Transfer Learning in High-dimensional Ising Models

# Transfer Learning in High-dimensional Ising Models
Source: [https://arxiv.org/html/2607.03005](https://arxiv.org/html/2607.03005)
###### Abstract

In high\-dimensional Ising model estimation, target sample sizes are often limited, and effectively using auxiliary binary datasets of unknown relevance remains challenging\. To address this, we propose Trans\-Ising, a transfer learning method that combines a loss\-based source screening rule with a two\-stage estimation procedure\. The method first identifies informative auxiliary sources using held\-out target pseudolikelihood to prevent negative transfer\. It then computes an initial estimator via pooled nodewiseℓ1\\ell\_\{1\}\-regularized logistic regression, followed by a target\-only correction step using a folded\-concave penalty\. Theoretically, we establish fixed\-nodeℓ2\\ell\_\{2\}andℓ1\\ell\_\{1\}error bounds, exact graph selection consistency, and the conditional consistency of the screening rule\. Through extensive simulations and real\-data analyses, we demonstrate that Trans\-Ising achieves lower estimation errors than both target\-only estimation and naive data pooling\.

Transfer learning, High\-dimensional inference, Ising model, Graphical model selection, Pseudolikelihood, Source detection

## 1Introduction

The Ising model has a long history in spatial statistics and statistical physics\(Besag,[1974](https://arxiv.org/html/2607.03005#bib.bib2),[1975](https://arxiv.org/html/2607.03005#bib.bib3)\)\. It is widely used for binary network data in psychometrics and related applications\(van Borkulo et al\.,[2014](https://arxiv.org/html/2607.03005#bib.bib43); Epskamp et al\.,[2018](https://arxiv.org/html/2607.03005#bib.bib17); Waldorp et al\.,[2019](https://arxiv.org/html/2607.03005#bib.bib46); Park et al\.,[2022](https://arxiv.org/html/2607.03005#bib.bib35); Brusco et al\.,[2023](https://arxiv.org/html/2607.03005#bib.bib6)\), and recent work has studied statistical inference and structure learning for Ising models\(Bhattacharya & Mukherjee,[2018](https://arxiv.org/html/2607.03005#bib.bib4); Lokhov et al\.,[2018](https://arxiv.org/html/2607.03005#bib.bib28); Meng et al\.,[2021](https://arxiv.org/html/2607.03005#bib.bib32)\)\. In this model, each variable represents a binary state, and the interaction network represents pairwise dependence\. Estimating this edge set is a common inferential target in these applications because edges are interpreted as conditional associations among binary variables\. A standard approach for high\-dimensional Ising model selection is nodewiseℓ1\\ell\_\{1\}\-regularized logistic regression\(Ravikumar et al\.,[2010](https://arxiv.org/html/2607.03005#bib.bib38)\)\. This neighborhood\-selection method avoids the intractable partition function and has become a common baseline in the Ising\-model literature\(Santhanam & Wainwright,[2012](https://arxiv.org/html/2607.03005#bib.bib40); Barber & Drton,[2015](https://arxiv.org/html/2607.03005#bib.bib1); Kuang et al\.,[2017](https://arxiv.org/html/2607.03005#bib.bib25); De Canditiis,[2020](https://arxiv.org/html/2607.03005#bib.bib15)\)\. For example, in cancer genomics, modeling mutation profiles requires estimating interactions amongp=200p=200genes using limited target observations, such asn0=160n\_\{0\}=160samples\. Whenppis comparable to or larger thann0n\_\{0\}, target\-only nodewise logistic regressions estimate sparse neighborhoods from limited target observations, which increases false\-positive and false\-negative edge errors\.

To reduce the target estimation error, transfer learning methods use related auxiliary datasets\(Pan & Yang,[2010](https://arxiv.org/html/2607.03005#bib.bib34); Fawaz et al\.,[2018](https://arxiv.org/html/2607.03005#bib.bib19); Cai & Wei,[2021](https://arxiv.org/html/2607.03005#bib.bib10); Cai & Pu,[2024](https://arxiv.org/html/2607.03005#bib.bib9); Cai et al\.,[2024a](https://arxiv.org/html/2607.03005#bib.bib8),[b](https://arxiv.org/html/2607.03005#bib.bib11); Weiss et al\.,[2016](https://arxiv.org/html/2607.03005#bib.bib47); Zhuang et al\.,[2021](https://arxiv.org/html/2607.03005#bib.bib52); Hosna et al\.,[2022](https://arxiv.org/html/2607.03005#bib.bib22)\)\. It has been thoroughly developed byLi et al\. \([2022](https://arxiv.org/html/2607.03005#bib.bib26)\)andTian & Feng \([2023](https://arxiv.org/html/2607.03005#bib.bib41)\)for high\-dimensional linear and generalized linear models, byKim et al\. \([2025](https://arxiv.org/html/2607.03005#bib.bib24)\)for benign\-overfitting linear regression, and byPark et al\. \([2025](https://arxiv.org/html/2607.03005#bib.bib37)\)for large\-scale low\-rank regression, with related approaches inLi et al\. \([2023](https://arxiv.org/html/2607.03005#bib.bib27)\)andZhao et al\. \([2026](https://arxiv.org/html/2607.03005#bib.bib50)\)for Gaussian graphical models\. For Gaussian graphical models, correction\-based transfer procedures reduce precision\-matrix estimation error when sources share structure with the target\(Zhao et al\.,[2026](https://arxiv.org/html/2607.03005#bib.bib50)\)\. Despite the growing availability of auxiliary binary datasets, there has been limited research on transfer learning for discrete graphical models such as the Ising model\.

Extending transfer learning theory to high\-dimensional Ising models presents technical difficulties\. Each nodewise Ising regression uses a signed logistic pseudolikelihood, and cross\-domain heterogeneity shifts the pooled population minimizer away from the target parameter\. This shift alters the edge\-selection behavior of standardℓ1\\ell\_\{1\}\-penalized estimators\. Because source relevance is unknown in practice, naive pooling increases the target validation risk and the target estimation error when incompatible sources are included\.

In this article, we focus on the problem of transfer learning for high\-dimensional Ising graph estimation and propose*Trans\-Ising*, a two\-stage procedure\. The proposed construction first obtains an initial estimator by nodewiseℓ1\\ell\_\{1\}\-regularized logistic regression using the target data and selected auxiliary samples, and then applies a target\-only correction step using a folded\-concave penalty\. In addition, we develop a loss\-based source screening rule using held\-out target pseudolikelihood to select informative sources and avoid increasing the target validation risk\.

Contributions\.This article makes several contributions to transfer learning for Ising model estimation in a high\-dimensional context:

- •Oracle Two\-Step Estimation and Error Bounds:For a known informative source set, we construct an oracle two\-step nodewise estimator using pooled logistic lasso initialization and target\-only correction\. Letsjs\_\{j\}denote the target neighborhood size,NNdenote the combined sample size from the target and informative auxiliary sources, andhjh\_\{j\}denote the source\-to\-target heterogeneity level\. Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)establishes fixed\-nodeℓ2\\ell\_\{2\}andℓ1\\ell\_\{1\}error bounds whose leading variance term issj​log⁡p/N\\sqrt\{s\_\{j\}\\log p/N\}and whose additional terms depend onhjh\_\{j\}\.
- •Dual\-Penalty Correction for Exact Support Recovery:We analyze a dual\-penalty correction step for support recovery in high\-dimensional Ising models\. While recent high\-dimensional transfer learning methods for generalized linear models\(Tian & Feng,[2023](https://arxiv.org/html/2607.03005#bib.bib41)\)established estimation guarantees for correction\-based estimators withℓ1\\ell\_\{1\}\-regularization, exact neighborhood recovery for high\-dimensional Ising models requires a separate support\-recovery analysis\. This is becauseℓ1\\ell\_\{1\}penalization introduces shrinkage on nonzero coefficients, and classical lasso selection analyses impose irrepresentable\-type conditions\(Zhao & Yu,[2006](https://arxiv.org/html/2607.03005#bib.bib51)\)\. Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)establishes fixed\-node exact neighborhood recovery for the oracle two\-step estimator under the stated beta\-min, separation, empirical curvature, local\-solution, and cone conditions, without imposing an irrepresentable condition\. In addition, Corollary[2](https://arxiv.org/html/2607.03005#Thmcorollary2)shows that the AND\-symmetrized estimator recovers the target edge set when the fixed\-node selection result holds uniformly over nodes\.
- •Data\-Driven Source Detection and Empirical Validation:We develop a data\-driven source detection algorithm constructed from out\-of\-sample pseudolikelihood loss\. Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3)establishes conditional recovery of a risk\-defined population informative\-source set under graph\-level separation, validation\-loss deviation, and adaptive\-threshold calibration conditions\. Finally, through simulation studies and real\-data analyses involving mutation, online\-transaction, and movie\-rating data, we show that Trans\-Ising attains a lower estimation error than target\-only estimation in the reported settings, and a lower error than naive pooling in most reported settings\. The simulations explicitly assess parameter estimation and graph recovery, while the real\-data analyses assess held\-out nodewise prediction error because the true interaction networks are unknown\.

Notation\.We define an Ising model on an undirected graphG=\(V,E\)G=\(V,E\), whereV=\{1,…,p\}V=\\\{1,\\dots,p\\\}denotes the node set andEEdenotes the edge set\. We work with the\{−1,1\}\\\{\-1,1\\\}coding\. LetX=\(X1,…,Xp\)∈\{−1,1\}pX=\(X\_\{1\},\\dots,X\_\{p\}\)\\in\\\{\-1,1\\\}^\{p\}denote a random configuration with realizationxx\. LetX\(0\)X^\{\(0\)\}denote the target dataset with sample sizen0n\_\{0\}\. We observeSSauxiliary datasets\{X\(s\)\}s=1S\\\{X^\{\(s\)\}\\\}\_\{s=1\}^\{S\}, whereX\(s\)X^\{\(s\)\}has sample sizensn\_\{s\}\. Letθ∗∈ℝp×p\\theta^\{\\ast\}\\in\\mathbb\{R\}^\{p\\times p\}denote the true target parameter, and letw\(s\)w^\{\(s\)\}denote the parameter of sourcess\. The target and source interaction matrices are symmetric, and each has zero diagonal\. The target edge set is defined as

E=\{\{j,k\}:1≤j<k≤p,θj​k∗≠0\}\.E=\\bigl\\\{\\\{j,k\\\}:1\\leq j<k\\leq p,\\ \\theta^\{\\ast\}\_\{jk\}\\neq 0\\bigr\\\}\.Letδ\(s\):=θ∗−w\(s\)\\delta^\{\(s\)\}:=\\theta^\{\\ast\}\-w^\{\(s\)\}denote the contrast matrix\. Let𝒜⊆\{1,…,S\}\\mathcal\{A\}\\subseteq\\\{1,\\dots,S\\\}denote the unknown set of informative sources\. We writenA:=∑s∈𝒜nsn\_\{A\}:=\\sum\_\{s\\in\\mathcal\{A\}\}n\_\{s\}andN:=n0\+nAN:=n\_\{0\}\+n\_\{A\}\. For nodej∈Vj\\in V, letθ∖j∗:=\(θj​k∗\)k≠j\\theta^\{\\ast\}\_\{\\setminus j\}:=\(\\theta^\{\\ast\}\_\{jk\}\)\_\{k\\neq j\}, letSj:=\{k≠j:θj​k∗≠0\}S\_\{j\}:=\\\{k\\neq j:\\theta^\{\\ast\}\_\{jk\}\\neq 0\\\}, and letsj:=\|Sj\|s\_\{j\}:=\|S\_\{j\}\|\. When writing logistic losses, we use the standard0/10/1response codingyj,i:=𝟏​\{xi​j\(0\)=1\}y\_\{j,i\}:=\\mathbf\{1\}\\\{x^\{\(0\)\}\_\{ij\}=1\\\}, which is equivalent to the\{−1,1\}\\\{\-1,1\\\}representation and avoids sign ambiguities in the likelihood expressions\. Letθ^\\hat\{\\theta\}denote the final estimate of the target interaction matrix\. Let∥⋅∥q\\\|\\cdot\\\|\_\{q\}and∥⋅∥F\\\|\\cdot\\\|\_\{F\}denote theℓq\\ell\_\{q\}norm and the Frobenius norm, respectively\. We writea≲ba\\lesssim bifa≤C​ba\\leq Cbfor an absolute constantC\>0C\>0independent of the sample sizes and dimensions, and we writea≍ba\\asymp bif botha≲ba\\lesssim bandb≲ab\\lesssim ahold\. We definea∧b:=min⁡\(a,b\)a\\wedge b:=\\min\(a,b\)\.

Organization\.Section[2](https://arxiv.org/html/2607.03005#S2)reviews related work and background on transfer learning and Ising model estimation\. Section[3](https://arxiv.org/html/2607.03005#S3)introduces the proposed algorithms, and Section[4](https://arxiv.org/html/2607.03005#S4)presents the theoretical analysis\. Section[5](https://arxiv.org/html/2607.03005#S5)presents simulations and the mutation\-data study\. Appendix[S\.2](https://arxiv.org/html/2607.03005#A2)presents additional real\-data analyses, and Section[6](https://arxiv.org/html/2607.03005#S6)concludes the article\.

## 2Background

This section reviews related work and summarizes the background used in our development\.

### 2\.1Related Work

Transfer learning in high\-dimensional regression encompasses selective information borrowing, two\-step correction, source detection, benign\-overfitting interpolation, and low\-rank multiple\-response estimation\(Li et al\.,[2022](https://arxiv.org/html/2607.03005#bib.bib26); Tian & Feng,[2023](https://arxiv.org/html/2607.03005#bib.bib41); Kim et al\.,[2025](https://arxiv.org/html/2607.03005#bib.bib24); Park et al\.,[2025](https://arxiv.org/html/2607.03005#bib.bib37)\)\. Lasso\-based neighborhood selection, introduced for Gaussian graphical models\(Meinshausen & Bühlmann,[2006](https://arxiv.org/html/2607.03005#bib.bib31)\), provides a standard estimator for high\-dimensional Ising structure learning via nodewiseℓ1\\ell\_\{1\}\-regularized logistic regression\(Ravikumar et al\.,[2010](https://arxiv.org/html/2607.03005#bib.bib38)\)\. Building on related correction ideas for Gaussian graphical models\(Li et al\.,[2023](https://arxiv.org/html/2607.03005#bib.bib27); Zhao et al\.,[2026](https://arxiv.org/html/2607.03005#bib.bib50)\),Tian & Feng \([2023](https://arxiv.org/html/2607.03005#bib.bib41)\)analyzed two\-step correction and source detection for high\-dimensional GLMs\. Other related contexts include time\-dependent spatially varying Gaussian models\(Greenewald et al\.,[2017](https://arxiv.org/html/2607.03005#bib.bib20)\)and heterogeneous logistic regression for grouped binary responses and not graph recovery\(Kim et al\.,[2023](https://arxiv.org/html/2607.03005#bib.bib23)\)\. While recent meta\-learning work exploredℓ1\\ell\_\{1\}\-regularized Ising support recovery under a random\-task model\(Xie & Honorio,[2024](https://arxiv.org/html/2607.03005#bib.bib48)\), our setting involves fixed domains with unknown relevance\. We use a source\-to\-target bias formulation for the signed\-design nodewise pseudolikelihood problem, augmenting the standard nodewise representation with loss\-based source screening and a target\-only folded\-concave correction on updated coefficients for support recovery\.

### 2\.2Transfer Learning

We briefly review the standard transfer learning method for high\-dimensional linear models\. Consider a target model:

Y\(0\)=X\(0\)​β\+ϵ\(0\),Y^\{\(0\)\}=X^\{\(0\)\}\\beta\+\\epsilon^\{\(0\)\},\(1\)whereY\(0\)∈ℝn0Y^\{\(0\)\}\\in\\mathbb\{R\}^\{n\_\{0\}\},X\(0\)∈ℝn0×pX^\{\(0\)\}\\in\\mathbb\{R\}^\{n\_\{0\}\\times p\}, andβ∈ℝp\\beta\\in\\mathbb\{R\}^\{p\}is the target parameter\.

We also observeSSauxiliary models

Y\(s\)=X\(s\)​w\(s\)\+ϵ\(s\),s=1,…,S,Y^\{\(s\)\}=X^\{\(s\)\}w^\{\(s\)\}\+\\epsilon^\{\(s\)\},\\quad s=1,\\ldots,S,\(2\)wherew\(s\)∈ℝpw^\{\(s\)\}\\in\\mathbb\{R\}^\{p\}is the parameter for thess\-th source\. A common structural assumption is that the contrastδ\(s\)=β−w\(s\)\\delta^\{\(s\)\}=\\beta\-w^\{\(s\)\}is “small” in a suitable sense, often inℓ1\\ell\_\{1\}norm\.

FollowingTian & Feng \([2023](https://arxiv.org/html/2607.03005#bib.bib41)\), one can define a transferring level via‖δ\(s\)‖1\\\|\\delta^\{\(s\)\}\\\|\_\{1\}and the level\-hhtransferring set

𝒜h=\{s:‖δ\(s\)‖1≤h\},\\mathcal\{A\}\_\{h\}=\\\{s:\\\|\\delta^\{\(s\)\}\\\|\_\{1\}\\leq h\\\},\(3\)wherehhquantifies how close a source must be to be useful\. In practice,𝒜h\\mathcal\{A\}\_\{h\}is unknown, and using non\-informative sources can cause negative transfer\. A recurring strategy in high dimensions is a two\-step approach: \(i\) construct a stable estimator using pooled data from informative sources, and \(ii\) correct the resulting bias using only the target data\. This logic underlies our Trans\-Ising procedure\.

### 2\.3Ising Models

The Ising model is a pairwise Markov Random Field that specifies dependence among binary variables through an undirected graph\. LetX=\(X1,…,Xp\)∈\{−1,1\}pX=\(X\_\{1\},\\dots,X\_\{p\}\)\\in\\\{\-1,1\\\}^\{p\}\. A common specification \(without external fields\) is

Pθ​\(x\)=1Z​\(θ\)​exp⁡\(∑\(i,j\)∈Eθi​j​xi​xj\),P\_\{\\theta\}\(x\)=\\frac\{1\}\{Z\(\\theta\)\}\\exp\\left\(\\sum\_\{\(i,j\)\\in E\}\\theta\_\{ij\}x\_\{i\}x\_\{j\}\\right\),\(4\)whereθi​j\\theta\_\{ij\}is the interaction strength andZ​\(θ\)Z\(\\theta\)is the partition function\. Direct likelihood\-based estimation is difficult becauseZ​\(θ\)Z\(\\theta\)involves a sum over2p2^\{p\}states\.

The conditional distribution of a single node has a logistic form\. Forj∈Vj\\in Vandxj∈\{−1,1\}x\_\{j\}\\in\\\{\-1,1\\\},

P​\(Xj=xj∣X∖j\)=exp⁡\(2​xj​∑k≠jθj​k​Xk\)1\+exp⁡\(2​xj​∑k≠jθj​k​Xk\)\.P\(X\_\{j\}=x\_\{j\}\\mid X\_\{\\setminus j\}\)=\\frac\{\\exp\\left\(2x\_\{j\}\\sum\_\{k\\neq j\}\\theta\_\{jk\}X\_\{k\}\\right\)\}\{1\+\\exp\\left\(2x\_\{j\}\\sum\_\{k\\neq j\}\\theta\_\{jk\}X\_\{k\}\\right\)\}\.\(5\)Equivalently,P​\(Xj=1∣X∖j\)=σ​\(2​∑k≠jθj​k​Xk\),P\(X\_\{j\}=1\\mid X\_\{\\setminus j\}\)=\\sigma\\\!\\Big\(2\\sum\_\{k\\neq j\}\\theta\_\{jk\}X\_\{k\}\\Big\),whereσ​\(u\)=\(1\+e−u\)−1\\sigma\(u\)=\(1\+e^\{\-u\}\)^\{\-1\}\. This identity motivates pseudolikelihood\-based estimation and the neighborhood\-selection approach\.

Given dataX∈\{−1,1\}n×pX\\in\\\{\-1,1\\\}^\{n\\times p\}, define the0/10/1responseyj,i:=𝟏​\{xi​j=1\}y\_\{j,i\}:=\\mathbf\{1\}\\\{x\_\{ij\}=1\\\}and the linear predictorηj,i​\(θ∖j\):=2​∑k≠jθj​k​xi​k\.\\eta\_\{j,i\}\(\\theta\_\{\\setminus j\}\):=2\\sum\_\{k\\neq j\}\\theta\_\{jk\}x\_\{ik\}\.Then the nodewise negative conditional log\-likelihood at nodejjcan be written as

ℓj​\(θ∖j;X\)=1n​∑i=1n\{log⁡\(1\+eηj,i​\(θ∖j\)\)−yj,i​ηj,i​\(θ∖j\)\}\.\\ell\_\{j\}\(\\theta\_\{\\setminus j\};X\)=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\Big\\\{\\log\\big\(1\+e^\{\\eta\_\{j,i\}\(\\theta\_\{\\setminus j\}\)\}\\big\)\-y\_\{j,i\}\\,\\eta\_\{j,i\}\(\\theta\_\{\\setminus j\}\)\\Big\\\}\.\(6\)The standardℓ1\\ell\_\{1\}\-regularized \(Lasso\) estimator\(Tibshirani,[1996](https://arxiv.org/html/2607.03005#bib.bib42); van de Geer,[2008](https://arxiv.org/html/2607.03005#bib.bib44)\)solves:

θ^∖j=argminθ∖j∈ℝp−1​\{ℓj​\(θ∖j;X\)\+λ‖θ∖j∥1\}\.\\hat\{\\theta\}\_\{\\setminus j\}=\\underset\{\\theta\_\{\\setminus j\}\\in\\mathbb\{R\}^\{p\-1\}\}\{\\operatorname\{argmin\}\}\\left\\\{\\ell\_\{j\}\(\\theta\_\{\\setminus j\};X\)\+\\lambda\\\|\\theta\_\{\\setminus j\}\\\|\_\{1\}\\right\\\}\.\(7\)Repeating this forj=1,…,pj=1,\\dots,pyields an \(asymmetric\) collection of neighborhood estimates, which can be symmetrized to recover an undirected graph\.

