\.□\\square

### VII\-EProof of Proposition[2](https://arxiv.org/html/2610.31702#Thmproposition2)

###### Proof\.

Sincerm2\>0r\_\{m\}^\{2\}\>0,P1is convex with a feasible set, i\.e\.,P1admits a unique global minimizer\. For any feasible weights satisfying∑m=1Mαm=1\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}=1, the Cauchy Schwarz inequality gives\(∑m=1Mαm2​rm2\)​\(∑m=1M1rm2\)≥\(∑m=1Mαm\)2=1\.\(\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}r\_\{m\}^\{2\}\)\(\\sum\_\{m=1\}^\{M\}\\frac\{1\}\{r\_\{m\}^\{2\}\}\)\\geq\(\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}\)^\{2\}=1\.Consequently,∑m=1Mαm2​rm2≥\(∑m=1M1rm2\)−1\.\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}^\{2\}r\_\{m\}^\{2\}\\geq\(\\sum\_\{m=1\}^\{M\}\\frac\{1\}\{r\_\{m\}^\{2\}\}\)^\{\-1\}\.Moreover, equality holds if and only ifαm​rm∝1/rm\\alpha\_\{m\}r\_\{m\}\\propto 1/r\_\{m\}, or equivalently,αm∝1/rm2\\alpha\_\{m\}\\propto 1/r\_\{m\}^\{2\}\. Enforcing the constraint∑m=1Mαm=1\\sum\_\{m=1\}^\{M\}\\alpha\_\{m\}=1therefore yieldsαm⋆=rm−2∑m′=1Mrm′−2=1rm2​\(∑m′=1M1rm′2\)−1,m∈\[M\]\.\\alpha\_\{m\}^\{\\star\}=\\frac\{r\_\{m\}^\{\-2\}\}\{\\sum\_\{m^\{\\prime\}=1\}^\{M\}r\_\{m^\{\\prime\}\}^\{\-2\}\}=\\frac\{1\}\{r\_\{m\}^\{2\}\}\(\\sum\_\{m^\{\\prime\}=1\}^\{M\}\\frac\{1\}\{r\_\{m^\{\\prime\}\}^\{2\}\}\)^\{\-1\},m\\in\[M\]\.Finally, definingκ≜\(∑m′=1M1/rm′2\)−1\\kappa\\triangleq\(\\sum\_\{m^\{\\prime\}=1\}^\{M\}1/r\_\{m^\{\\prime\}\}^\{2\}\)^\{\-1\}givesαm⋆=κ/rm2\\alpha\_\{m\}^\{\\star\}=\\kappa/r\_\{m\}^\{2\}, which is exactly \([19](https://arxiv.org/html/2610.31702#S5.E19)\)\. Sincerm2\>0r\_\{m\}^\{2\}\>0, we haveαm⋆\>0\\alpha\_\{m\}^\{\\star\}\>0for everymm, so all constraints inP1are satisfied\.□\\square

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