This pseudolikelihood/logistic representation is the starting point for our transfer learning method\. Trans\-Ising keeps the nodewise logistic model and uses auxiliary datasets through a selective pooling step followed by a target\-only refinement\.

## 3Methodology

### 3\.1Oracle Trans\-Ising Algorithm with Known Informative Sources

We first describe an oracle version of our procedure that assumes*the informative source set𝒜\\mathcal\{A\}is known*\.

Initial Estimation Usingℓ1\\ell\_\{1\}\-Regularized Logistic Regression\.The initial step pools the samples from informative auxiliary datasets\{X\(s\)\}s∈𝒜\\\{X^\{\(s\)\}\\\}\_\{s\\in\\mathcal\{A\}\}with the primary datasetX\(0\)X^\{\(0\)\}\. Letn0n\_\{0\}be the sample size of the primary data,nsn\_\{s\}be the sample size for source datasets∈𝒜s\\in\\mathcal\{A\}, and letnA=∑s∈𝒜nsn\_\{A\}=\\sum\_\{s\\in\\mathcal\{A\}\}n\_\{s\}\. PutN=n0\+nAN=n\_\{0\}\+n\_\{A\}, and writenr=n0n\_\{r\}=n\_\{0\}whenr=0r=0andnr=nsn\_\{r\}=n\_\{s\}whenr=s∈𝒜r=s\\in\\mathcal\{A\}\.

As established in Section[2](https://arxiv.org/html/2607.03005#S2), estimation can be decomposed intoppneighborhood selection problems\. Letf​\(z\):=log⁡\(1\+e−z\)f\(z\):=\\log\(1\+e^\{\-z\}\), and letλw\>0\\lambda\_\{w\}\>0denote the Step 1 regularization parameter\. For each nodej∈Vj\\in V, we estimate its neighborhood parametersw∖j∈ℝp−1w\_\{\\setminus j\}\\in\\mathbb\{R\}^\{p\-1\}by solving the pooled nodewise logistic lasso:

w^∖jA=argminw∖j∈ℝp−1\{1N∑r∈\{0\}∪𝒜∑i=1nrf\(2∑k≠jwj​kxi​j\(r\)xi​k\(r\)\)\\displaystyle\\hat\{w\}^\{A\}\_\{\\setminus j\}=\\underset\{w\_\{\\setminus j\}\\in\\mathbb\{R\}^\{p\-1\}\}\{\\operatorname\{argmin\}\}\\Bigg\\\{\\frac\{1\}\{N\}\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}f\\left\(2\\sum\_\{k\\neq j\}w\_\{jk\}x\_\{ij\}^\{\(r\)\}x\_\{ik\}^\{\(r\)\}\\right\)\(8\)\+λw∑k≠j\|wj​k\|\},\\displaystyle\+\\lambda\_\{w\}\\sum\_\{k\\neq j\}\|w\_\{jk\}\|\\Bigg\\\},Repeating this for allj=1,…,pj=1,\\ldots,pyields an initial \(typically asymmetric\) matrix estimate, denotedw^A\\hat\{w\}^\{A\}\.

Bias Correction Using Primary Data\.Althoughw^A\\hat\{w\}^\{A\}uses auxiliary data, it can be biased when the source and target models differ\. To reduce this bias, we apply a target\-only correction\. Letλδ\>0\\lambda\_\{\\delta\}\>0be the correction regularization parameter, and letPλ​\(⋅\)P\_\{\\lambda\}\(\\cdot\)be the SCAD penalty with penalty levelλ\\lambda\. For each nodejj, define

Qj\(δ∖j;w^∖jA\):=\{1n0∑i=1n0f\(2∑k≠j\(w^j​kA\+δj​k\)xi​j\(0\)xi​k\(0\)\)\\displaystyle Q\_\{j\}\(\\delta\_\{\\setminus j\};\\hat\{w\}^\{A\}\_\{\\setminus j\}\)=\\Bigg\\\{\\frac\{1\}\{n\_\{0\}\}\\sum\_\{i=1\}^\{n\_\{0\}\}f\\left\(2\\sum\_\{k\\neq j\}\(\\hat\{w\}\_\{jk\}^\{A\}\+\\delta\_\{jk\}\)x\_\{ij\}^\{\(0\)\}x\_\{ik\}^\{\(0\)\}\\right\)\(9\)\+λδ∑k≠j\|δj​k\|\+∑k≠jPλ\(w^j​kA\+δj​k\)\},\\displaystyle\+\\lambda\_\{\\delta\}\\sum\_\{k\\neq j\}\|\\delta\_\{jk\}\|\+\\sum\_\{k\\neq j\}P\_\{\\lambda\}\(\\hat\{w\}^\{A\}\_\{jk\}\+\\delta\_\{jk\}\)\\Bigg\\\},Letδ^∖jA\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}denote the local solution of \([9](https://arxiv.org/html/2607.03005#S3.E9)\) returned by the optimization routine analyzed in Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\.δ^A\\hat\{\\delta\}^\{A\}adjustsw^A\\hat\{w\}^\{A\}toward the target distribution using onlyX\(0\)X^\{\(0\)\}\.

Dual Penalty for Bias Correction\.The pooled initializerw^A\\hat\{w\}^\{A\}can suffer from shrinkage bias on strong edges\. A naive application of Lasso in the correction step, as in previous GLM transfer methods\(Tian & Feng,[2023](https://arxiv.org/html/2607.03005#bib.bib41)\), would impose double shrinkage on large coefficients and make it difficult to distinguish true signals from transfer bias\. We therefore use a dual\-penalty correction in \([9](https://arxiv.org/html/2607.03005#S3.E9)\): theℓ1\\ell\_\{1\}penalty acts on the*correction*δ∖j\\delta\_\{\\setminus j\}to encourage sparse adjustments, while a folded\-concave SCAD penalty is applied to the*updated coefficients*w^j​kA\+δj​k\\hat\{w\}^\{A\}\_\{jk\}\+\\delta\_\{jk\}\. For coefficients above the SCAD threshold, the SCAD derivative is zero, which reduces additional shrinkage on strong signals and supports the selection argument in Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)\(Fan & Li,[2001](https://arxiv.org/html/2607.03005#bib.bib18)\)\. See Appendix[S\.5\.1](https://arxiv.org/html/2607.03005#A5.SS1)for the explicit piecewise form\. In Section[4](https://arxiv.org/html/2607.03005#S4), Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)establishes nodewise support recovery and sign consistency for the resulting estimator under its stated small\-error, beta\-min, separation, empirical curvature, local\-solution, and cone conditions\.

Optimization and Implementation\.Step 2 leads to a nonsmooth, nonconvex objective because it combines anℓ1\\ell\_\{1\}penalty on the correction with a folded\-concave SCAD penalty on the updated coefficients\. We optimize Step 2 using the standard*local linear approximation \(LLA\)*scheme for SCAD, which replaces the nonconvex penalty by a sequence of weightedℓ1\\ell\_\{1\}surrogates\.

For each nodejj, letQ​\(δ\):=Qj​\(δ;w^∖jA\)Q\(\\delta\):=Q\_\{j\}\(\\delta;\\hat\{w\}^\{A\}\_\{\\setminus j\}\)denote the objective in \([9](https://arxiv.org/html/2607.03005#S3.E9)\)\. Hereℓ0,j\\ell\_\{0,j\}denotes the target signed\-design nodewise loss for nodejj; its explicit form is stated in Section[4](https://arxiv.org/html/2607.03005#S4)\. Putϑ:=w^∖jA\+δ\\vartheta:=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\deltaand rewrite the objective as

minϑ∈ℝp−1⁡ℓ0,j​\(ϑ\)\+λδ​‖ϑ−w^∖jA‖1\+∑k≠jPλ​\(ϑk\)\.\\min\_\{\\vartheta\\in\\mathbb\{R\}^\{p\-1\}\}\\ \\ell\_\{0,j\}\(\\vartheta\)\+\\lambda\_\{\\delta\}\\\|\\vartheta\-\\hat\{w\}^\{A\}\_\{\\setminus j\}\\\|\_\{1\}\+\\sum\_\{k\\neq j\}P\_\{\\lambda\}\(\\vartheta\_\{k\}\)\.At iterationtt, LLA linearizesPλP\_\{\\lambda\}atϑ\(t\)\\vartheta^\{\(t\)\}via weightsωk\(t\):=Pλ′​\(\|ϑk\(t\)\|\)\\omega^\{\(t\)\}\_\{k\}:=P^\{\\prime\}\_\{\\lambda\}\(\|\\vartheta^\{\(t\)\}\_\{k\}\|\), yielding the convex surrogate

minϑ⁡ℓ0,j​\(ϑ\)\+∑k≠j\(λδ​\|ϑk−w^j​kA\|\+ωk\(t\)​\|ϑk\|\)\.\\min\_\{\\vartheta\}\\ \\ell\_\{0,j\}\(\\vartheta\)\+\\sum\_\{k\\neq j\}\\Big\(\\lambda\_\{\\delta\}\\,\|\\vartheta\_\{k\}\-\\hat\{w\}^\{A\}\_\{jk\}\|\+\\omega^\{\(t\)\}\_\{k\}\|\\vartheta\_\{k\}\|\\Big\)\.We solve this surrogate by proximal gradient\. We stop when the coordinate KKT residual satisfies

dist∞​\(0,∂Q​\(δ^∖jA\)\):=infξ∈∂Q​\(δ^∖jA\)‖ξ‖∞≤εn,j,\\mathrm\{dist\}\_\{\\infty\}\\bigl\(0,\\partial Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\\bigr\):=\\inf\_\{\\xi\\in\\partial Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\}\\\|\\xi\\\|\_\{\\infty\}\\leq\\varepsilon\_\{n,j\},as in Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\.

Algorithm 1Oracle Trans\-Ising Algorithm1:Input: Primary data

X\(0\)X^\{\(0\)\}and informative auxiliary data

\{X\(s\)\}s∈𝒜\\\{X^\{\(s\)\}\\\}\_\{s\\in\\mathcal\{A\}\}\.

2:Output: Final coefficient estimate

θ^\\hat\{\\theta\}\.

3:Step 1 \(Initial Estimation\)\.For each node

j∈Vj\\in V, compute

w^∖jA\\hat\{w\}^\{A\}\_\{\\setminus j\}by solving \([8](https://arxiv.org/html/2607.03005#S3.E8)\)\.

4:Step 2 \(Bias Correction\)\.For each node

j∈Vj\\in V, compute

δ^∖jA\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}by solving \([9](https://arxiv.org/html/2607.03005#S3.E9)\)\.

5:Step 3 \(Final Estimation\)\.Assemble the asymmetric intermediate matrices

w^A\\hat\{w\}^\{A\}and

δ^A\\hat\{\\delta\}^\{A\}from the node\-wise estimates; compute

θ^asym=w^A\+δ^A,\\hat\{\\theta\}^\{\\text\{asym\}\}=\\hat\{w\}^\{A\}\+\\hat\{\\delta\}^\{A\},and apply the symmetrization rule \(AND rule\) to

θ^asym\\hat\{\\theta\}^\{\\text\{asym\}\}to obtain the final symmetric estimate

θ^\\hat\{\\theta\}\.

Final Parameter Estimation\.The final interaction coefficients are obtained by combining the initial estimate and the correction\. For each node pair\(j,k\)\(j,k\),

θ^j​kasym=w^j​kA\+δ^j​kA\.\\hat\{\\theta\}^\{\\text\{asym\}\}\_\{jk\}=\\hat\{w\}^\{A\}\_\{jk\}\+\\hat\{\\delta\}^\{A\}\_\{jk\}\.This yields a complete, but still potentially asymmetric, matrixθ^asym\\hat\{\\theta\}^\{\\text\{asym\}\}\. A final symmetrization step is applied using the AND rule:

θ^j​k=θ^k​j=\{\(θ^j​kasym\+θ^k​jasym\)2if​θ^j​kasym≠0&θ^k​jasym≠0,0,otherwise,\\hat\{\\theta\}\_\{jk\}=\\hat\{\\theta\}\_\{kj\}=\\begin\{cases\}\\frac\{\(\\hat\{\\theta\}^\{\\text\{asym\}\}\_\{jk\}\+\\hat\{\\theta\}^\{\\text\{asym\}\}\_\{kj\}\)\}\{2\}&\\begin\{aligned\} &\\text\{if \}\\hat\{\\theta\}^\{\\text\{asym\}\}\_\{jk\}\\neq 0\\ \\&\\ \\hat\{\\theta\}^\{\\text\{asym\}\}\_\{kj\}\\neq 0,\\end\{aligned\}\\\\ 0,&\\text\{otherwise\},\\end\{cases\}which produces the final undirected graph estimate\.

### 3\.2Trans\-Ising Algorithm with Informative Source Detection

The oracle algorithm assumes the informative source set𝒜\\mathcal\{A\}is known; in practice it is not, and naively pooling all auxiliary datasets can cause*negative transfer*\. We therefore select informative sources by comparing each source’s effect on held\-out target*pseudolikelihood*\.

For a configurationx∈\{−1,1\}px\\in\\\{\-1,1\\\}^\{p\}and a parameter matrixΘ\\Theta, writeΘj,∖j:=\(Θj​k\)k≠j\\Theta\_\{j,\\setminus j\}:=\(\\Theta\_\{jk\}\)\_\{k\\neq j\}and define

q​\(xj∣x∖j;Θj,∖j\)\\displaystyle q\(x\_\{j\}\\mid x\_\{\\setminus j\};\\Theta\_\{j,\\setminus j\}\):=exp⁡\{2​xj​∑k≠jΘj​k​xk\}1\+exp⁡\{2​xj​∑k≠jΘj​k​xk\},\\displaystyle:=\\frac\{\\exp\\\{2x\_\{j\}\\sum\_\{k\\neq j\}\\Theta\_\{jk\}x\_\{k\}\\\}\}\{1\+\\exp\\\{2x\_\{j\}\\sum\_\{k\\neq j\}\\Theta\_\{jk\}x\_\{k\}\\\}\},ℓpseudo​\(x;Θ\)\\displaystyle\\ell^\{\\mathrm\{pseudo\}\}\(x;\\Theta\):=−∑j=1plog⁡q​\(xj∣x∖j;Θj,∖j\)\.\\displaystyle:=\-\\sum\_\{j=1\}^\{p\}\\log q\(x\_\{j\}\\mid x\_\{\\setminus j\};\\Theta\_\{j,\\setminus j\}\)\.
For a dataset of sizennand a parameter estimateθ^\\hat\{\\theta\}, define the \(negative\) pseudolikelihood loss

ℒnpseudo​\(θ^\)\\displaystyle\\mathcal\{L\}\_\{n\}^\{\\text\{pseudo\}\}\(\\hat\{\\theta\}\)=−1n​∑i=1n∑j=1plog⁡q​\(xi​j∣xi,∖j;θ^j,∖j\)\\displaystyle=\-\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{p\}\\log q\(x\_\{ij\}\\mid x\_\{i,\\setminus j\};\\hat\{\\theta\}\_\{j,\\setminus j\}\)\(10\)=1n​∑i=1n∑j=1pf​\(2​∑k≠jθ^j​k​xi​j​xi​k\),\\displaystyle=\\frac\{1\}\{n\}\\sum\_\{i=1\}^\{n\}\\sum\_\{j=1\}^\{p\}f\\\!\\left\(2\\sum\_\{k\\neq j\}\\hat\{\\theta\}\_\{jk\}\\,x\_\{ij\}x\_\{ik\}\\right\),which measures how well the implied conditional distributions fit the target data\. We keep sources whose inclusion does not increase the held\-out target loss beyond a tolerance threshold, and then run the oracle Trans\-Ising on the selected set\.

Algorithm 2Trans\-Ising with Source Detection1:Input:Primary data

X\(0\)X^\{\(0\)\}and auxiliary datasets

\{X\(s\)\}s=1S\\\{X^\{\(s\)\}\\\}\_\{s=1\}^\{S\}\.

2:Output:Final graph estimate

θ^\\hat\{\\theta\}\.

3:Randomly split

X\(0\)X^\{\(0\)\}into 2 folds:

\{X\(0\)​\[1\],X\(0\)​\[2\]\}\\\{X^\{\(0\)\[1\]\},X^\{\(0\)\[2\]\}\\\}\.

4:for

r=1r=1to

22do

5:Compute estimate in \([11](https://arxiv.org/html/2607.03005#S3.E11)\)

6:Compute baseline loss in \([13](https://arxiv.org/html/2607.03005#S3.E13)\)

7:for

s=1s=1to

SSdo

8:Compute estimate in \([12](https://arxiv.org/html/2607.03005#S3.E12)\)

9:Compute combined loss in \([14](https://arxiv.org/html/2607.03005#S3.E14)\)

10:endfor

11:endfor

12:Compute average baseline loss

ℒ¯0=12​∑r=12ℒ^0\(r\)\\bar\{\\mathcal\{L\}\}\_\{0\}=\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\hat\{\\mathcal\{L\}\}\_\{0\}^\{\(r\)\}\.

13:Compute variance:

σ^2=12−1​∑r=12\(ℒ^0\(r\)−ℒ¯0\)2\\hat\{\\sigma\}^\{2\}=\\frac\{1\}\{2\-1\}\\sum\_\{r=1\}^\{2\}\(\\hat\{\\mathcal\{L\}\}\_\{0\}^\{\(r\)\}\-\\bar\{\\mathcal\{L\}\}\_\{0\}\)^\{2\}\.

14:Set threshold

τ←Cτ​σ^\\tau\\leftarrow C\_\{\\tau\}\\,\\hat\{\\sigma\}⊳\\trianglerightCτ\>0C\_\{\\tau\}\>0\.

15:for

s=1s=1to

SSdo

16:Compute scores as in \([15](https://arxiv.org/html/2607.03005#S3.E15)\)

17:endfor

18:Select informative auxiliary datasets as in \([16](https://arxiv.org/html/2607.03005#S3.E16)\)\.

19:Run Algorithm[1](https://arxiv.org/html/2607.03005#alg1)on

X\(0\)X^\{\(0\)\}and

\{X\(s\)\}s∈𝒜^\\\{X^\{\(s\)\}\\\}\_\{s\\in\\hat\{\\mathcal\{A\}\}\}\.

Data Splitting and Pseudolikelihood Loss Calculation\.We perform two\-fold cross\-validation on the target dataX\(0\)X^\{\(0\)\}, splitting it into\{X\(0\)​\[1\],X\(0\)​\[2\]\}\\\{X^\{\(0\)\[1\]\},X^\{\(0\)\[2\]\}\\\}\. For each foldr∈\{1,2\}r\\in\\\{1,2\\\}, letn0,r:=\|X\(0\)​\[r\]\|n\_\{0,r\}:=\|X^\{\(0\)\[r\]\}\|\. We compute with nodewise logistic lasso:

θ^\(0\)​\[−r\]\\displaystyle\\hat\{\\theta\}^\{\(0\)\[\-r\]\}:estimate on​X\(0\)∖X\(0\)​\[r\],\\displaystyle:\\text\{ estimate on \}X^\{\(0\)\}\\setminus X^\{\(0\)\[r\]\},\(11\)θ^\(0\+s\)​\[−r\]\\displaystyle\\hat\{\\theta\}^\{\(0\+s\)\[\-r\]\}:estimate on​\(X\(0\)∖X\(0\)​\[r\]\)∪X\(s\)\.\\displaystyle:\\text\{ estimate on \}\(X^\{\(0\)\}\\setminus X^\{\(0\)\[r\]\}\)\\cup X^\{\(s\)\}\.\(12\)Then we evaluate both estimates on the held\-out validation foldX\(0\)​\[r\]X^\{\(0\)\[r\]\}:

ℒ^0\(r\)\\displaystyle\\hat\{\\mathcal\{L\}\}\_\{0\}^\{\(r\)\}=1n0,r​∑x∈X\(0\)​\[r\]ℓpseudo​\(x;θ^\(0\)​\[−r\]\),\\displaystyle=\\frac\{1\}\{n\_\{0,r\}\}\\sum\_\{x\\in X^\{\(0\)\[r\]\}\}\\ell^\{\\mathrm\{pseudo\}\}\(x;\\hat\{\\theta\}^\{\(0\)\[\-r\]\}\),\(13\)ℒ^s\(r\)\\displaystyle\\hat\{\\mathcal\{L\}\}\_\{s\}^\{\(r\)\}=1n0,r​∑x∈X\(0\)​\[r\]ℓpseudo​\(x;θ^\(0\+s\)​\[−r\]\)\.\\displaystyle=\\frac\{1\}\{n\_\{0,r\}\}\\sum\_\{x\\in X^\{\(0\)\[r\]\}\}\\ell^\{\\mathrm\{pseudo\}\}\(x;\\hat\{\\theta\}^\{\(0\+s\)\[\-r\]\}\)\.\(14\)
Informative Source Selection\.We define the auxiliary compatibility scoreΔs\\Delta\_\{s\}as the average difference in pseudolikelihood loss across folds:

Δs=12​∑r=12\(ℒ^s\(r\)−ℒ^0\(r\)\)\.\\Delta\_\{s\}=\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\left\(\\hat\{\\mathcal\{L\}\}\_\{s\}^\{\(r\)\}\-\\hat\{\\mathcal\{L\}\}\_\{0\}^\{\(r\)\}\\right\)\.\(15\)A negativeΔs\\Delta\_\{s\}indicates that includingX\(s\)X^\{\(s\)\}improves predictive performance on the target fold\. A positive value suggests potential negative transfer\.

To account for statistical fluctuations, we allow a tolerance band\. Specifically, we use an adaptive thresholdτ=Cτ​σ^\\tau=C\_\{\\tau\}\\,\\hat\{\\sigma\}, whereσ^\\hat\{\\sigma\}is the empirical standard deviation of\{ℒ^0\(r\)\}r=12\\\{\\hat\{\\mathcal\{L\}\}\_\{0\}^\{\(r\)\}\\\}\_\{r=1\}^\{2\}, and select

𝒜^=\{s:Δs<τ\}\.\\hat\{\\mathcal\{A\}\}=\\\{s:\\Delta\_\{s\}<\\tau\\\}\.\(16\)
Final Oracle Trans\-Ising Estimation\.The final graph estimate is obtained by applying Algorithm[1](https://arxiv.org/html/2607.03005#alg1)using the full target datasetX\(0\)X^\{\(0\)\}and the selected sources\{X\(s\)\}s∈𝒜^\\\{X^\{\(s\)\}\\\}\_\{s\\in\\hat\{\\mathcal\{A\}\}\}\. This loss\-based selection is adapted fromTian & Feng \([2023](https://arxiv.org/html/2607.03005#bib.bib41)\)\.

## 4Theoretical Analysis

We establish the theoretical properties of our algorithms for a fixed nodejj\. Our analysis uses high\-dimensional M\-estimation theory for nodewise logistic Ising selection\(van de Geer,[2008](https://arxiv.org/html/2607.03005#bib.bib44); Ravikumar et al\.,[2010](https://arxiv.org/html/2607.03005#bib.bib38); Bühlmann & van de Geer,[2011](https://arxiv.org/html/2607.03005#bib.bib7); Negahban et al\.,[2012](https://arxiv.org/html/2607.03005#bib.bib33)\), transfer\-learning arguments\(Li et al\.,[2022](https://arxiv.org/html/2607.03005#bib.bib26); Tian & Feng,[2023](https://arxiv.org/html/2607.03005#bib.bib41); Park et al\.,[2025](https://arxiv.org/html/2607.03005#bib.bib37)\), and folded\-concave penalties\(Fan & Li,[2001](https://arxiv.org/html/2607.03005#bib.bib18); Zhang,[2010](https://arxiv.org/html/2607.03005#bib.bib49)\)\.

Nodewise logistic loss \(signed\-design form\)\.For a datasetX\(r\)=\(xi​ℓ\(r\)\)1≤i≤nr,1≤ℓ≤pX^\{\(r\)\}=\(x^\{\(r\)\}\_\{i\\ell\}\)\_\{1\\leq i\\leq n\_\{r\},\\,1\\leq\\ell\\leq p\}withxi​ℓ\(r\)∈\{−1,1\}x^\{\(r\)\}\_\{i\\ell\}\\in\\\{\-1,1\\\}, fix a nodej∈Vj\\in Vand define the signed covariateszi,ℓ\(r\):=2​xi​j\(r\)​xi​ℓ\(r\)z^\{\(r\)\}\_\{i,\\ell\}:=2\\,x^\{\(r\)\}\_\{ij\}\\,x^\{\(r\)\}\_\{i\\ell\},ℓ∈V∖\{j\}\\ell\\in V\\setminus\\\{j\\\},zi\(r\)∈ℝp−1z^\{\(r\)\}\_\{i\}\\in\\mathbb\{R\}^\{p\-1\}\. Foru∈ℝp−1u\\in\\mathbb\{R\}^\{p\-1\}, define the nodewise logistic lossℓj\(r\)​\(u\):=1nr​∑i=1nrf​\(zi\(r\)⊤​u\)\\ell^\{\(r\)\}\_\{j\}\(u\):=\\frac\{1\}\{n\_\{r\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}f\\\!\\big\(z^\{\(r\)\\top\}\_\{i\}u\\big\), wheref​\(t\):=log⁡\(1\+e−t\)f\(t\):=\\log\(1\+e^\{\-t\}\)\. We writeℓ0,j​\(⋅\):=ℓj\(0\)​\(⋅\)\\ell\_\{0,j\}\(\\cdot\):=\\ell^\{\(0\)\}\_\{j\}\(\\cdot\)for the target loss\.

Pooled objective and population quantities\.For informative set𝒜\\mathcal\{A\}, letN:=n0\+∑s∈𝒜nsN:=n\_\{0\}\+\\sum\_\{s\\in\\mathcal\{A\}\}n\_\{s\}and define weightsαr:=nr/N\\alpha\_\{r\}:=n\_\{r\}/Nforr∈\{0\}∪𝒜r\\in\\\{0\\\}\\cup\\mathcal\{A\}\. Define the pooled empirical lossℓA,j​\(u\):=∑r∈\{0\}∪𝒜αr​ℓj\(r\)​\(u\),\\ell\_\{A,j\}\(u\):=\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\alpha\_\{r\}\\,\\ell^\{\(r\)\}\_\{j\}\(u\),and the population risksℒr,j​\(u\):=𝔼​\[ℓj\(r\)​\(u\)\]\\mathcal\{L\}\_\{r,j\}\(u\):=\\mathbb\{E\}\[\\ell^\{\(r\)\}\_\{j\}\(u\)\]andℒA,j​\(u\):=𝔼​\[ℓA,j​\(u\)\]\\mathcal\{L\}\_\{A,j\}\(u\):=\\mathbb\{E\}\[\\ell\_\{A,j\}\(u\)\]\. LetwA,∖j∗:=arg⁡minu∈ℝp−1⁡ℒA,j​\(u\)\.w^\{\\ast\}\_\{A,\\setminus j\}:=\\arg\\min\_\{u\\in\\mathbb\{R\}^\{p\-1\}\}\\,\\mathcal\{L\}\_\{A,j\}\(u\)\.The pooled\-to\-target transfer bias isδ∖j∗:=θ∖j∗−wA,∖j∗\.\\delta^\{\\ast\}\_\{\\setminus j\}:=\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\.We quantify cross\-domain heterogeneity by the source\-to\-targetℓ1\\ell\_\{1\}proximity levelhjh\_\{j\}in Assumption[2](https://arxiv.org/html/2607.03005#Thmassumption2)\. Under Assumption[3](https://arxiv.org/html/2607.03005#Thmassumption3), Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)shows that‖δ∖j∗‖1≤C​hj\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq Ch\_\{j\}for some constantC\>0C\>0\.

Two\-step estimator\.The oracle two\-step estimator at nodejjis defined as follows\.

w^∖jA=arg⁡minu∈ℝp−1⁡\{ℓA,j​\(u\)\+λw​‖u‖1\},\\hat\{w\}^\{A\}\_\{\\setminus j\}=\\arg\\min\_\{u\\in\\mathbb\{R\}^\{p\-1\}\}\\Big\\\{\\ell\_\{A,j\}\(u\)\+\\lambda\_\{w\}\\\|u\\\|\_\{1\}\\Big\\\},\(17\)For Step 2, define

Qj\(δ;w^∖jA\):=\{ℓ0,j\(w^∖jA\+δ\)\+λδ∥δ∥1\\displaystyle Q\_\{j\}\(\\delta;\\hat\{w\}^\{A\}\_\{\\setminus j\}\)=\\Big\\\{\\ell\_\{0,j\}\(\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\delta\)\+\\lambda\_\{\\delta\}\\\|\\delta\\\|\_\{1\}\(18\)\+∑k≠jPλ\(w^j​kA\+δj​k\)\},\\displaystyle\+\\sum\_\{k\\neq j\}P\_\{\\lambda\}\(\\hat\{w\}^\{A\}\_\{jk\}\+\\delta\_\{jk\}\)\\Big\\\},Letδ^∖jA\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}denote the local solution returned by the optimization routine and analyzed under Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\. We setθ^∖j:=w^∖jA\+δ^∖jA\\hat\{\\theta\}\_\{\\setminus j\}:=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\. This vector is the asymmetric nodewise estimate used in the fixed\-node analysis; Algorithm[1](https://arxiv.org/html/2607.03005#alg1)applies the AND symmetrization after the nodewise estimates are computed\.

Matrix norms\.For a matrixMM, define‖M‖1:=maxk​∑ℓ\|Mℓ​k\|\\\|M\\\|\_\{1\}:=\\max\_\{k\}\\sum\_\{\\ell\}\|M\_\{\\ell k\}\|\(inducedℓ1\\ell\_\{1\}norm, max column sum\),‖M‖∞:=maxℓ​∑k\|Mℓ​k\|\\\|M\\\|\_\{\\infty\}:=\\max\_\{\\ell\}\\sum\_\{k\}\|M\_\{\\ell k\}\|\(inducedℓ∞\\ell\_\{\\infty\}norm, max row sum\),‖M‖max:=maxℓ,k⁡\|Mℓ​k\|\\\|M\\\|\_\{\\max\}:=\\max\_\{\\ell,k\}\|M\_\{\\ell k\}\|\(entrywise max norm\)\. Then for any vectorxx,‖M​x‖∞≤‖M‖∞​‖x‖1\\\|Mx\\\|\_\{\\infty\}\\leq\\\|M\\\|\_\{\\infty\}\\,\\\|x\\\|\_\{1\}and‖M​x‖∞≤‖M‖max​‖x‖1\\\|Mx\\\|\_\{\\infty\}\\leq\\\|M\\\|\_\{\\max\}\\,\\\|x\\\|\_\{1\}\.

Target design representation\.Define the signed design matrixZ∖j\(0\):=2​diag​\(x⋅j\(0\)\)​X∖j\(0\)∈ℝn0×\(p−1\),Z^\{\(0\)\}\_\{\\setminus j\}:=2\\,\\mathrm\{diag\}\\\!\\big\(x^\{\(0\)\}\_\{\\cdot j\}\\big\)\\,X^\{\(0\)\}\_\{\\setminus j\}\\in\\mathbb\{R\}^\{n\_\{0\}\\times\(p\-1\)\},uj,i​\(ϑ\):=\[Z∖j\(0\)\]i,⋅​ϑ,u\_\{j,i\}\(\\vartheta\):=\\big\[Z^\{\(0\)\}\_\{\\setminus j\}\\big\]\_\{i,\\cdot\}\\,\\vartheta,and letσ​\(u\)=\(1\+e−u\)−1\\sigma\(u\)=\(1\+e^\{\-u\}\)^\{\-1\}andp^j,i​\(ϑ\):=σ​\(uj,i​\(ϑ\)\)\\widehat\{p\}\_\{j,i\}\(\\vartheta\):=\\sigma\(u\_\{j,i\}\(\\vartheta\)\)\. With this convention,

ℓ0,j​\(ϑ\)=1n0​∑i=1n0\{log⁡\(1\+euj,i​\(ϑ\)\)−uj,i​\(ϑ\)\}\\displaystyle\\ell\_\{0,j\}\(\\vartheta\)=\\frac\{1\}\{n\_\{0\}\}\\sum\_\{i=1\}^\{n\_\{0\}\}\\Big\\\{\\log\\big\(1\+e^\{u\_\{j,i\}\(\\vartheta\)\}\\big\)\-u\_\{j,i\}\(\\vartheta\)\\Big\\\}=1n0​∑i=1n0f​\(uj,i​\(ϑ\)\),\\displaystyle=\\frac\{1\}\{n\_\{0\}\}\\sum\_\{i=1\}^\{n\_\{0\}\}f\\\!\\big\(u\_\{j,i\}\(\\vartheta\)\\big\),and if we setyj†:=𝟏n0y^\{\\dagger\}\_\{j\}:=\\mathbf\{1\}\_\{n\_\{0\}\}\(the all\-ones vector\), then the gradient and Hessian can be written as

∇ℓ0,j​\(ϑ\)=1n0​Z∖j\(0\)⊤​\(p^j​\(ϑ\)−yj†\),\\nabla\\ell\_\{0,j\}\(\\vartheta\)=\\frac\{1\}\{n\_\{0\}\}\\,Z^\{\(0\)\\top\}\_\{\\setminus j\}\\big\(\\widehat\{p\}\_\{j\}\(\\vartheta\)\-y^\{\\dagger\}\_\{j\}\\big\),∇2ℓ0,j​\(ϑ\)=1n0​Z∖j\(0\)⊤​Wj​\(ϑ\)​Z∖j\(0\),\\nabla^\{2\}\\ell\_\{0,j\}\(\\vartheta\)=\\frac\{1\}\{n\_\{0\}\}\\,Z^\{\(0\)\\top\}\_\{\\setminus j\}\\,W\_\{j\}\(\\vartheta\)\\,Z^\{\(0\)\}\_\{\\setminus j\},wherep^j​\(ϑ\)=\(p^j,1​\(ϑ\),…,p^j,n0​\(ϑ\)\)⊤\\widehat\{p\}\_\{j\}\(\\vartheta\)=\(\\widehat\{p\}\_\{j,1\}\(\\vartheta\),\\dots,\\widehat\{p\}\_\{j,n\_\{0\}\}\(\\vartheta\)\)^\{\\top\}andWj​\(ϑ\)=diag​\(p^j,i​\(ϑ\)​\(1−p^j,i​\(ϑ\)\)\)W\_\{j\}\(\\vartheta\)=\\mathrm\{diag\}\\\!\\big\(\\widehat\{p\}\_\{j,i\}\(\\vartheta\)\(1\-\\widehat\{p\}\_\{j,i\}\(\\vartheta\)\)\\big\)\.

### 4\.1Rates and Selection Consistency of Algorithm[1](https://arxiv.org/html/2607.03005#alg1)

Assumptions and Lemmas\.All technical assumptions, lemmas and their discussions used in this section are stated in Appendix[S\.3](https://arxiv.org/html/2607.03005#A3)and[S\.4](https://arxiv.org/html/2607.03005#A4)\. Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1)–[4](https://arxiv.org/html/2607.03005#Thmcondition4)are theorem\-local inputs for empirical curvature, local optimization, and the Step 2 cone bound; they are separated from the primitive model and population assumptions\.

###### Theorem 1\(Transfer\-typeℓ2\\ell\_\{2\}andℓ1\\ell\_\{1\}rates for Oracle Trans\-Ising\)\.

Consider Algorithm[1](https://arxiv.org/html/2607.03005#alg1)for a fixed nodejj\. Suppose Assumptions[1](https://arxiv.org/html/2607.03005#Thmassumption1),[2](https://arxiv.org/html/2607.03005#Thmassumption2),[3](https://arxiv.org/html/2607.03005#Thmassumption3), and[4](https://arxiv.org/html/2607.03005#Thmassumption4)hold\. Suppose Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1),[2](https://arxiv.org/html/2607.03005#Thmcondition2),[3](https://arxiv.org/html/2607.03005#Thmcondition3), and[4](https://arxiv.org/html/2607.03005#Thmcondition4)hold simultaneously with probability at least1−C1​p−C21\-C\_\{1\}p^\{\-C\_\{2\}\}\. Choose tuning parameters

λw=Cw​log⁡pN,λδ=Cδ​log⁡pn0,λ=Cλ​log⁡pn0,\\displaystyle\\lambda\_\{w\}=C\_\{w\}\\sqrt\{\\frac\{\\log p\}\{N\}\},\\quad\\lambda\_\{\\delta\}=C\_\{\\delta\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\},\\quad\\lambda=C\_\{\\lambda\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\},whereλ\\lambdais the SCAD penalty level in \([18](https://arxiv.org/html/2607.03005#S4.E18)\) andCw,Cδ,Cλ\>0C\_\{w\},C\_\{\\delta\},C\_\{\\lambda\}\>0are sufficiently large absolute constants\. Assume further that

log⁡pn0=o​\(1\),sj​log⁡pN\+hj≲λδ,\\displaystyle\\frac\{\\log p\}\{n\_\{0\}\}=o\(1\),\\quad s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\\lesssim\\lambda\_\{\\delta\},sj​log⁡pN\+log⁡pN​hj2=o​\(1\)\.\\displaystyle s\_\{j\}\\frac\{\\log p\}\{N\}\+\\frac\{\\log p\}\{N\}h\_\{j\}^\{2\}=o\(1\)\.Define

rn,j:=C\[\\displaystyle r\_\{n,j\}=C\\bigg\[sj​log⁡pN\+\(\[\(log⁡pN\)1/4​hj1/2\]∧hj\)\\displaystyle\\sqrt\{\\frac\{s\_\{j\}\\log p\}\{N\}\}\+\\Big\(\\big\[\(\\tfrac\{\\log p\}\{N\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\big\]\\wedge h\_\{j\}\\Big\)\+\(\[\(log⁡pn0\)1/4hj1/2\]∧hj\)\]\\displaystyle\+\\Big\(\\big\[\(\\tfrac\{\\log p\}\{n\_\{0\}\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\big\]\\wedge h\_\{j\}\\Big\)\\bigg\]for a sufficiently large constantC\>0C\>0\. Then, for constantsC0,c0\>0C\_\{0\},c\_\{0\}\>0, with probability at least1−C0​p−c01\-C\_\{0\}p^\{\-c\_\{0\}\},

‖θ^∖j−θ∖j∗‖2\\displaystyle\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{2\}≤rn,j,‖θ^∖j−θ∖j∗‖1≲sj​log⁡pN\+hj\.\\displaystyle\\leq r\_\{n,j\},\\qquad\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\.

Decomposition of the error bound\.Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)decomposes the final error into:

sj​log⁡pN⏟pooled variance reduction\+\(\(log⁡pN\)1/4​hj1/2∧hj\)⏟Step 1 heterogeneity/approximation\\displaystyle\\underbrace\{\\sqrt\{\\frac\{s\_\{j\}\\log p\}\{N\}\}\}\_\{\\text\{pooled variance reduction\}\}\+\\underbrace\{\\Big\(\(\\tfrac\{\\log p\}\{N\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\wedge h\_\{j\}\\Big\)\}\_\{\\text\{Step~1 heterogeneity/approximation\}\}\+\(\(log⁡pn0\)1/4​hj1/2∧hj\)⏟Step 2 correction limit\.\\displaystyle\\qquad\\qquad\+\\underbrace\{\\Big\(\(\\tfrac\{\\log p\}\{n\_\{0\}\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\wedge h\_\{j\}\\Big\)\}\_\{\\text\{Step~2 correction limit\}\}\.Compared to the target\-only rateOp​\(sj​log⁡p/n0\)O\_\{p\}\(\\sqrt\{s\_\{j\}\\log p/n\_\{0\}\}\), the bound above is smaller whenN≫n0N\\gg n\_\{0\}and the two heterogeneity terms are smaller than the variance termsj​log⁡p/N\\sqrt\{s\_\{j\}\\log p/N\}\. In particular, in the idealized homogeneous casehj=0h\_\{j\}=0, the bound reduces to the pooled rate up to constants,sj​log⁡p/N\\sqrt\{s\_\{j\}\\log p/N\}\. Ashjh\_\{j\}grows, the heterogeneity terms dominate the variance term and a largeNNno longer reduces the bound, consistent with the negative transfer of naive pooling under distribution shift\.

Leta\>2a\>2denote the SCAD shape parameter in the selection analysis, and letεn,j\\varepsilon\_\{n,j\}denote the Step 2 coordinate KKT tolerance in Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\.

###### Theorem 2\(Support recovery via SCAD regularization\)\.

Letθ^∖j=w^∖jA\+δ^∖jA\\hat\{\\theta\}\_\{\\setminus j\}=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}be the estimator from \([17](https://arxiv.org/html/2607.03005#S4.E17)\)–\([18](https://arxiv.org/html/2607.03005#S4.E18)\)\. Define the estimated neighborhood by the*nonzero pattern*

S^j:=\{k≠j:θ^j​k≠0\}\.\\hat\{S\}\_\{j\}:=\\\{k\\neq j:\\ \\hat\{\\theta\}\_\{jk\}\\neq 0\\\}\.Assume the conditions of Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)hold\. Letrn,jr\_\{n,j\}be defined as in Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)\. Suppose

rn,j≤λ,r\_\{n,j\}\\leq\\lambda,λ−λδ−εn,j≥Csel​\(log⁡pn0\+sj​log⁡pN\+hj\)\\lambda\-\\lambda\_\{\\delta\}\-\\varepsilon\_\{n,j\}\\geq C\_\{\\mathrm\{sel\}\}\\left\(\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}\+s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\\right\)for a sufficiently large constantCsel\>0C\_\{\\mathrm\{sel\}\}\>0, and suppose

\|θj​m∗\|≥a​λ\+rn,jfor all​m∈Sj\.\|\\theta^\{\\ast\}\_\{jm\}\|\\geq a\\lambda\+r\_\{n,j\}\\qquad\\text\{for all \}m\\in S\_\{j\}\.Then,

ℙ​\(S^j=Sj,sign⁡\(θ^j​k\)=sign⁡\(θj​k∗\)​for all​k∈Sj\)→1\.\\mathbb\{P\}\\\!\\left\(\\begin\{array\}\[\]\{c\}\\hat\{S\}\_\{j\}=S\_\{j\},\\\\ \\operatorname\{sign\}\(\\hat\{\\theta\}\_\{jk\}\)=\\operatorname\{sign\}\(\\theta^\{\\ast\}\_\{jk\}\)\\ \\text\{for all \}k\\in S\_\{j\}\\end\{array\}\\right\)\\to 1\.

Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)establishes exact neighborhood recovery and sign consistency under a small\-error regime, a separation condition betweenλ\\lambdaand\(λδ,εn,j\)\(\\lambda\_\{\\delta\},\\varepsilon\_\{n,j\}\), and a beta\-min condition\. Compared with lasso, SCAD reduces shrinkage on large signals at the cost of nonconvex optimization and a computable stationary point \(Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\)\.

![Refer to caption](https://arxiv.org/html/2607.03005v1/x1.png)Figure 1:Average relative errors for Naive\-LogLasso \(red\), Oracle Trans\-Ising \(green\), Pooled\-Trans\-Ising \(cyan\), and Trans\-Ising \(purple\), across four graph structures at perturbation levels100​σ∈\{1,10,20\}100\\sigma\\in\\\{1,10,20\\\}, withp=200p=200\.
### 4\.2Consistency of Informative Source Detection

While the Oracle Trans\-Ising estimator uses a known informative source set𝒜\\mathcal\{A\}, Algorithm[2](https://arxiv.org/html/2607.03005#alg2)selects sources from held\-out target pseudolikelihood\. We state the detection result for the graph\-level, risk\-defined source set used by the validation rule\.

For a parameter matrixΘ\\Theta, define the target population pseudolikelihood risk

ℒ0pseudo​\(Θ\):=∑j=1pℒ0,j​\(Θj,∖j\)\.\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\Theta\):=\\sum\_\{j=1\}^\{p\}\\mathcal\{L\}\_\{0,j\}\(\\Theta\_\{j,\\setminus j\}\)\.For sourcessand foldr∈\{1,2\}r\\in\\\{1,2\\\}, putn0,−r:=n0−n0,rn\_\{0,\-r\}:=n\_\{0\}\-n\_\{0,r\}, wheren0,rn\_\{0,r\}is the validation\-fold size in the source\-detection split, and define

α0​r,s:=n0,−rn0,−r\+ns,αs​r:=nsn0,−r\+ns\.\\alpha\_\{0r,s\}:=\\frac\{n\_\{0,\-r\}\}\{n\_\{0,\-r\}\+n\_\{s\}\},\\qquad\\alpha\_\{sr\}:=\\frac\{n\_\{s\}\}\{n\_\{0,\-r\}\+n\_\{s\}\}\.For eachj=1,…,pj=1,\\ldots,p, define the fold\-specific rowwise population pooled minimizerW\(0\+s\),∗,rW^\{\(0\+s\),\\ast,r\}through

w∖j\(0\+s\),∗,r\\displaystyle w^\{\(0\+s\),\\ast,r\}\_\{\\setminus j\}:=arg⁡minu∈ℝp−1⁡\{α0​r,s​ℒ0,j​\(u\)\+αs​r​ℒs,j​\(u\)\}\.\\displaystyle=\\arg\\min\_\{u\\in\\mathbb\{R\}^\{p\-1\}\}\\left\\\{\\alpha\_\{0r,s\}\\mathcal\{L\}\_\{0,j\}\(u\)\+\\alpha\_\{sr\}\\mathcal\{L\}\_\{s,j\}\(u\)\\right\\\}\.Define

ℰ​\(s\):=12​∑r=12ℒ0pseudo​\(W\(0\+s\),∗,r\)−ℒ0pseudo​\(θ∗\)\.\\mathcal\{E\}\(s\):=\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(W^\{\(0\+s\),\\ast,r\}\)\-\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\theta^\{\\ast\}\)\.Fix a separation constantcgap\>0c\_\{\\mathrm\{gap\}\}\>0and a sequenceνn\>0\\nu\_\{n\}\>0\. The population informative set at level\(cgap,νn\)\(c\_\{\\mathrm\{gap\}\},\\nu\_\{n\}\)is

𝒜h:=\{s∈\{1,…,S\}:ℰ​\(s\)≤cgap​νn\},\\mathcal\{A\}\_\{h\}:=\\\{s\\in\\\{1,\\ldots,S\\\}:\\mathcal\{E\}\(s\)\\leq c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\\\},
###### Theorem 3\(Consistency of source detection\)\.

Suppose Assumption[1](https://arxiv.org/html/2607.03005#Thmassumption1)and Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)hold\. Let𝒜^\\hat\{\\mathcal\{A\}\}be the set selected by Algorithm[2](https://arxiv.org/html/2607.03005#alg2)with thresholdτ=Cτ​σ^\\tau=C\_\{\\tau\}\\hat\{\\sigma\}\. Then

ℙ​\(𝒜^=𝒜h\)→1\.\\mathbb\{P\}\(\\hat\{\\mathcal\{A\}\}=\\mathcal\{A\}\_\{h\}\)\\to 1\.

###### Corollary 1\(Oracle properties after source detection\)\.

Fix a nodejjand defineNh:=n0\+∑s∈𝒜hnsN\_\{h\}:=n\_\{0\}\+\\sum\_\{s\\in\\mathcal\{A\}\_\{h\}\}n\_\{s\}\. Suppose that the assumptions and conditions of Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3)hold\. Suppose, in addition, that the assumptions, theorem\-local conditions, tuning choices, and rate conditions of Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)hold with𝒜\\mathcal\{A\}andNNreplaced by𝒜h\\mathcal\{A\}\_\{h\}andNhN\_\{h\}, respectively\. Letθ^∖jdet\\hat\{\\theta\}^\{\\mathrm\{det\}\}\_\{\\setminus j\}be the asymmetric nodewise estimator obtained before the AND symmetrization step in the final run of Algorithm[1](https://arxiv.org/html/2607.03005#alg1)inside Algorithm[2](https://arxiv.org/html/2607.03005#alg2)\. Define, for a sufficiently large constantC\>0C\>0,

rn,j\(h\):=C\[sj​log⁡pNh\\displaystyle r\_\{n,j\}^\{\(h\)\}=\{\}C\\bigg\[\\sqrt\{\\frac\{s\_\{j\}\\log p\}\{N\_\{h\}\}\}\+\(\(log⁡pNh\)1/4​hj1/2∧hj\)\\displaystyle\+\\Big\(\\big\(\\frac\{\\log p\}\{N\_\{h\}\}\\big\)^\{1/4\}h\_\{j\}^\{1/2\}\\wedge h\_\{j\}\\Big\)\+\(\(log⁡pn0\)1/4hj1/2∧hj\)\]\.\\displaystyle\+\\Big\(\\big\(\\frac\{\\log p\}\{n\_\{0\}\}\\big\)^\{1/4\}h\_\{j\}^\{1/2\}\\wedge h\_\{j\}\\Big\)\\bigg\]\.Then, with probability tending to one,

‖θ^∖jdet−θ∖j∗‖2≤rn,j\(h\),‖θ^∖jdet−θ∖j∗‖1≲sj​log⁡pNh\+hj\.\\\|\\hat\{\\theta\}^\{\\mathrm\{det\}\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{2\}\\leq r\_\{n,j\}^\{\(h\)\},\\qquad\\\|\\hat\{\\theta\}^\{\\mathrm\{det\}\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\_\{h\}\}\}\+h\_\{j\}\.If the conditions of Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)also hold with𝒜\\mathcal\{A\}andNNreplaced by𝒜h\\mathcal\{A\}\_\{h\}andNhN\_\{h\}, respectively, and

S^jdet:=\{k≠j:θ^j​kdet≠0\},\\hat\{S\}\_\{j\}^\{\\mathrm\{det\}\}:=\\\{k\\neq j:\\hat\{\\theta\}^\{\\mathrm\{det\}\}\_\{jk\}\\neq 0\\\},then, with probability tending to one,

S^jdet=Sj,sign⁡\(θ^j​kdet\)=sign⁡\(θj​k∗\)for all​k∈Sj\.\\hat\{S\}\_\{j\}^\{\\mathrm\{det\}\}=S\_\{j\},\\qquad\\operatorname\{sign\}\(\\hat\{\\theta\}^\{\\mathrm\{det\}\}\_\{jk\}\)=\\operatorname\{sign\}\(\\theta^\{\\ast\}\_\{jk\}\)\\quad\\text\{for all \}k\\in S\_\{j\}\.

###### Proof\.

On the event𝒜^=𝒜h\\hat\{\\mathcal\{A\}\}=\\mathcal\{A\}\_\{h\}, Algorithm[2](https://arxiv.org/html/2607.03005#alg2)runs Algorithm[1](https://arxiv.org/html/2607.03005#alg1)with source set𝒜h\\mathcal\{A\}\_\{h\}\. Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3)givesℙ​\(𝒜^=𝒜h\)→1\\mathbb\{P\}\(\\hat\{\\mathcal\{A\}\}=\\mathcal\{A\}\_\{h\}\)\\to 1\. Intersecting this event with the high\-probability events in Theorems[1](https://arxiv.org/html/2607.03005#Thmtheorem1)and[2](https://arxiv.org/html/2607.03005#Thmtheorem2), with𝒜\\mathcal\{A\}andNNreplaced by𝒜h\\mathcal\{A\}\_\{h\}andNhN\_\{h\}, gives the two claims\. This completes the proof\. ∎

Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3)is a conditional result for the screening statistic used in Algorithm[2](https://arxiv.org/html/2607.03005#alg2)\. It shows that graph\-level separation, validation\-loss deviation, and adaptive\-threshold calibration imply exact recovery of𝒜h\\mathcal\{A\}\_\{h\}\. Proposition[2](https://arxiv.org/html/2607.03005#Thmproposition2)gives sufficient validation\-stability conditions under which Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)holds\.

## 5Experiments

### 5\.1Simulation Study

We empirically evaluate Trans\-Ising on synthetic Ising models and real\-world applications\. We assess whether Trans\-Ising \(i\) reduces estimation error relative to target\-only learning, \(ii\) preserves this reduction through data\-driven source detection when heterogeneous sources are present, and \(iii\) reduces prediction error on real data\. We report relative Frobenius error for parameter estimation and edge recovery performance via Precision–Recall \(PR\) curves \(details are in Appendix[S\.1](https://arxiv.org/html/2607.03005#A1)\)\.

Simulation setup\.We generate a target dataset withn0=160n\_\{0\}=160samples andS=12S=12auxiliary datasets each withns=300n\_\{s\}=300samples in a high\-dimensional regime \(p=200\>n0p=200\>n\_\{0\}\)\. The target interaction matrixθtrue\\theta\_\{\\text\{true\}\}is drawn from one of four sparse graph topologies \(Appendix[S\.1\.2](https://arxiv.org/html/2607.03005#A1.SS2); Figure[5](https://arxiv.org/html/2607.03005#A1.F5)\)\. For each run,\|𝒜\|∈\{0,3,6,9,12\}\|\\mathcal\{A\}\|\\in\\\{0,3,6,9,12\\\}auxiliary sources are*informative*, generated by perturbingθtrue\\theta\_\{\\text\{true\}\}with symmetric Gaussian noise at levelsσ∈\{0\.01,0\.1,0\.2\}\\sigma\\in\\\{0\.01,0\.1,0\.2\\\}\. The remaining sources are generated from a structurally heterogeneous graph to induce negative transfer\. All data are sampled usingIsingSampler\(Epskamp,[2025](https://arxiv.org/html/2607.03005#bib.bib16)\)\. We average results over 100 independent simulation runs\.

We compare four methods: \(i\)Naive\-LogLasso\(target\-only nodewise logistic lasso\), \(ii\)Pooled\-Trans\-Ising\(naively pooling all sources; i\.e\., Algorithm[1](https://arxiv.org/html/2607.03005#alg1)with all auxiliaries\), \(iii\)Oracle Trans\-Ising\(informative set known\), and \(iv\)Trans\-Ising\(data\-driven source detection\)\. Hyperparameters are tuned proportionally to theoretical rates via cross\-validation; full tuning details are in Appendix[S\.1\.3](https://arxiv.org/html/2607.03005#A1.SS3)\.

Estimation errors\.Figure[1](https://arxiv.org/html/2607.03005#S4.F1)reports the average relative Frobenius errors across four graph structures\. As the number of informative sources\|𝒜\|\|\\mathcal\{A\}\|increases, both Oracle Trans\-Ising and Trans\-Ising reduce error, consistent with variance reduction from additional compatible samples\. The Trans\-Ising curve closely tracks the oracle curve across the settings we considered\. In contrast, pooling all sources without screening can increase error when heterogeneous \(non\-informative\) sources are present, illustrating negative transfer in this regime\.

Edge recovery\.We evaluate support recovery using PR curves, which are typically more informative than ROC curves under graph sparsity \(Appendix[S\.1\.1](https://arxiv.org/html/2607.03005#A1.SS1)\)\. We fix\|𝒜\|=3\|\\mathcal\{A\}\|=3informative sources and99heterogeneous sources, and vary the perturbation levelσ∈\{0\.01,0\.1,0\.2\}\\sigma\\in\\\{0\.01,0\.1,0\.2\\\}\. In Figure[2](https://arxiv.org/html/2607.03005#S5.F2), Trans\-Ising attains PR performance comparable to the oracle benchmark across the graph families we considered\. Naive pooling can yield lower precision at the recall values shown in Figure[2](https://arxiv.org/html/2607.03005#S5.F2), consistent with additional false positives induced by heterogeneous sources\. ROC curves are also reported in Appendix[S\.1\.4](https://arxiv.org/html/2607.03005#A1.SS4)for completeness\.

![Refer to caption](https://arxiv.org/html/2607.03005v1/x2.png)Figure 2:Averaged PR curves for Naive\-LogLasso \(green\), Oracle Trans\-Ising \(black\), Pooled\-Trans\-Ising \(red\), and Trans\-Ising \(cyan, dashed\), across four graph structures at perturbation levels100​σ∈\{1,10,20\}100\\sigma\\in\\\{1,10,20\\\}, withp=200p=200\.
### 5\.2Real Data Study 1: Mutation Data Analysis

To evaluate the practical effectiveness of our proposed method, we conduct a real data study using the mutation data from the Cancer Dependency Map \(DepMap\) portal’sdepmap\_mutationCallsdataset \(release 22Q2\)\(Broad DepMap,[2022](https://arxiv.org/html/2607.03005#bib.bib5)\)\. Interaction\-network inference is a common goal in high\-dimensional biological data analysis\(Marbach et al\.,[2012](https://arxiv.org/html/2607.03005#bib.bib29)\)\. High\-dimensional binary\-response modeling for Cancer Cell\-Line Encyclopedia data has also been studied through low\-rank multiple\-response logistic regression and its joint Ising extension\(Park et al\.,[2024](https://arxiv.org/html/2607.03005#bib.bib36)\)\. The data contain binary mutation indicators for over 18,000 genes across 1,771 cancer cell lines\. We select the 200 most frequently mutated genes in the target training data\. We treat each cancer type with at least 20 samples as a target study, and use the non\-target groups with at least 20 samples as auxiliary studies\.

Because the ground\-truth interaction network is unknown, we evaluate via prediction\. For each target cancer, we run 5\-fold cross\-validation and report the misclassification error for predicting held\-out mutation statuses\. Figure[3](https://arxiv.org/html/2607.03005#S5.F3)shows relative misclassification error compared to the Naive\-LogLasso baseline \(normalized to 1\.0\)\. Trans\-Ising consistently improves over the target\-only baseline across cancer types, while Pooled\-Trans\-Ising is less reliable and mostly performs worse than Trans\-Ising\.

To assess transferability across cancer types, Appendix[S\.2\.2](https://arxiv.org/html/2607.03005#A2.SS2)reports the detected informative sources for each target\. The number and identity of informative sources are heterogeneous, supporting the need for selective transfer and not indiscriminate pooling\. Additional real data studies are provided in Appendix[S\.2\.3](https://arxiv.org/html/2607.03005#A2.SS3)–[S\.2\.4](https://arxiv.org/html/2607.03005#A2.SS4)\.

![Refer to caption](https://arxiv.org/html/2607.03005v1/x3.png)Figure 3:Relative misclassification error rates for Pooled\-Trans\-Ising \(red\) and Trans\-Ising \(cyan\) across different target cancers\. The error is relative to the Naive\-LogLasso baseline \(y=1\.0\)\.

## 6Discussion

Trans\-Ising provides a transfer learning method for high\-dimensional Ising models that mitigates negative transfer through loss\-based screening and dual\-penalty correction\. Theoretically, our error bounds separate pooled variance reduction from cross\-domain heterogeneity \(hjh\_\{j\}\), and achieve exact support recovery without imposing restrictive irrepresentable conditions\. Empirically, Trans\-Ising consistently outperforms target\-only and naive\-pooling baselines\.

A limitation is that cross\-validated screening becomes computationally costly when many auxiliaries are available\. Future work may explore computationally efficient and scalable screening methods\. In addition, when parameters exhibit known group structure, structured regularization such as the logistic group lasso may improve interpretability and stability\(Meier et al\.,[2008](https://arxiv.org/html/2607.03005#bib.bib30)\)\.

## Acknowledgements

Joonho Kim and Seyoung Park’s work was supported by the National Research Foundation of Korea \(NRF\) grant funded by the MSIT \(No\. RS\-2025\-00517793\) and by the Yonsei University Research Fund of 2025\-22\-0071\.

## Impact Statement

This paper presents work whose goal is to advance the field of Machine Learning\. We do not identify specific societal consequences that require separate discussion here\.

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APPENDIX

## Appendix S\.1Additional Experimental Details

### S\.1\.1Evaluation Metrics

#### Frobenius norm and relative error\.

We evaluate estimation accuracy using the relative error under the Frobenius norm:

‖θ^−θtrue‖F‖θtrue‖F,\\frac\{\\\|\\hat\{\\theta\}\-\\theta\_\{\\text\{true\}\}\\\|\_\{F\}\}\{\\\|\\theta\_\{\\text\{true\}\}\\\|\_\{F\}\},whereθ^\\hat\{\\theta\}is the estimated interaction matrix andθtrue\\theta\_\{\\text\{true\}\}is the ground\-truth matrix\.

#### ROC and PR curves for graph recovery\.

We evaluate edge recovery via Receiver Operating Characteristic \(ROC\) and Precision–Recall \(PR\) curves by treating edge detection as a binary classification task\. For each pair\(j,k\)\(j,k\), we use the absolute value of the estimated parameter\|θ^j​k\|\|\\hat\{\\theta\}\_\{jk\}\|as a score, with larger scores indicating higher confidence of an edge\. The ROC curve summarizes the tradeoff between

TPR=TPTP\+FNandFPR=FPFP\+TN,\\text\{TPR\}=\\frac\{\\text\{TP\}\}\{\\text\{TP\}\+\\text\{FN\}\}\\quad\\text\{and\}\\quad\\text\{FPR\}=\\frac\{\\text\{FP\}\}\{\\text\{FP\}\+\\text\{TN\}\},whereas the PR curve summarizes the tradeoff between precision

Precision=TPTP\+FP\\text\{Precision\}=\\frac\{\\text\{TP\}\}\{\\text\{TP\}\+\\text\{FP\}\}and recall \(TPR\)\. We use AUC \(ROC\) and AUCPR \(PR\) as summary statistics\.

PR curves are often more informative than ROC curves when positives are much rarer than negatives, as in sparse graph recovery\(Davis & Goadrich,[2006](https://arxiv.org/html/2607.03005#bib.bib14); Saito & Rehmsmeier,[2015](https://arxiv.org/html/2607.03005#bib.bib39)\)\. In such settings, the large number of true negatives can make ROC summaries less sensitive to false positive edge discoveries, whereas PR curves focus on precision among selected edges\.

### S\.1\.2Simulation Design and Settings

We generate Ising model data withp=200p=200variables\. The ground\-truth interaction matrixθtrue\\theta\_\{\\text\{true\}\}follows one of four sparse graph structures \(Figure[5](https://arxiv.org/html/2607.03005#A1.F5)\)\. For all structures,diag​\(θtrue\)=0\\mathrm\{diag\}\(\\theta\_\{\\text\{true\}\}\)=0\.

#### \(i\) Sparse block star graph\.

Theppnodes are divided into disjoint blocks of size 10\. Within each block, one hub node is connected to a random subset of 5 spoke nodes\. Edge weights are sampled fromUnif​\(0\.5,1\.5\)\\mathrm\{Unif\}\(0\.5,1\.5\)\.

#### \(ii\) Block\-wise star graph\.

Theppnodes are divided into disjoint blocks of size 10\. Within each block, one node is designated as a hub and connected to all other 9 nodes\. Edge weights are sampled fromUnif​\(0\.5,1\.5\)\\mathrm\{Unif\}\(0\.5,1\.5\)\.

#### \(iii\) Chain graph\.

A path graph where only adjacent pairs\(i,i\+1\)\(i,i\+1\)have nonzero interactions\. Edge weights are sampled fromUnif​\(0\.5,1\.5\)\\mathrm\{Unif\}\(0\.5,1\.5\)\.

#### \(iv\) Random sparse graph\.

We first generateθraw∼𝒩​\(0,1\)p×p\\theta\_\{\\text\{raw\}\}\\sim\\mathcal\{N\}\(0,1\)^\{p\\times p\}and symmetrize as\(θraw\+θraw⊤\)/4\(\\theta\_\{\\text\{raw\}\}\+\\theta\_\{\\text\{raw\}\}^\{\\top\}\)/4, yielding off\-diagonal entries distributed as𝒩​\(0,1/8\)\\mathcal\{N\}\(0,1/8\)\. We set the diagonal to zero and induce sparsity by thresholding: off\-diagonal entries with absolute value below 0\.7 are set to zero\.

#### Target and auxiliary sampling\.

We create a primary dataset withn0=160n\_\{0\}=160samples andS=12S=12auxiliary datasets withns=300n\_\{s\}=300samples each\. All Ising samples are generated usingIsingSamplerfrom theIsingSamplerR package\(Epskamp,[2025](https://arxiv.org/html/2607.03005#bib.bib16)\)\. The number of informative sources\|𝒜\|\|\\mathcal\{A\}\|is varied across\{0,3,6,9,12\}\\\{0,3,6,9,12\\\}\. Informative auxiliaries are generated by adding symmetric Gaussian noiseΔ\(s\)\\Delta^\{\(s\)\}toθtrue\\theta\_\{\\text\{true\}\}, where off\-diagonal entries satisfyΔj​k\(s\)∼𝒩​\(0,σ2\)\\Delta^\{\(s\)\}\_\{jk\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\)andσ∈\{0\.01,0\.1,0\.2\}\\sigma\\in\\\{0\.01,0\.1,0\.2\\\}\. Non\-informative auxiliaries are generated from a structurally heterogeneous graph: a sparse binary matrix is sampled withP​\(θi​j=1\)=0\.05P\(\\theta\_\{ij\}=1\)=0\.05, multiplied by standard normal noise, and symmetrized using the element\-wise maximum\.

### S\.1\.3Tuning details

Regularization parameters are tuned proportionally to theoretical rates by cross\-validation\. ForOracle Trans\-Ising, putNorc:=n0\+∑s∈𝒜nsN\_\{\\mathrm\{orc\}\}:=n\_\{0\}\+\\sum\_\{s\\in\\mathcal\{A\}\}n\_\{s\}\. ForTrans\-Ising, after source detection, putNsel:=n0\+∑s∈𝒜^nsN\_\{\\mathrm\{sel\}\}:=n\_\{0\}\+\\sum\_\{s\\in\\hat\{\\mathcal\{A\}\}\}n\_\{s\}for the final estimation step\. ForPooled\-Trans\-Ising, putNpool:=n0\+∑s=1SnsN\_\{\\mathrm\{pool\}\}:=n\_\{0\}\+\\sum\_\{s=1\}^\{S\}n\_\{s\}\.

In Step 1, we tune the penalty over the grid

λw∈\{\{1\.1,0\.9,0\.7,0\.5,0\.3,0\.2,0\.1\}​log⁡p/Norc,Oracle Trans\-Ising,\{1\.1,0\.9,0\.7,0\.5,0\.3,0\.2,0\.1\}​log⁡p/Nsel,Trans\-Ising,\{1\.1,0\.9,0\.7,0\.5,0\.3,0\.2,0\.1\}​log⁡p/Npool,Pooled\-Trans\-Ising\.\\lambda\_\{w\}\\in\\begin\{cases\}\\\{1\.1,0\.9,0\.7,0\.5,0\.3,0\.2,0\.1\\\}\\sqrt\{\\log p/N\_\{\\mathrm\{orc\}\}\},&\\text\{Oracle Trans\-Ising\},\\\\ \\\{1\.1,0\.9,0\.7,0\.5,0\.3,0\.2,0\.1\\\}\\sqrt\{\\log p/N\_\{\\mathrm\{sel\}\}\},&\\text\{Trans\-Ising\},\\\\ \\\{1\.1,0\.9,0\.7,0\.5,0\.3,0\.2,0\.1\\\}\\sqrt\{\\log p/N\_\{\\mathrm\{pool\}\}\},&\\text\{Pooled\-Trans\-Ising\}\.\\end\{cases\}
In Step 2, we first select a base penalty

λbase∈\{1\.5,1\.3,1\.1,0\.9,0\.7,0\.5,0\.3\}​log⁡pn0\\lambda\_\{\\mathrm\{base\}\}\\in\\\{1\.5,1\.3,1\.1,0\.9,0\.7,0\.5,0\.3\\\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}by cross\-validation, and then set

λδ=0\.5​λbase,λ=0\.5​λbase\.\\lambda\_\{\\delta\}=0\.5\\,\\lambda\_\{\\mathrm\{base\}\},\\qquad\\lambda=0\.5\\,\\lambda\_\{\\mathrm\{base\}\}\.
The adaptive threshold parameter for source detection is set toCτ=1/2C\_\{\\tau\}=1/2inτ=Cτ​σ^\\tau=C\_\{\\tau\}\\,\\hat\{\\sigma\}for all simulations\. ForNaive\-LogLasso, we tune

λnaive∈\{1\.5,1\.4,1\.3,1\.2,1\.1,1\.0,0\.9,0\.8,0\.7,0\.6,0\.5\}​log⁡pn0\\lambda\_\{\\mathrm\{naive\}\}\\in\\\{1\.5,1\.4,1\.3,1\.2,1\.1,1\.0,0\.9,0\.8,0\.7,0\.6,0\.5\\\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}by cross\-validation\. All reported metrics are averaged over 100 independent trials for each condition\.

### S\.1\.4ROC curves for edge recovery

For completeness, we also provide the ROC curves \(Figure[5](https://arxiv.org/html/2607.03005#A1.F5)\) for the edge recovery experiments described in Section[5\.1](https://arxiv.org/html/2607.03005#S5.SS1)\. We fix\|𝒜\|=3\|\\mathcal\{A\}\|=3informative sources and 9 non\-informative sources and varyσ∈\{0\.01,0\.1,0\.2\}\\sigma\\in\\\{0\.01,0\.1,0\.2\\\}\. The score for each potential edge is\|θ^j​k\|\|\\hat\{\\theta\}\_\{jk\}\|\.

![Refer to caption](https://arxiv.org/html/2607.03005v1/x4.png)Figure 4:Underlying network structures forp=200p=200\.
![Refer to caption](https://arxiv.org/html/2607.03005v1/x5.png)Figure 5:Averaged ROC curves for edge recovery across four graph structures at perturbation levels100​σ∈\{1,10,20\}100\\sigma\\in\\\{1,10,20\\\}\.

## Appendix S\.2Additional Real Data Studies

### S\.2\.1Exploratory Data Analysis of the Mutation Data

For better understanding of the target dataset, we perform an exploratory data analysis \(EDA\) to characterize the mutation landscape of thedepmap\_mutationCallsdataset\.

Figure[6](https://arxiv.org/html/2607.03005#A2.F6)shows the most frequently mutated genes for each primary disease\. Mutation frequencies vary across primary diseases\. Some genes such as TTN appear frequently across cancer types, whereas the ranking and presence of other genes depend on the primary disease\. For instance, KRAS is a top mutated gene in Pancreatic Cancer but not in the other reported cancer types\. These patterns motivate source screening before pooling cancer\-type datasets\.

### S\.2\.2Supplementary Analysis on Mutation Data

To further check the potential for transfer learning across different cancer types, Table[2](https://arxiv.org/html/2607.03005#A2.T2)summarizes the outcomes of the source detection procedure for each target disease analyzed\. The procedure selects sources according to the change in held\-out pseudolikelihood loss\. The number of informative sources varies across cancer types\. For instance, Brain Cancer and Ovarian Cancer select more auxiliary datasets, whereas Skin Cancer has few informative sources\. These results describe source\-screening behavior and do not identify specific biological mechanisms\.

![Refer to caption](https://arxiv.org/html/2607.03005v1/x6.png)Figure 6:Exploratory data analysis of the DepMap mutation data\. Mutation frequencies vary across primary diseases\.Table 1:Disease to Index MappingTable 2:Informative Source Indices for each Target Disease \(DepMap Mutation Data\)\.
### S\.2\.3Real Data Study 2: Online Transaction Analysis

We also conduct a real data study on e\-commerce transaction data\. We use theUCI Online Retaildataset from theUCI Machine Learning Repository, which contains transactions from a UK\-based online retailer across customer countries\(Chen et al\.,[2012](https://arxiv.org/html/2607.03005#bib.bib13); Chen,[2015](https://arxiv.org/html/2607.03005#bib.bib12)\)\.

We frame this as an Ising model estimation problem, where each item is a node, and each invoice is a sample\. The binary state \(1 or 0\) represents whether an item was purchased in that transaction\. The resulting Ising graph represents conditional associations among item\-purchase indicators\.

#### Experimental Setup\.

We define each country as a separate domain\. One country with a sufficient number of invoices \(e\.g\., Germany, France\) is designated as thetargetstudy, and other countries with sample size greater than5050serve as theauxiliarysources\. We exclude the “United Kingdom” dataset because, after removing cancellation invoices and nonpositive\-quantity records, it contains18,78618\{,\}786of20,72820\{,\}728cleaned invoices \(90\.6%90\.6\\%\) and would dominate the pooled analysis\. These are invoice\-level counts after applying the two filters and before selecting target and auxiliary countries\.

For each target country, we select thep=200p=200most frequently purchased items within that country to define the nodes of our graph\. All auxiliary datasets are then projected onto this same 200\-item space\. We then apply ourTrans\-Isingalgorithm and thePooled\-Trans\-Isingmethod\.

Here we use 5\-fold cross\-validation on the target data\. We use the same nodewise conditional prediction rule as in the DepMap study, and we report the results as the relative misclassification error standardized against theNaive\-LogLassobaseline\.

#### Results\.

Figure[8](https://arxiv.org/html/2607.03005#A2.F8)shows the performance of Trans\-Ising \(cyan bars\) and Pooled\-Trans\-Ising \(red bars\) across nine target countries\.

![Refer to caption](https://arxiv.org/html/2607.03005v1/x7.png)Figure 7:Relative misclassification error rates for Pooled\-Trans\-Ising and Trans\-Ising on the Online Retail dataset\. The error is relative to the Naive\-LogLasso baseline \(y=1\.0y=1\.0\)\.
![Refer to caption](https://arxiv.org/html/2607.03005v1/x8.png)Figure 8:Relative misclassification error rates for Pooled\-Trans\-Ising and Trans\-Ising on the MovieLens 1M dataset\. The error is relative to the Naive\-LogLasso baseline \(y=1\.0y=1\.0\)\.

The Online Retail results are consistent with the DepMap study for the reported targets\. Trans\-Ising yields relative misclassification error below 1\.0 for the reported targets, indicating improved prediction compared with the target\-only baseline\.

Compared with pooling all auxiliary countries, the screened estimator attains relative misclassification error comparable to or below Pooled\-Trans\-Ising for the reported targets, while both methods remain below the target\-only baseline in Figure[8](https://arxiv.org/html/2607.03005#A2.F8)\. Table[3](https://arxiv.org/html/2607.03005#A2.T3)summarizes the selected sources for each target\. In this dataset, most reported targets select many available auxiliary countries; this is an empirical observation for the included countries and not a general claim about online transaction data\.

Table 3:Informative Source Countries for each Target Country \(Online Retail data\)\.

### S\.2\.4Real Data Study 3: Movie Data Analysis

We also apply Trans\-Ising to collaborative filtering using theMovieLens 1Mdataset\(Harper & Konstan,[2015](https://arxiv.org/html/2607.03005#bib.bib21)\)\. This dataset contains 1 million user\-item ratings\. We formulate this as a problem of learning user preference networks\.

We define a binary“like”event as any rating≥4\\geq 4\. Each user is treated as a sample, and each movie is a node\. The binary state \(1 or 0\) indicates whether a user “liked” a specific movie\. The resulting Ising graph represents conditional associations among movie\-preference indicators\.

#### Experimental Setup\.

We use user demographics to define distinct domains\. For this study, we select userAgeas the domain variable\. The dataset provides discrete age brackets \(1, 18, 25, 35, 45, 50, 56\)\. We designate one age group as thetargetstudy and all other age groups \(with\>200\>200users\) asauxiliarysources\.

Following the same methodology as in the previous studies, we select thep=100p=100most “liked” movies based on the preferences within the target age group\. All auxiliary age group datasets are then aligned to thisp=100p=100movie space\. We again use the same nodewise conditional prediction rule and 5\-fold cross\-validation to compareTrans\-IsingandPooled\-Trans\-Ising, reporting the relative misclassification error against theNaive\-LogLassobaseline\.

#### Results\.

Figure[8](https://arxiv.org/html/2607.03005#A2.F8)reports 5\-fold cross\-validated misclassification error normalized by the target\-only baseline\. Trans\-Ising gives a smaller relative misclassification error than naive pooling for most of the reported target age groups\.

Separately, to provide a descriptive view of which age groups the screening rule deems compatible, we also report the selected source sets obtained by running the screening step on the full sample \(Table[4](https://arxiv.org/html/2607.03005#A2.T4)\)\. This full\-sample analysis is not used for out\-of\-sample evaluation; it is included to summarize empirical patterns in estimated transferability across age groups within this dataset\.

Table 4:Informative source age groups for each target age group \(MovieLens 1M\)\.The three real\-data analyses show how source screening changes prediction error for cancer genomics, online transactions, and movie ratings in the reported settings\.

## Appendix S\.3Technical Assumptions for Section[4](https://arxiv.org/html/2607.03005#S4)

###### Assumption 1\(Independent samples across domains\)\.

For eachr∈\{0,1,…,S\}r\\in\\\{0,1,\\ldots,S\\\}, the observationsX1\(r\),…,Xnr\(r\)X^\{\(r\)\}\_\{1\},\\ldots,X^\{\(r\)\}\_\{n\_\{r\}\}are independent copies from the Ising distribution with parameterθ\(r\)\\theta^\{\(r\)\}\. The samples are independent across different domainsrr\. Hereθ\(0\)=θ∗\\theta^\{\(0\)\}=\\theta^\{\\ast\}, andθ\(r\)=w\(r\)\\theta^\{\(r\)\}=w^\{\(r\)\}forr≥1r\\geq 1\.

###### Assumption 2\(Sparsity and source\-to\-target proximity\)\.

Fix a nodejjand letSj=\{m≠j:θj​m∗≠0\}S\_\{j\}=\\\{m\\neq j:\\theta^\{\\ast\}\_\{jm\}\\neq 0\\\}withsj=\|Sj\|s\_\{j\}=\|S\_\{j\}\|\. Assume the target neighborhood is sparse:sj=o​\(n0log⁡p\)\.s\_\{j\}=o\\\!\\left\(\\frac\{n\_\{0\}\}\{\\log p\}\\right\)\.For each informative sourcek∈𝒜k\\in\\mathcal\{A\}, letw∖j\(k\)w^\{\(k\)\}\_\{\\setminus j\}denote the population nodewise parameter at nodejj\. Assume the informative sources areℓ1\\ell\_\{1\}\-close to the target:

‖θ∖j∗−w∖j\(k\)‖1≤hj,∀k∈𝒜,\\\|\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\(k\)\}\_\{\\setminus j\}\\\|\_\{1\}\\leq h\_\{j\},\\qquad\\forall k\\in\\mathcal\{A\},wherehj≥0h\_\{j\}\\geq 0may depend onjjand is allowed to tend to zero as\(n0,p\)\(n\_\{0\},p\)grow\.

###### Assumption 3\(Information comparability across sources\)\.

There exist a convex set𝒰⊂ℝp−1\\mathcal\{U\}\\subset\\mathbb\{R\}^\{p\-1\}and a constantC\>0C\>0such that:

1. 1\.\(Local comparability\)for allk∈\{0\}∪𝒜k\\in\\\{0\\\}\\cup\\mathcal\{A\}and allu,v∈𝒰u,v\\in\\mathcal\{U\}, ‖\(∇2ℒA,j​\(u\)\)−1​∇2ℒk,j​\(v\)‖1≤C\.\\Big\\\|\\Big\(\\nabla^\{2\}\\mathcal\{L\}\_\{A,j\}\(u\)\\Big\)^\{\-1\}\\nabla^\{2\}\\mathcal\{L\}\_\{k,j\}\(v\)\\Big\\\|\_\{1\}\\leq C\.In particular,∇2ℒA,j​\(u\)\\nabla^\{2\}\\mathcal\{L\}\_\{A,j\}\(u\)is invertible for allu∈𝒰u\\in\\mathcal\{U\}\.
2. 2\.\(Local well\-posedness of population minimizers\)The population minimizers of our analysis lie in𝒰\\mathcal\{U\}: θ∖j∗∈𝒰,w∖j\(k\)∈𝒰∀k∈𝒜,wA,∖j∗∈𝒰,\\theta^\{\\ast\}\_\{\\setminus j\}\\in\\mathcal\{U\},\\qquad w^\{\(k\)\}\_\{\\setminus j\}\\in\\mathcal\{U\}\\ \\ \\forall k\\in\\mathcal\{A\},\\qquad w^\{\\ast\}\_\{A,\\setminus j\}\\in\\mathcal\{U\},wherew∖j\(k\):=arg⁡minu⁡ℒk,j​\(u\)w^\{\(k\)\}\_\{\\setminus j\}:=\\arg\\min\_\{u\}\\mathcal\{L\}\_\{k,j\}\(u\)andwA,∖j∗:=arg⁡minu⁡ℒA,j​\(u\)w^\{\\ast\}\_\{A,\\setminus j\}:=\\arg\\min\_\{u\}\\mathcal\{L\}\_\{A,j\}\(u\)\.
3. 3\.\(Integrated comparability\)The bound in \(i\) also holds when∇2ℒA,j​\(u\)\\nabla^\{2\}\\mathcal\{L\}\_\{A,j\}\(u\)and∇2ℒk,j​\(v\)\\nabla^\{2\}\\mathcal\{L\}\_\{k,j\}\(v\)are replaced by their averages over line segments in𝒰\\mathcal\{U\}\.

###### Condition 1\(Theorem\-local pooled RSC and localization\)\.

There exist constantsκA\>0\\kappa\_\{A\}\>0,τA≥0\\tau\_\{A\}\\geq 0, and a radiusrA\>0r\_\{A\}\>0such that for all

Δ∈𝒞A:=\{Δ:‖ΔSjc‖1≤3​‖ΔSj‖1\+4​‖\(wA,∖j∗\)Sjc‖1\},\\Delta\\in\\mathcal\{C\}\_\{A\}:=\\Big\\\{\\Delta:\\\|\\Delta\_\{S\_\{j\}^\{c\}\}\\\|\_\{1\}\\leq 3\\\|\\Delta\_\{S\_\{j\}\}\\\|\_\{1\}\+4\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\\|\_\{1\}\\Big\\\},and allϑ\\varthetasatisfying‖ϑ−wA,∖j∗‖1≤rA\\\|\\vartheta\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\leq r\_\{A\},

Δ⊤​∇2ℓA,j​\(ϑ\)​Δ≥κA​‖Δ‖22−τA​log⁡pN​‖Δ‖12\.\\Delta^\{\\top\}\\nabla^\{2\}\\ell\_\{A,j\}\(\\vartheta\)\\,\\Delta\\ \\geq\\ \\kappa\_\{A\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-\\tau\_\{A\}\\frac\{\\log p\}\{N\}\\\|\\Delta\\\|\_\{1\}^\{2\}\.In addition,

‖w^∖jA−wA,∖j∗‖1≤rA\.\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\leq r\_\{A\}\.This implies that the segment betweenw^∖jA\\hat\{w\}^\{A\}\_\{\\setminus j\}andwA,∖j∗w^\{\\ast\}\_\{A,\\setminus j\}is contained in\{ϑ:‖ϑ−wA,∖j∗‖1≤rA\}\\\{\\vartheta:\\\|\\vartheta\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\leq r\_\{A\}\\\}\.

###### Condition 2\(Theorem\-local target RSC for bias correction\)\.

LetCh≥1C\_\{h\}\\geq 1be chosen to satisfy

‖δ∖j∗‖1≤Ch​hj,\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq C\_\{h\}h\_\{j\},which is available under Assumptions[2](https://arxiv.org/html/2607.03005#Thmassumption2)and[3](https://arxiv.org/html/2607.03005#Thmassumption3)by Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1), and definetj:=⌈Ch​hjλδ⌉\.t\_\{j\}:=\\left\\lceil\\frac\{C\_\{h\}\\,h\_\{j\}\}\{\\lambda\_\{\\delta\}\}\\right\\rceil\.There exist constantsκ0\>0\\kappa\_\{0\}\>0,τ0≥0\\tau\_\{0\}\\geq 0, and a radiusr0\>0r\_\{0\}\>0such that for any index setT⊆V∖\{j\}T\\subseteq V\\setminus\\\{j\\\}with\|T\|≤tj\|T\|\\leq t\_\{j\}, for all

Δ∈𝒞T,h:=\{Δ:‖ΔTc‖1≤3​‖ΔT‖1\+3​Ch​hj\},\\Delta\\in\\mathcal\{C\}\_\{T,h\}:=\\\{\\Delta:\\\|\\Delta\_\{T^\{c\}\}\\\|\_\{1\}\\leq 3\\\|\\Delta\_\{T\}\\\|\_\{1\}\+3C\_\{h\}h\_\{j\}\\\},and allϑ\\varthetasatisfying‖ϑ−θ∖j∗‖1≤r0\\\|\\vartheta\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq r\_\{0\},

Δ⊤​∇2ℓ0,j​\(ϑ\)​Δ≥κ0​‖Δ‖22−τ0​log⁡pn0​‖Δ‖12\.\\Delta^\{\\top\}\\nabla^\{2\}\\ell\_\{0,j\}\(\\vartheta\)\\,\\Delta\\ \\geq\\ \\kappa\_\{0\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-\\tau\_\{0\}\\frac\{\\log p\}\{n\_\{0\}\}\\\|\\Delta\\\|\_\{1\}^\{2\}\.

###### Assumption 4\(Bounded nodewise fields and non\-degenerate curvature\)\.

The Ising variables satisfyxi​k∈\{−1,1\}x\_\{ik\}\\in\\\{\-1,1\\\}\. There exists a finite constantM<∞M<\\inftysuch that, for the target parameterθ∗\\theta^\{\\ast\}and all informative\-source parameters\{w\(s\)\}s∈𝒜\\\{w^\{\(s\)\}\\\}\_\{s\\in\\mathcal\{A\}\},

maxr∈\{0\}∪𝒜⁡maxj∈V⁡maxx∈\{−1,1\}p⁡2​\|∑k≠jθj​k\(r\)​xk\|≤M,\\max\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\ \\max\_\{j\\in V\}\\ \\max\_\{x\\in\\\{\-1,1\\\}^\{p\}\}\\ 2\\left\|\\sum\_\{k\\neq j\}\\theta^\{\(r\)\}\_\{jk\}\\,x\_\{k\}\\right\|\\ \\leq\\ M,whereθ\(0\):=θ∗\\theta^\{\(0\)\}:=\\theta^\{\\ast\}andθ\(r\):=w\(r\)\\theta^\{\(r\)\}:=w^\{\(r\)\}forr∈𝒜r\\in\\mathcal\{A\}\. For any signed linear predictor whose absolute value is bounded byMM, define

ρ0:=inf\|t\|≤Mf′′​\(t\)\>0,f​\(t\)=log⁡\(1\+e−t\)\.\\rho\_\{0\}\\;:=\\;\\inf\_\{\|t\|\\leq M\}f^\{\\prime\\prime\}\(t\)\\;\>\\;0,\\qquad f\(t\)=\\log\(1\+e^\{\-t\}\)\.Curvature over the empirical neighborhoods used in the proofs is imposed separately in Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1)and[2](https://arxiv.org/html/2607.03005#Thmcondition2)\.

###### Condition 3\(Theorem\-local stationary local solution in Step 2\)\.

LetQ​\(δ\):=Qj​\(δ;w^∖jA\)Q\(\\delta\):=Q\_\{j\}\(\\delta;\\hat\{w\}^\{A\}\_\{\\setminus j\}\)denote the Step 2 objective in \([18](https://arxiv.org/html/2607.03005#S4.E18)\)\. Assume the optimization routine returns a pointδ^∖jA\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}and a toleranceεn,j≥0\\varepsilon\_\{n,j\}\\geq 0such that:

1. 1\.\(Approximate stationarity\) dist∞​\(0,∂Q​\(δ^∖jA\)\):=infξ∈∂Q​\(δ^∖jA\)‖ξ‖∞≤εn,j\.\\mathrm\{dist\}\_\{\\infty\}\\bigl\(0,\\partial Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\\bigr\):=\\inf\_\{\\xi\\in\\partial Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\}\\\|\\xi\\\|\_\{\\infty\}\\leq\\varepsilon\_\{n,j\}\.Equivalently, for each coordinatek≠jk\\neq j, there existgj​k∈∂\|δ^j​kA\|g\_\{jk\}\\in\\partial\|\\hat\{\\delta\}^\{A\}\_\{jk\}\|andhj​k∈∂Pλ​\(w^j​kA\+δ^j​kA\)h\_\{jk\}\\in\\partial P\_\{\\lambda\}\(\\hat\{w\}^\{A\}\_\{jk\}\+\\hat\{\\delta\}^\{A\}\_\{jk\}\)with\|gj​k\|≤1\|g\_\{jk\}\|\\leq 1such that \|\[∇ℓ0,j​\(w^∖jA\+δ^∖jA\)\]k\+λδ​gj​k\+hj​k\|≤εn,j\.\\left\|\\Big\[\\nabla\\ell\_\{0,j\}\(\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\\Big\]\_\{k\}\+\\lambda\_\{\\delta\}\\,g\_\{jk\}\+h\_\{jk\}\\right\|\\leq\\varepsilon\_\{n,j\}\.
2. 2\.\(Local minimality in a neighborhood ofδ∗\\delta^\{\\ast\}\)There exists a radiusrδ\>0r\_\{\\delta\}\>0such that‖δ^∖jA−δ∖j∗‖1≤rδ\\\|\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq r\_\{\\delta\}and Q​\(δ^∖jA\)≤Q​\(δ\)for all​δ​satisfying​‖δ−δ∖j∗‖1≤rδ\.Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\\leq Q\(\\delta\)\\qquad\\text\{for all \}\\delta\\text\{ satisfying \}\\\|\\delta\-\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq r\_\{\\delta\}\.
3. 3\.\(RSC neighborhood applicability\)Both the oracle\-shifted point and the returned estimator lie in the target\-RSC neighborhood in Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2), i\.e\., ‖w^∖jA\+δ∖j∗−θ∖j∗‖1≤r0,‖w^∖jA\+δ^∖jA−θ∖j∗‖1≤r0,\\big\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\delta^\{\\ast\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\big\\\|\_\{1\}\\leq r\_\{0\},\\qquad\\big\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\big\\\|\_\{1\}\\leq r\_\{0\},wherer0r\_\{0\}is the radius in Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2)\.

###### Condition 4\(Theorem\-local Step 2 cone condition\)\.

Letv=δ^∖jA−δ∖j∗v=\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\delta^\{\\ast\}\_\{\\setminus j\}andT=\{k≠j:\|δj​k∗\|\>λδ\}T=\\\{k\\neq j:\|\\delta^\{\\ast\}\_\{jk\}\|\>\\lambda\_\{\\delta\}\\\}\. The Step 2 error satisfies

‖vTc‖1≤3​‖vT‖1\+3​Ch​hj\.\\\|v\_\{T^\{c\}\}\\\|\_\{1\}\\leq 3\\\|v\_\{T\}\\\|\_\{1\}\+3C\_\{h\}h\_\{j\}\.

###### Assumption 5\(Tuning and Beta\-min for Oracle Property\)\.

LetPλ​\(⋅\)P\_\{\\lambda\}\(\\cdot\)be the SCAD penalty with parametera\>2a\>2\. Choose tuning parameters as

λw=cw​log⁡pN,λδ=cδ​log⁡pn0,λ=cλ​log⁡pn0,\\lambda\_\{w\}=c\_\{w\}\\sqrt\{\\frac\{\\log p\}\{N\}\},\\qquad\\lambda\_\{\\delta\}=c\_\{\\delta\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\},\\qquad\\lambda=c\_\{\\lambda\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\},wherecw,cδ,cλc\_\{w\},c\_\{\\delta\},c\_\{\\lambda\}are sufficiently large\. Define the target estimation error ratern,jr\_\{n,j\}as in Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)\.

We assume the beta\-min condition:

\|θj​m∗\|≥a​λ\+rn,jfor all​m∈Sj\.\|\\theta^\{\\ast\}\_\{jm\}\|\\ \\geq\\ a\\lambda\+r\_\{n,j\}\\qquad\\text\{for all \}m\\in S\_\{j\}\.Remark: This condition ensures that the true signals are large enough to fall into the constant penalty region of SCAD \(wherePλ′​\(⋅\)=0P^\{\\prime\}\_\{\\lambda\}\(\\cdot\)=0\) even after accounting for estimation error, which avoids the irrepresentable condition\(Zhao & Yu,[2006](https://arxiv.org/html/2607.03005#bib.bib51); Wainwright,[2009](https://arxiv.org/html/2607.03005#bib.bib45)\)\.

###### Condition 5\(Theorem\-local source\-detection separation and threshold calibration\)\.

For the constantcgap\>0c\_\{\\mathrm\{gap\}\}\>0, the sequenceνn\>0\\nu\_\{n\}\>0, and the set𝒜h\\mathcal\{A\}\_\{h\}defined before Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3), there exist a constantC\>0C\>0and a sequenceϵn=o​\(νn\)\\epsilon\_\{n\}=o\(\\nu\_\{n\}\)such that:

1. 1\.\(Informative sources\)For everys∈𝒜hs\\in\\mathcal\{A\}\_\{h\}, ℰ​\(s\)≤cgap​νn\.\\mathcal\{E\}\(s\)\\leq c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\.
2. 2\.\(Non\-informative sources\)For everys∉𝒜hs\\notin\\mathcal\{A\}\_\{h\}, ℰ​\(s\)≥\(1\+cgap\)​νn\.\\mathcal\{E\}\(s\)\\geq\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\.
3. 3\.\(Validation\-loss deviation\) max1≤s≤S⁡\|Δs−ℰ​\(s\)\|≤C​ϵn\\max\_\{1\\leq s\\leq S\}\|\\Delta\_\{s\}\-\\mathcal\{E\}\(s\)\|\\leq C\\epsilon\_\{n\}with probability tending to one\.
4. 4\.\(Adaptive\-threshold calibration\)The thresholdτ=Cτ​σ^\\tau=C\_\{\\tau\}\\hat\{\\sigma\}in Algorithm[2](https://arxiv.org/html/2607.03005#alg2)satisfies cgap​νn\+C​ϵn<Cτ​σ^<\(1\+cgap\)​νn−C​ϵnc\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\+C\\epsilon\_\{n\}<C\_\{\\tau\}\\hat\{\\sigma\}<\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\-C\\epsilon\_\{n\}with probability tending to one\.

Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1)–[4](https://arxiv.org/html/2607.03005#Thmcondition4)and Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)are theorem\-local high\-level inputs\. They are not asserted to follow from Assumptions[1](https://arxiv.org/html/2607.03005#Thmassumption1)–[4](https://arxiv.org/html/2607.03005#Thmassumption4)alone\. The following propositions give sufficient conditions for the empirical curvature and validation\-loss deviation inputs\. The Step 2 local\-solution and cone inputs remain theorem\-local algorithmic conditions unless a separate optimization argument verifies Conditions[3](https://arxiv.org/html/2607.03005#Thmcondition3)and[4](https://arxiv.org/html/2607.03005#Thmcondition4)\.

### S\.3\.1Sufficient Conditions for Theorem\-Local Inputs

For a fixed nodejjand each domainr∈\{0\}∪𝒜r\\in\\\{0\\\}\\cup\\mathcal\{A\}, letZ∖j\(r\)∈ℝnr×\(p−1\)Z\_\{\\setminus j\}^\{\(r\)\}\\in\\mathbb\{R\}^\{n\_\{r\}\\times\(p\-1\)\}denote the signed design matrix whoseii\-th row is\(zi\(r\)\)⊤\(z\_\{i\}^\{\(r\)\}\)^\{\\top\}\. Recall thatℓA,j\\ell\_\{A,j\}andℓ0,j\\ell\_\{0,j\}are the pooled and target losses\. Define

Σ^A:=∑r∈\{0\}∪𝒜αr​\(Z∖j\(r\)\)⊤​Z∖j\(r\)nr,Σ^0:=\(Z∖j\(0\)\)⊤​Z∖j\(0\)n0,\\widehat\{\\Sigma\}\_\{A\}:=\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\alpha\_\{r\}\\frac\{\(Z\_\{\\setminus j\}^\{\(r\)\}\)^\{\\top\}Z\_\{\\setminus j\}^\{\(r\)\}\}\{n\_\{r\}\},\\qquad\\widehat\{\\Sigma\}\_\{0\}:=\\frac\{\(Z\_\{\\setminus j\}^\{\(0\)\}\)^\{\\top\}Z\_\{\\setminus j\}^\{\(0\)\}\}\{n\_\{0\}\},and letΣA:=𝔼​Σ^A\\Sigma\_\{A\}:=\\mathbb\{E\}\\widehat\{\\Sigma\}\_\{A\}andΣ0:=𝔼​Σ^0\\Sigma\_\{0\}:=\\mathbb\{E\}\\widehat\{\\Sigma\}\_\{0\}\. For radiirA,r0,rδ\>0r\_\{A\},r\_\{0\},r\_\{\\delta\}\>0, define the local sets

𝒰A​\(rA\):=\{ϑ:‖ϑ−wA,∖j∗‖1≤rA\},𝒰0​\(r0\):=\{ϑ:‖ϑ−θ∖j∗‖1≤r0\}\.\\mathcal\{U\}\_\{A\}\(r\_\{A\}\):=\\\{\\vartheta:\\\|\\vartheta\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\leq r\_\{A\}\\\},\\qquad\\mathcal\{U\}\_\{0\}\(r\_\{0\}\):=\\\{\\vartheta:\\\|\\vartheta\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq r\_\{0\}\\\}\.Let𝒢j\\mathcal\{G\}\_\{j\}denote the event on which the two empirical Gram inequalities in \([19](https://arxiv.org/html/2607.03005#A3.E19)\) below hold\.

###### Assumption 6\(Primitive bounded\-design curvature inputs\)\.

There exist constantsMA,M0<∞M\_\{A\},M\_\{0\}<\\inftyandγA,γ0\>0\\gamma\_\{A\},\\gamma\_\{0\}\>0such that:

1. 1\.\(Local bounded fields\)For everyϑ∈𝒰A​\(rA\)\\vartheta\\in\\mathcal\{U\}\_\{A\}\(r\_\{A\}\)and everyx∈\{−1,1\}px\\in\\\{\-1,1\\\}^\{p\}, \|2​xj​∑k≠jϑk​xk\|≤MA\.\\left\|2x\_\{j\}\\sum\_\{k\\neq j\}\\vartheta\_\{k\}x\_\{k\}\\right\|\\leq M\_\{A\}\.For everyϑ∈𝒰0​\(r0\)\\vartheta\\in\\mathcal\{U\}\_\{0\}\(r\_\{0\}\)and everyx∈\{−1,1\}px\\in\\\{\-1,1\\\}^\{p\}, \|2​xj​∑k≠jϑk​xk\|≤M0\.\\left\|2x\_\{j\}\\sum\_\{k\\neq j\}\\vartheta\_\{k\}x\_\{k\}\\right\|\\leq M\_\{0\}\.
2. 2\.\(Population restricted eigenvalues\)For everyΔ∈𝒞A\\Delta\\in\\mathcal\{C\}\_\{A\}, Δ⊤​ΣA​Δ≥γA​‖Δ‖22\.\\Delta^\{\\top\}\\Sigma\_\{A\}\\Delta\\geq\\gamma\_\{A\}\\\|\\Delta\\\|\_\{2\}^\{2\}\.For every index setT⊆V∖\{j\}T\\subseteq V\\setminus\\\{j\\\}with\|T\|≤tj\|T\|\\leq t\_\{j\}and everyΔ∈𝒞T,h\\Delta\\in\\mathcal\{C\}\_\{T,h\}, Δ⊤​Σ0​Δ≥γ0​‖Δ‖22\.\\Delta^\{\\top\}\\Sigma\_\{0\}\\Delta\\geq\\gamma\_\{0\}\\\|\\Delta\\\|\_\{2\}^\{2\}\.
3. 3\.\(Sparse signed\-design tails\)Put mA:=⌊cm​Nlog⁡p⌋,m0:=⌊cm​n0log⁡p⌋m\_\{A\}:=\\left\\lfloor c\_\{m\}\\frac\{N\}\{\\log p\}\\right\\rfloor,\\qquad m\_\{0\}:=\\left\\lfloor c\_\{m\}\\frac\{n\_\{0\}\}\{\\log p\}\\right\\rfloorfor an absolute constantcm\>0c\_\{m\}\>0chosen such that the support union bound in the proof of Proposition[1](https://arxiv.org/html/2607.03005#Thmproposition1)is bounded byC​p−cCp^\{\-c\}\. There exists a constantK<∞K<\\inftysuch that, for everyr∈\{0\}∪𝒜r\\in\\\{0\\\}\\cup\\mathcal\{A\}and every vectorvvwith‖v‖2=1\\\|v\\\|\_\{2\}=1and‖v‖0≤2​\(mA∨m0\)\\\|v\\\|\_\{0\}\\leq 2\(m\_\{A\}\\vee m\_\{0\}\), ‖v⊤​zi\(r\)‖ψ2≤K,\\\|v^\{\\top\}z\_\{i\}^\{\(r\)\}\\\|\_\{\\psi\_\{2\}\}\\leq K,where∥⋅∥ψ2\\\|\\cdot\\\|\_\{\\psi\_\{2\}\}denotes the sub\-Gaussian Orlicz norm\.
4. 4\.\(Sample\-size scaling\) mA→∞,m0→∞,log⁡pN→0,log⁡pn0→0\.m\_\{A\}\\to\\infty,\\qquad m\_\{0\}\\to\\infty,\\qquad\\frac\{\\log p\}\{N\}\\to 0,\\qquad\\frac\{\\log p\}\{n\_\{0\}\}\\to 0\.

###### Condition 6\(Sufficient local estimator and LLA inputs\)\.

1. 1\.\(Local estimator containment\)On𝒢j\\mathcal\{G\}\_\{j\}, the Step 1 solution satisfies ‖w^∖jA−wA,∖j∗‖1≤rA\.\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\leq r\_\{A\}\.
2. 2\.\(LLA local\-basin condition\)LetT=\{k≠j:\|δj​k∗\|\>λδ\}T=\\\{k\\neq j:\|\\delta^\{\\ast\}\_\{jk\}\|\>\\lambda\_\{\\delta\}\\\}and ℬδ,j:=\{δ:‖δ−δ∖j∗‖1≤rδ,‖\(δ−δ∖j∗\)Tc‖1≤3​‖\(δ−δ∖j∗\)T‖1\+3​Ch​hj\}\.\\mathcal\{B\}\_\{\\delta,j\}:=\\left\\\{\\delta:\\\|\\delta\-\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq r\_\{\\delta\},\\ \\\|\(\\delta\-\\delta^\{\\ast\}\_\{\\setminus j\}\)\_\{T^\{c\}\}\\\|\_\{1\}\\leq 3\\\|\(\\delta\-\\delta^\{\\ast\}\_\{\\setminus j\}\)\_\{T\}\\\|\_\{1\}\+3C\_\{h\}h\_\{j\}\\right\\\}\.The Step 2 LLA routine is initialized inℬδ,j\\mathcal\{B\}\_\{\\delta,j\}, all surrogate problems are solved to coordinate KKT toleranceεn,j\\varepsilon\_\{n,j\}, the iterates remain inℬδ,j\\mathcal\{B\}\_\{\\delta,j\}, and the returned pointδ^∖jA\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}satisfies the original Step 2 KKT residual bound dist∞​\(0,∂Q​\(δ^∖jA\)\)≤εn,j\.\\mathrm\{dist\}\_\{\\infty\}\\bigl\(0,\\partial Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\\bigr\)\\leq\\varepsilon\_\{n,j\}\.In addition, the returned point satisfies Q​\(δ^∖jA\)≤Q​\(δ\)for all​δ​such that​‖δ−δ∖j∗‖1≤rδ\.Q\(\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\)\\leq Q\(\\delta\)\\qquad\\text\{for all \}\\delta\\text\{ such that \}\\\|\\delta\-\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq r\_\{\\delta\}\.In addition, ‖w^∖jA\+δ∖j∗−θ∖j∗‖1≤r0,‖w^∖jA\+δ^∖jA−θ∖j∗‖1≤r0\.\\big\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\delta^\{\\ast\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\big\\\|\_\{1\}\\leq r\_\{0\},\\qquad\\big\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\big\\\|\_\{1\}\\leq r\_\{0\}\.

###### Proposition 1\(Verification of empirical curvature inputs\)\.

Suppose Assumptions[1](https://arxiv.org/html/2607.03005#Thmassumption1),[2](https://arxiv.org/html/2607.03005#Thmassumption2),[3](https://arxiv.org/html/2607.03005#Thmassumption3),[4](https://arxiv.org/html/2607.03005#Thmassumption4), and[6](https://arxiv.org/html/2607.03005#Thmassumption6)and Condition6\(i\) hold\. Then Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1)and[2](https://arxiv.org/html/2607.03005#Thmcondition2)hold with probability at least1−C​p−c1\-Cp^\{\-c\}for some constantsC,c\>0C,c\>0\. On this event, Condition6\(ii\) implies Conditions[3](https://arxiv.org/html/2607.03005#Thmcondition3)and[4](https://arxiv.org/html/2607.03005#Thmcondition4)\.

###### Proof\.

We first verify the empirical Gram bounds\. Forq=Aq=A, setΣ^q=Σ^A\\widehat\{\\Sigma\}\_\{q\}=\\widehat\{\\Sigma\}\_\{A\},Σq=ΣA\\Sigma\_\{q\}=\\Sigma\_\{A\},nq=Nn\_\{q\}=N,mq=mAm\_\{q\}=m\_\{A\}, andγq=γA\\gamma\_\{q\}=\\gamma\_\{A\}\. Forq=0q=0, setΣ^q=Σ^0\\widehat\{\\Sigma\}\_\{q\}=\\widehat\{\\Sigma\}\_\{0\},Σq=Σ0\\Sigma\_\{q\}=\\Sigma\_\{0\},nq=n0n\_\{q\}=n\_\{0\},mq=m0m\_\{q\}=m\_\{0\}, andγq=γ0\\gamma\_\{q\}=\\gamma\_\{0\}\. For a fixedq∈\{A,0\}q\\in\\\{A,0\\\}and a fixed supportBBwith\|B\|≤2​mq\|B\|\\leq 2m\_\{q\}, let𝕊B\\mathbb\{S\}\_\{B\}be the unit sphere onBBand let𝒩B\\mathcal\{N\}\_\{B\}be a1/41/4\-net of𝕊B\\mathbb\{S\}\_\{B\}with\|𝒩B\|≤9\|B\|\|\\mathcal\{N\}\_\{B\}\|\\leq 9^\{\|B\|\}\. For anyu∈𝒩Bu\\in\\mathcal\{N\}\_\{B\}, the variables\(u⊤​zi\(r\)\)2−𝔼​\{\(u⊤​zi\(r\)\)2\}\(u^\{\\top\}z\_\{i\}^\{\(r\)\}\)^\{2\}\-\\mathbb\{E\}\\\{\(u^\{\\top\}z\_\{i\}^\{\(r\)\}\)^\{2\}\\\}are independent and sub\-exponential with norm bounded by a constant that depends only onKK\. Forq=Aq=Athese variables are summed overr∈\{0\}∪𝒜r\\in\\\{0\\\}\\cup\\mathcal\{A\}with weight1/N1/N, and forq=0q=0they are summed over the target sample with weight1/n01/n\_\{0\}\. Bernstein’s inequality gives

Pr⁡\(\|u⊤​\(Σ^q−Σq\)​u\|\>γq/16\)≤2​exp⁡\(−c​nq\)\\Pr\\left\(\\left\|u^\{\\top\}\(\\widehat\{\\Sigma\}\_\{q\}\-\\Sigma\_\{q\}\)u\\right\|\>\\gamma\_\{q\}/16\\right\)\\leq 2\\exp\(\-cn\_\{q\}\)for a constantc\>0c\>0that depends only onKKandγq\\gamma\_\{q\}\. A union bound over the nets and over all supports with\|B\|≤2​mq\|B\|\\leq 2m\_\{q\}, together withmq≤cm​nq/log⁡pm\_\{q\}\\leq c\_\{m\}n\_\{q\}/\\log pand a sufficiently smallcmc\_\{m\}, gives, with probability at least1−C​p−c1\-Cp^\{\-c\},

sup‖u‖2=1,‖u‖0≤2​mq\|u⊤​\(Σ^q−Σq\)​u\|≤γq/4\\sup\_\{\\\|u\\\|\_\{2\}=1,\\ \\\|u\\\|\_\{0\}\\leq 2m\_\{q\}\}\\left\|u^\{\\top\}\(\\widehat\{\\Sigma\}\_\{q\}\-\\Sigma\_\{q\}\)u\\right\|\\leq\\gamma\_\{q\}/4simultaneously forq=Aq=Aandq=0q=0\. To pass from sparse vectors to all vectors, fixΔ\\Deltaand order its coordinates by decreasing absolute value\. LetS0,S1,…S\_\{0\},S\_\{1\},\\ldotsbe consecutive blocks of sizemqm\_\{q\}\. The sparse bound above and polarization imply

\|a⊤​\(Σ^q−Σq\)​b\|≤C​γq​‖a‖2​‖b‖2\\left\|a^\{\\top\}\(\\widehat\{\\Sigma\}\_\{q\}\-\\Sigma\_\{q\}\)b\\right\|\\leq C\\gamma\_\{q\}\\\|a\\\|\_\{2\}\\\|b\\\|\_\{2\}whenever\|supp⁡\(a\)∪supp⁡\(b\)\|≤2​mq\|\\operatorname\{supp\}\(a\)\\cup\\operatorname\{supp\}\(b\)\|\\leq 2m\_\{q\}\. Because

∑ℓ≥1‖ΔSℓ‖2≤mq−1/2​‖Δ‖1,\\sum\_\{\\ell\\geq 1\}\\\|\\Delta\_\{S\_\{\\ell\}\}\\\|\_\{2\}\\leq m\_\{q\}^\{\-1/2\}\\\|\\Delta\\\|\_\{1\},expandingΔ⊤​\(Σ^q−Σq\)​Δ\\Delta^\{\\top\}\(\\widehat\{\\Sigma\}\_\{q\}\-\\Sigma\_\{q\}\)\\Deltaover the blocks gives

\|Δ⊤​\(Σ^q−Σq\)​Δ\|≤γq2​‖Δ‖22\+C​log⁡pnq​‖Δ‖12\.\\left\|\\Delta^\{\\top\}\(\\widehat\{\\Sigma\}\_\{q\}\-\\Sigma\_\{q\}\)\\Delta\\right\|\\leq\\frac\{\\gamma\_\{q\}\}\{2\}\\\|\\Delta\\\|\_\{2\}^\{2\}\+C\\frac\{\\log p\}\{n\_\{q\}\}\\\|\\Delta\\\|\_\{1\}^\{2\}\.Combining this deviation bound with Assumption[6](https://arxiv.org/html/2607.03005#Thmassumption6)\(ii\) gives constantsC,c\>0C,c\>0such that, with probability at least1−C​p−c1\-Cp^\{\-c\},

Δ⊤​Σ^A​Δ\\displaystyle\\Delta^\{\\top\}\\widehat\{\\Sigma\}\_\{A\}\\Delta≥γA2​‖Δ‖22−C​log⁡pN​‖Δ‖12for all​Δ∈𝒞A,\\displaystyle\\geq\\frac\{\\gamma\_\{A\}\}\{2\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-C\\frac\{\\log p\}\{N\}\\\|\\Delta\\\|\_\{1\}^\{2\}\\qquad\\text\{for all \}\\Delta\\in\\mathcal\{C\}\_\{A\},\(19\)Δ⊤​Σ^0​Δ\\displaystyle\\Delta^\{\\top\}\\widehat\{\\Sigma\}\_\{0\}\\Delta≥γ02​‖Δ‖22−C​log⁡pn0​‖Δ‖12\\displaystyle\\geq\\frac\{\\gamma\_\{0\}\}\{2\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-C\\frac\{\\log p\}\{n\_\{0\}\}\\\|\\Delta\\\|\_\{1\}^\{2\}for allΔ∈𝒞T,h\\Delta\\in\\mathcal\{C\}\_\{T,h\}and allT⊆V∖\{j\}T\\subseteq V\\setminus\\\{j\\\}satisfying\|T\|≤tj\|T\|\\leq t\_\{j\}\.

For everyϑ∈𝒰A​\(rA\)\\vartheta\\in\\mathcal\{U\}\_\{A\}\(r\_\{A\}\), Assumption[6](https://arxiv.org/html/2607.03005#Thmassumption6)\(i\) gives

f′′​\{\(zi\(r\)\)⊤​ϑ\}≥ρA:=inf\|u\|≤MAf′′​\(u\)\>0\.f^\{\\prime\\prime\}\\\{\(z\_\{i\}^\{\(r\)\}\)^\{\\top\}\\vartheta\\\}\\geq\\rho\_\{A\}:=\\inf\_\{\|u\|\\leq M\_\{A\}\}f^\{\\prime\\prime\}\(u\)\>0\.This and \([19](https://arxiv.org/html/2607.03005#A3.E19)\) imply

Δ⊤​∇2ℓA,j​\(ϑ\)​Δ≥ρA​Δ⊤​Σ^A​Δ≥ρA​γA2​‖Δ‖22−C​ρA​log⁡pN​‖Δ‖12\\Delta^\{\\top\}\\nabla^\{2\}\\ell\_\{A,j\}\(\\vartheta\)\\Delta\\geq\\rho\_\{A\}\\Delta^\{\\top\}\\widehat\{\\Sigma\}\_\{A\}\\Delta\\geq\\frac\{\\rho\_\{A\}\\gamma\_\{A\}\}\{2\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-C\\rho\_\{A\}\\frac\{\\log p\}\{N\}\\\|\\Delta\\\|\_\{1\}^\{2\}for allΔ∈𝒞A\\Delta\\in\\mathcal\{C\}\_\{A\}and allϑ∈𝒰A​\(rA\)\\vartheta\\in\\mathcal\{U\}\_\{A\}\(r\_\{A\}\)\. Together with Condition6\(i\), this verifies Condition[1](https://arxiv.org/html/2607.03005#Thmcondition1)withκA=ρA​γA/2\\kappa\_\{A\}=\\rho\_\{A\}\\gamma\_\{A\}/2andτA=C​ρA\\tau\_\{A\}=C\\rho\_\{A\}\. The same argument withρ0,loc:=inf\|u\|≤M0f′′​\(u\)\>0\\rho\_\{0,\\mathrm\{loc\}\}:=\\inf\_\{\|u\|\\leq M\_\{0\}\}f^\{\\prime\\prime\}\(u\)\>0verifies Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2)withκ0=ρ0,loc​γ0/2\\kappa\_\{0\}=\\rho\_\{0,\\mathrm\{loc\}\}\\gamma\_\{0\}/2andτ0=C​ρ0,loc\\tau\_\{0\}=C\\rho\_\{0,\\mathrm\{loc\}\}\.

It remains to relate Condition6\(ii\) to the Step 2 theorem\-local inputs\. Condition6\(ii\) gives the original\-objective coordinate KKT tolerance, the local minimality condition, and the target\-RSC neighborhood condition in Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\. Because the returned point belongs toℬδ,j\\mathcal\{B\}\_\{\\delta,j\}, the vectorv=δ^∖jA−δ∖j∗v=\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\delta^\{\\ast\}\_\{\\setminus j\}satisfies the cone bound in Condition[4](https://arxiv.org/html/2607.03005#Thmcondition4)\. This completes the proof\. ∎

For source detection, letW~\(0\),r\\widetilde\{W\}^\{\(0\),r\}denote the target\-only training\-fold estimator used in foldrrof Algorithm[2](https://arxiv.org/html/2607.03005#alg2), and letW~\(0\+s\),r\\widetilde\{W\}^\{\(0\+s\),r\}denote the corresponding target\-plus\-source estimator for sourcess\. Putnminval:=minr=1,2⁡n0,rn\_\{\\min\}^\{\\mathrm\{val\}\}:=\\min\_\{r=1,2\}n\_\{0,r\}\.

###### Condition 7\(Sufficient validation inputs\)\.

There exist sequencesbn≥0b\_\{n\}\\geq 0,σn\>0\\sigma\_\{n\}\>0, andϵn\\epsilon\_\{n\}such that

ϵn=bn\+p​log⁡\(S∨p\)nminval,ϵn=o​\(νn\)\.\\epsilon\_\{n\}=b\_\{n\}\+p\\sqrt\{\\frac\{\\log\(S\\vee p\)\}\{n\_\{\\min\}^\{\\mathrm\{val\}\}\}\},\\qquad\\epsilon\_\{n\}=o\(\\nu\_\{n\}\)\.With probability at least1−C​p−c1\-Cp^\{\-c\}, the following conditions hold:

1. 1\.\(Uniform bounded fitted fields\)For every nodejj, every foldrr, and every fitted matrix Θ∈\{W~\(0\),r\}∪\{W~\(0\+s\),r:1≤s≤S\},\\Theta\\in\\\{\\widetilde\{W\}^\{\(0\),r\}\\\}\\cup\\\{\\widetilde\{W\}^\{\(0\+s\),r\}:1\\leq s\\leq S\\\},the nodewise signed linear predictor satisfies supx∈\{−1,1\}p\|2​xj​∑k≠jΘj​k​xk\|≤Bv\\sup\_\{x\\in\\\{\-1,1\\\}^\{p\}\}\\left\|2x\_\{j\}\\sum\_\{k\\neq j\}\\Theta\_\{jk\}x\_\{k\}\\right\|\\leq B\_\{v\}for a finite constantBvB\_\{v\}\.
2. 2\.\(Training\-estimator risk stability\) \|12​∑r=12ℒ0pseudo​\(W~\(0\),r\)−ℒ0pseudo​\(θ∗\)\|≤bn\\left\|\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\widetilde\{W\}^\{\(0\),r\}\)\-\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\theta^\{\\ast\}\)\\right\|\\leq b\_\{n\}and max1≤s≤S⁡\|12​∑r=12ℒ0pseudo​\(W~\(0\+s\),r\)−12​∑r=12ℒ0pseudo​\(W\(0\+s\),∗,r\)\|≤bn\.\\max\_\{1\\leq s\\leq S\}\\left\|\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\widetilde\{W\}^\{\(0\+s\),r\}\)\-\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(W^\{\(0\+s\),\\ast,r\}\)\\right\|\\leq b\_\{n\}\.
3. 3\.\(Population separation\)For everys∈𝒜hs\\in\\mathcal\{A\}\_\{h\},ℰ​\(s\)≤cgap​νn\\mathcal\{E\}\(s\)\\leq c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}, and for everys∉𝒜hs\\notin\\mathcal\{A\}\_\{h\},ℰ​\(s\)≥\(1\+cgap\)​νn\\mathcal\{E\}\(s\)\\geq\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\.
4. 4\.\(Threshold stability\) \|σ^−σn\|≤bn\|\\hat\{\\sigma\}\-\\sigma\_\{n\}\|\\leq b\_\{n\}and, for a sufficiently large constantCΔ\>0C\_\{\\Delta\}\>0, cgap​νn\+CΔ​ϵn<Cτ​\(σn−bn\)≤Cτ​\(σn\+bn\)<\(1\+cgap\)​νn−CΔ​ϵn\.c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\+C\_\{\\Delta\}\\epsilon\_\{n\}<C\_\{\\tau\}\(\\sigma\_\{n\}\-b\_\{n\}\)\\leq C\_\{\\tau\}\(\\sigma\_\{n\}\+b\_\{n\}\)<\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\-C\_\{\\Delta\}\\epsilon\_\{n\}\.

###### Proposition 2\(Verification of source\-detection inputs\)\.

Suppose Assumption[1](https://arxiv.org/html/2607.03005#Thmassumption1)and Condition[7](https://arxiv.org/html/2607.03005#Thmcondition7)hold, and the validation folds are independent of the fitted training\-fold estimators conditional on the training folds\. Then Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)holds with probability tending to one\.

###### Proof\.

For each foldr∈\{1,2\}r\\in\\\{1,2\\\}, condition on the training sampleX\(0\)∖X\(0\)​\[r\]X^\{\(0\)\}\\setminus X^\{\(0\)\[r\]\}, all source samples, and the fitted matrices\{W~\(0\),r\}∪\{W~\(0\+s\),r:1≤s≤S\}\.\\\{\\widetilde\{W\}^\{\(0\),r\}\\\}\\cup\\\{\\widetilde\{W\}^\{\(0\+s\),r\}:1\\leq s\\leq S\\\}\.The validation observations inX\(0\)​\[r\]X^\{\(0\)\[r\]\}are independent of these fitted matrices\. For any fitted matrixΘ\\Thetain Condition[7](https://arxiv.org/html/2607.03005#Thmcondition7)\(i\), the validation pseudolikelihood contribution∑j=1pf​\(2​Xi​j\(0\)​∑k≠jΘj​k​Xi​k\(0\)\)\\sum\_\{j=1\}^\{p\}f\\left\(2X\_\{ij\}^\{\(0\)\}\\sum\_\{k\\neq j\}\\Theta\_\{jk\}X\_\{ik\}^\{\(0\)\}\\right\)is bounded byp​log⁡\(1\+exp⁡\(Bv\)\)p\\log\(1\+\\exp\(B\_\{v\}\)\)\. Hoeffding’s inequality applied conditionally for each fixed fold and fitted matrix, followed by a union bound over the two folds and theS\+1S\+1fitted losses in each fold, implies that, with conditional probability at least1−C​\(S\+1\)​\(S∨p\)−c1\-C\(S\+1\)\(S\\vee p\)^\{\-c\},

max1≤s≤S⁡\|Δs−\{12​∑r=12ℒ0pseudo​\(W~\(0\+s\),r\)−12​∑r=12ℒ0pseudo​\(W~\(0\),r\)\}\|≤C​p​log⁡\(S∨p\)nminval\.\\max\_\{1\\leq s\\leq S\}\\left\|\\Delta\_\{s\}\-\\left\\\{\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\widetilde\{W\}^\{\(0\+s\),r\}\)\-\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\mathcal\{L\}^\{\\mathrm\{pseudo\}\}\_\{0\}\(\\widetilde\{W\}^\{\(0\),r\}\)\\right\\\}\\right\|\\leq Cp\\sqrt\{\\frac\{\\log\(S\\vee p\)\}\{n\_\{\\min\}^\{\\mathrm\{val\}\}\}\}\.Combining this inequality with Condition[7](https://arxiv.org/html/2607.03005#Thmcondition7)\(ii\) gives

max1≤s≤S⁡\|Δs−ℰ​\(s\)\|≤CΔ​ϵn\\max\_\{1\\leq s\\leq S\}\|\\Delta\_\{s\}\-\\mathcal\{E\}\(s\)\|\\leq C\_\{\\Delta\}\\epsilon\_\{n\}for a sufficiently large constantCΔC\_\{\\Delta\}\. Condition[7](https://arxiv.org/html/2607.03005#Thmcondition7)\(iii\) gives the informative\-source and non\-informative\-source parts of Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)\. Condition[7](https://arxiv.org/html/2607.03005#Thmcondition7)\(iv\) gives

cgap​νn\+CΔ​ϵn<Cτ​σ^<\(1\+cgap\)​νn−CΔ​ϵn\.c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\+C\_\{\\Delta\}\\epsilon\_\{n\}<C\_\{\\tau\}\\hat\{\\sigma\}<\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\-C\_\{\\Delta\}\\epsilon\_\{n\}\.Combining the event in Condition[7](https://arxiv.org/html/2607.03005#Thmcondition7)with the validation concentration event gives probability at least1−C​p−c−C​\(S\+1\)​\(S∨p\)−c\.1\-Cp^\{\-c\}\-C\(S\+1\)\(S\\vee p\)^\{\-c\}\.After increasing the numerical constant in the deviation bound, the exponentcccan be chosen larger than11, and this probability tends to one\. Becauseϵn=o​\(νn\)\\epsilon\_\{n\}=o\(\\nu\_\{n\}\), the four parts of Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)hold with probability tending to one\. This completes the proof\. ∎

Assumptions[2](https://arxiv.org/html/2607.03005#Thmassumption2),[3](https://arxiv.org/html/2607.03005#Thmassumption3),[4](https://arxiv.org/html/2607.03005#Thmassumption4), and[5](https://arxiv.org/html/2607.03005#Thmassumption5), together with Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1),[2](https://arxiv.org/html/2607.03005#Thmcondition2),[3](https://arxiv.org/html/2607.03005#Thmcondition3),[4](https://arxiv.org/html/2607.03005#Thmcondition4), and[5](https://arxiv.org/html/2607.03005#Thmcondition5), combine standard conditions for high\-dimensional Ising structure learning and folded\-concave regularization with specific conditions required to control cross\-domain transfer\. Below we interpret these assumptions and conditions, discuss their validity, and clarify their roles in our theoretical analysis\.

#### Assumption[2](https://arxiv.org/html/2607.03005#Thmassumption2)\(sparsity and source\-to\-target proximity\)\.

The sparsity scalingsj=o​\(n0/log⁡p\)s\_\{j\}=o\(n\_\{0\}/\\log p\)is the usual regime in which nodewise logistic lasso can consistently estimate neighborhoods in Ising models\(Ravikumar et al\.,[2010](https://arxiv.org/html/2607.03005#bib.bib38); Negahban et al\.,[2012](https://arxiv.org/html/2607.03005#bib.bib33)\)\. The proximity levelhjh\_\{j\}quantifies how close an informative source is to the target inℓ1\\ell\_\{1\}\. When sources are generated by perturbations of the target interactions of the order used in our simulations,hjh\_\{j\}has the corresponding order and transfer can reduce the target estimation error\. Whenhjh\_\{j\}is larger, naive pooling can incur negative transfer\. This assumption is imposed only for sources in the informative set𝒜\\mathcal\{A\}; the full Trans\-Ising algorithm attempts to identify such sources via loss\-based screening\.

#### Assumption[3](https://arxiv.org/html/2607.03005#Thmassumption3)\(information comparability across sources\)\.

This condition requires the local curvature \(Hessian\) of each informative source risk to be comparable to the pooled risk curvature within a neighborhood ofθ∖j∗\\theta^\{\\ast\}\_\{\\setminus j\}\. This condition excludes sources whose conditional distributions are nearly deterministic or whose local Fisher information degenerates relative to the pooled task\. In bounded\-degree Ising models with parameters lying in a common bounded neighborhood, such local comparability is typically mild and ensures that the pooled population minimizerwA,∖j∗w^\{\\ast\}\_\{A,\\setminus j\}stays close toθ∖j∗\\theta^\{\\ast\}\_\{\\setminus j\}\(Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)\)\.

#### Condition[1](https://arxiv.org/html/2607.03005#Thmcondition1)\(pooled RSC on a shifted cone\)\.

Restricted strong convexity \(RSC\) is the standard curvature condition used in high\-dimensional M\-estimation rates\. The cone here is*shifted*becausewA,∖j∗w^\{\\ast\}\_\{A,\\setminus j\}is not assumed to be exactly supported onSjS\_\{j\}; instead, its off\-support mass‖\(wA,∖j∗\)Sjc‖1\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\\|\_\{1\}acts as an approximation error\. Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)implies this mass is controlled byhjh\_\{j\}\. The shift is of orderhjh\_\{j\}when informative sources are close to the target\.

#### Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2)\(target RSC for bias correction and effective sparsity\)\.

Step 2 estimates a correctionδ\\deltausing only target data, and its analysis requires target curvature\. The index budgettj=⌈Ch​hj/λδ⌉t\_\{j\}=\\lceil C\_\{h\}h\_\{j\}/\\lambda\_\{\\delta\}\\rceilbounds the*effective sparsity*ofδ∖j∗\\delta^\{\\ast\}\_\{\\setminus j\}: since we only assume‖δ∖j∗‖1≲hj\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim h\_\{j\}, the set of coordinates larger thanλδ\\lambda\_\{\\delta\}can have size at most on the order ofhj/λδh\_\{j\}/\\lambda\_\{\\delta\}\. This means that the RSC condition is only needed on cones associated with setsTTof size≤tj\\leq t\_\{j\}, which is weaker whenhj/λδh\_\{j\}/\\lambda\_\{\\delta\}is of smaller order thansjs\_\{j\}\. Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)uses the lower\-curvature constantκ0\>0\\kappa\_\{0\}\>0together with the tolerance term in Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2)\. A stronger bound such asκ0\>1/\(a−1\)\\kappa\_\{0\}\>1/\(a\-1\)is sufficient for analyses that combine the SCAD concavity term with the target curvature, but it is not needed for Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)\.

#### Assumption[4](https://arxiv.org/html/2607.03005#Thmassumption4)\(bounded nodewise fields and non\-degenerate curvature\)\.

Assumption[4](https://arxiv.org/html/2607.03005#Thmassumption4)imposes a uniform bound on the nodewise linear predictors \(equivalently, a row\-wiseℓ1\\ell\_\{1\}bound such as2​‖θj,∖j\(r\)‖1≤M2\\\|\\theta^\{\(r\)\}\_\{j,\\setminus j\}\\\|\_\{1\}\\leq Mfor alljjandrr\)\. This condition holds, for example, under bounded degree together with uniformly bounded edge weights, but it is more general and directly controls the range of the logistic argument\. It prevents near\-separation and ensures that the logistic curvaturef′′​\(⋅\)f^\{\\prime\\prime\}\(\\cdot\)is bounded away from zero on the relevant neighborhood, supporting local RSC and concentration arguments\(Ravikumar et al\.,[2010](https://arxiv.org/html/2607.03005#bib.bib38); Negahban et al\.,[2012](https://arxiv.org/html/2607.03005#bib.bib33)\)\.

#### Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\(computable stationary local solution in Step 2\)\.

Because Step 2 is nonconvex and nonsmooth, global optimality is generally intractable\. We therefore analyze a practically computable solution returned by an optimization routine \(e\.g\., LLA\) that achieves approximate stationarity \(KKT residual control\) and lies in a local basin aroundδ∖j∗\\delta^\{\\ast\}\_\{\\setminus j\}\. This is a common way to obtain theoretical guarantees for folded\-concave penalties: the statistical analysis proceeds on a high\-probability event where such a local solution exists and is reached from a reasonable initialization \(e\.g\., initial estimator atδ=0\\delta=0\)\.

#### Assumption[5](https://arxiv.org/html/2607.03005#Thmassumption5)\(tuning and beta\-min for exact selection\)\.

The tuning sequencesλw≍log⁡p/N\\lambda\_\{w\}\\asymp\\sqrt\{\\log p/N\}andλδ,λ≍log⁡p/n0\\lambda\_\{\\delta\},\\lambda\\asymp\\sqrt\{\\log p/n\_\{0\}\}match the usual stochastic fluctuation scales in the pooled and target\-only objectives\. The separationcλ\>cδc\_\{\\lambda\}\>c\_\{\\delta\}is needed to ensure that, on null coordinates, the SCAD subgradient in its linear region can dominate theℓ1\\ell\_\{1\}correction penalty and the optimization tolerance, enabling false\-positive control in Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)\. Finally, the beta\-min condition ensures true nonzero interactions are large enough to survive the combined estimation error and the SCAD transition region \(in particular, exceeding the SCAD thresholda​λa\\lambda\), which is the standard condition for exact support recovery\.

#### Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)\(separability for consistent source detection\)\.

Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)ensures that the “signal” distinguishing informative sources from non\-informative ones dominates the “noise” induced by finite\-sample stochastic fluctuations\. Specifically, it requires that the gap in population excess risk between the true source set𝒜h\\mathcal\{A\}\_\{h\}and heterogeneous sources is sufficiently large relative to the concentration rate of the empirical pseudolikelihood\. This condition guarantees that the cross\-validation based screening rule can consistently identify𝒜h\\mathcal\{A\}\_\{h\}with high probability\.

## Appendix S\.4Lemmas for Section[4](https://arxiv.org/html/2607.03005#S4)

###### Lemma 1\(Implication for pooled\-to\-target bias\)\.

Under Assumptions[2](https://arxiv.org/html/2607.03005#Thmassumption2)and[3](https://arxiv.org/html/2607.03005#Thmassumption3), there exists a constantC\>0C\>0such that

‖δ∖j∗‖1=‖θ∖j∗−wA,∖j∗‖1≤C​hj\.\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}=\\\|\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\ \\leq\\ C\\,h\_\{j\}\.

Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)bounds the*population*pooled\-to\-target discrepancy by‖δ∖j∗‖1≲hj\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim h\_\{j\}\. That is, the pooled population optimum cannot drift far from the target when the informative sources areℓ1\\ell\_\{1\}\-close\. Whenhjh\_\{j\}is large, this bound becomes loose, which corresponds to the negative\-transfer regime\.

###### Lemma 2\(Uniform max\-norm bound for logistic Hessians\)\.

For the signed\-design logistic lossℓj\(r\)​\(u\)=1nr​∑i=1nrf​\(zi\(r\)⊤​u\)\\ell^\{\(r\)\}\_\{j\}\(u\)=\\frac\{1\}\{n\_\{r\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}f\(z^\{\(r\)\\top\}\_\{i\}u\)withzi,ℓ\(r\)∈\{−2,2\}z^\{\(r\)\}\_\{i,\\ell\}\\in\\\{\-2,2\\\}, we have for allu∈ℝp−1u\\in\\mathbb\{R\}^\{p\-1\},‖∇2ℓj\(r\)​\(u\)‖max≤1\.\\\|\\nabla^\{2\}\\ell^\{\(r\)\}\_\{j\}\(u\)\\\|\_\{\\max\}\\leq 1\.

This implies that, for allu,v∈ℝp−1u,v\\in\\mathbb\{R\}^\{p\-1\},

‖∇ℓj\(r\)​\(u\)−∇ℓj\(r\)​\(v\)‖∞≤‖u−v‖1\.\\\|\\nabla\\ell^\{\(r\)\}\_\{j\}\(u\)\-\\nabla\\ell^\{\(r\)\}\_\{j\}\(v\)\\\|\_\{\\infty\}\\leq\\\|u\-v\\\|\_\{1\}\.The same bounds hold forℓ0,j\\ell\_\{0,j\},ℓA,j\\ell\_\{A,j\}and the population risksℒr,j\\mathcal\{L\}\_\{r,j\}andℒA,j\\mathcal\{L\}\_\{A,j\}\.

###### Lemma 3\(Score concentration\)\.

Under Assumptions[1](https://arxiv.org/html/2607.03005#Thmassumption1)and[4](https://arxiv.org/html/2607.03005#Thmassumption4), there exist constantscA,c0,c\>0c\_\{A\},c\_\{0\},c\>0such that, with probability at least1−p−c1\-p^\{\-c\},

‖∇ℓA,j​\(wA,∖j∗\)‖∞≤cA​log⁡pN,‖∇ℓ0,j​\(θ∖j∗\)‖∞≤c0​log⁡pn0\.\\displaystyle\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\\|\_\{\\infty\}\\ \\leq\\ c\_\{A\}\\sqrt\{\\frac\{\\log p\}\{N\}\},\\qquad\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\\ \\leq\\ c\_\{0\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}\.

###### Lemma 4\(Step 1 error\)\.

Under Assumptions[2](https://arxiv.org/html/2607.03005#Thmassumption2),[3](https://arxiv.org/html/2607.03005#Thmassumption3),[4](https://arxiv.org/html/2607.03005#Thmassumption4), Condition[1](https://arxiv.org/html/2607.03005#Thmcondition1), and Lemma[3](https://arxiv.org/html/2607.03005#Thmlemma3), withλw=Cw​\(log⁡p\)/N\\lambda\_\{w\}=C\_\{w\}\\sqrt\{\(\\log p\)/N\}for a sufficiently large absolute constantCw\>0C\_\{w\}\>0and

sj​log⁡pN\+log⁡pN​hj2=o​\(1\),s\_\{j\}\\frac\{\\log p\}\{N\}\+\\frac\{\\log p\}\{N\}h\_\{j\}^\{2\}=o\(1\),the Step 1 estimator satisfies, w\.h\.p\.,

‖w^∖jA−wA,∖j∗‖2≲sj​log⁡pN\+\(λw​‖\(wA,∖j∗\)Sjc‖1∧‖\(wA,∖j∗\)Sjc‖1\),\\displaystyle\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{2\}\\ \\lesssim\\ \\sqrt\{\\frac\{s\_\{j\}\\log p\}\{N\}\}\+\\Big\(\\sqrt\{\\lambda\_\{w\}\\,\\big\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\big\\\|\_\{1\}\}\\ \\wedge\\ \\big\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\big\\\|\_\{1\}\\Big\),‖w^∖jA−wA,∖j∗‖1≲sj​log⁡pN\+‖\(wA,∖j∗\)Sjc‖1\.\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\ \\lesssim\\ s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+\\big\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\big\\\|\_\{1\}\.In particular, Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)gives‖\(wA,∖j∗\)Sjc‖1≲hj\\big\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\big\\\|\_\{1\}\\lesssim h\_\{j\}, which implies

‖w^∖jA−wA,∖j∗‖2≲sj​log⁡pN\+\(\[\(log⁡pN\)1/4​hj1/2\]∧hj\),‖w^∖jA−wA,∖j∗‖1≲sj​log⁡pN\+hj\.\\displaystyle\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{2\}\\ \\lesssim\\ \\sqrt\{\\frac\{s\_\{j\}\\log p\}\{N\}\}\+\\Big\(\\big\[\(\\tfrac\{\\log p\}\{N\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\big\]\\ \\wedge\\ h\_\{j\}\\Big\),\\qquad\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\ \\lesssim\\ s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\.

Lemma[4](https://arxiv.org/html/2607.03005#Thmlemma4)yields a Step 1 error decomposition into a variance termsj​log⁡p/N\\sqrt\{s\_\{j\}\\log p/N\}that depends on the total informative sample sizeNN, and a heterogeneity/approximation term controlled by‖\(wA,∖j∗\)Sjc‖1≲hj\\\|\(w^\{\\ast\}\_\{A,\\setminus j\}\)\_\{S\_\{j\}^\{c\}\}\\\|\_\{1\}\\lesssim h\_\{j\}\(Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)\)\. This shows that pooling stabilizes the initializer whenN≫n0N\\gg n\_\{0\}andhjh\_\{j\}is of smaller order than the variance reduction\.

###### Lemma 5\(Step 2 error\)\.

Suppose Assumptions[1](https://arxiv.org/html/2607.03005#Thmassumption1),[2](https://arxiv.org/html/2607.03005#Thmassumption2),[3](https://arxiv.org/html/2607.03005#Thmassumption3), and[4](https://arxiv.org/html/2607.03005#Thmassumption4), and Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1),[2](https://arxiv.org/html/2607.03005#Thmcondition2),[3](https://arxiv.org/html/2607.03005#Thmcondition3), and[4](https://arxiv.org/html/2607.03005#Thmcondition4)hold\. Choose tuning levels

λδ=Cδ​log⁡pn0,λ=Cλ​log⁡pn0,\\lambda\_\{\\delta\}=C\_\{\\delta\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\},\\qquad\\lambda=C\_\{\\lambda\}\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\},whereλ\\lambdais the SCAD penalty level in \([18](https://arxiv.org/html/2607.03005#S4.E18)\) andCδ,Cλ\>0C\_\{\\delta\},C\_\{\\lambda\}\>0are sufficiently large absolute constants\. Assume in addition thatlog⁡p/n0=o​\(1\)\\log p/n\_\{0\}=o\(1\),sj​log⁡p/N\+hj≲λδs\_\{j\}\\sqrt\{\\log p/N\}\+h\_\{j\}\\lesssim\\lambda\_\{\\delta\}, and the Step 1 estimate satisfies, w\.h\.p\.,

‖w^∖jA−wA,∖j∗‖1≲λδ\.\\\|\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}\\ \\lesssim\\ \\lambda\_\{\\delta\}\.Letδ^∖jA\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}be the stationary local minimizer returned by Step 2 satisfying Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\. Then, with high probability,

‖δ^∖jA−δ∖j∗‖2≲\(\[\(log⁡pn0\)1/4​hj1/2\]∧hj\),‖δ^∖jA−δ∖j∗‖1≲hj\.\\\|\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{2\}\\ \\lesssim\\ \\Big\(\\big\[\(\\tfrac\{\\log p\}\{n\_\{0\}\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\big\]\\wedge h\_\{j\}\\Big\),\\ \\\|\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\ \\lesssim\\ h\_\{j\}\.

Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)shows that Step 2 estimates the cross\-domain correctionδ∖j∗\\delta^\{\\ast\}\_\{\\setminus j\}using only target data; its rates depend on\(n0,p\)\(n\_\{0\},p\)and the heterogeneity scalehjh\_\{j\}\. The neighborhood sparsitysjs\_\{j\}does not enter the Step 2 rate above\. Whenhjh\_\{j\}is not of the same order aslog⁡p/n0\\sqrt\{\\log p/n\_\{0\}\}or smaller, the conditionsj​log⁡p/N\+hj≲λδs\_\{j\}\\sqrt\{\\log p/N\}\+h\_\{j\}\\lesssim\\lambda\_\{\\delta\}can fail, and Step 2 no longer gives the rate in Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)\.

## Appendix S\.5Additional Details

### S\.5\.1SCAD penalty

We use the Smoothly Clipped Absolute Deviation \(SCAD\) penalty\(Fan & Li,[2001](https://arxiv.org/html/2607.03005#bib.bib18)\):

Pλ​\(θ\)=\{λ​\|θ\|,if​\|θ\|≤λ,−θ2\+2​a​λ​\|θ\|−λ22​\(a−1\),λ<\|θ\|≤a​λ,\(a\+1\)​λ22,if​\|θ\|\>a​λ,\(a\>2\)\.P\_\{\\lambda\}\(\\theta\)=\\begin\{cases\}\\lambda\|\\theta\|,&\\text\{if \}\|\\theta\|\\leq\\lambda,\\\\\[3\.00003pt\] \\dfrac\{\-\\theta^\{2\}\+2a\\lambda\|\\theta\|\-\\lambda^\{2\}\}\{2\(a\-1\)\},&\\lambda<\|\\theta\|\\leq a\\lambda,\\\\\[3\.00003pt\] \\dfrac\{\(a\+1\)\\lambda^\{2\}\}\{2\},&\\text\{if \}\|\\theta\|\>a\\lambda,\\end\{cases\}\\quad\(a\>2\)\.\(20\)

### S\.5\.2Remarks and Corollaries for Section[4](https://arxiv.org/html/2607.03005#S4)

###### Corollary 2\(Graph recovery after AND symmetrization\)\.

Suppose the conclusion of Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)holds uniformly for all nodesj=1,…,pj=1,\\dots,p\. LetE^\\hat\{E\}be the edge set obtained by the AND rule applied to the asymmetric nodewise estimates\. Thenℙ​\(E^=E\)→1\\mathbb\{P\}\(\\hat\{E\}=E\)\\to 1\.

### S\.5\.3Explicit conditions for Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)

For completeness, we restate the explicit sufficient conditions used for selection consistency\. Letrn,jr\_\{n,j\}denote the nodewise error bound in Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)\. In Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2), we require:

1. 1\.\(Small\-error regime for SCAD linear part\) the nodewise error boundrn,jr\_\{n,j\}in Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)satisfiesrn,j≤λ,r\_\{n,j\}\\leq\\lambda,which implies that, for anyk∈Sjck\\in S\_\{j\}^\{c\}, we have\|θ^j​k\|≤λ\|\\hat\{\\theta\}\_\{jk\}\|\\leq\\lambdaon the high\-probability event\.
2. 2\.\(Separation between SCAD and correction penalty\) there exists a sufficiently large universal constantCsel\>0C\_\{\\mathrm\{sel\}\}\>0such that λ−λδ−εn,j≥Csel​\(log⁡pn0\+sj​log⁡pN\+hj\)\.\\lambda\-\\lambda\_\{\\delta\}\-\\varepsilon\_\{n,j\}\\ \\geq\\ C\_\{\\mathrm\{sel\}\}\\left\(\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}\+s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\\right\)\.\(21\)
3. 3\.\(Beta\-min\) Assumption[5](https://arxiv.org/html/2607.03005#Thmassumption5)holds; that is, \|θj​k∗\|≥a​λ\+rn,jfor all​k∈Sj\.\|\\theta^\{\\ast\}\_\{jk\}\|\\geq a\\lambda\+r\_\{n,j\}\\qquad\\text\{for all \}k\\in S\_\{j\}\.

## Appendix S\.6Proofs

### S\.6\.1Proof of Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)

###### Proof\.

By definition,wA,∖j∗w^\{\\ast\}\_\{A,\\setminus j\}satisfies the population first\-order condition∇ℒA,j​\(wA,∖j∗\)=0\\nabla\\mathcal\{L\}\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)=0, i\.e\.,

0=∑r∈\{0\}∪𝒜αr​∇ℒr,j​\(wA,∖j∗\)\.0=\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\alpha\_\{r\}\\,\\nabla\\mathcal\{L\}\_\{r,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\.For the target domainr=0r=0,θ∖j∗\\theta^\{\\ast\}\_\{\\setminus j\}minimizesℒ0,j\\mathcal\{L\}\_\{0,j\}, which implies∇ℒ0,j​\(θ∖j∗\)=0\\nabla\\mathcal\{L\}\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)=0\. For each sources∈𝒜s\\in\\mathcal\{A\}, letw∖j\(s\)w^\{\(s\)\}\_\{\\setminus j\}denote the nodewise parameter, where∇ℒs,j​\(w∖j\(s\)\)=0\\nabla\\mathcal\{L\}\_\{s,j\}\(w^\{\(s\)\}\_\{\\setminus j\}\)=0\. By the integral form of Taylor’s theorem,

∇ℒs,j​\(θ∖j∗\)=H¯s​\(θ∖j∗−w∖j\(s\)\),H¯s:=∫01∇2ℒs,j​\(w∖j\(s\)\+t​\(θ∖j∗−w∖j\(s\)\)\)​𝑑t\.\\nabla\\mathcal\{L\}\_\{s,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)=\\bar\{H\}\_\{s\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\(s\)\}\_\{\\setminus j\}\),\\qquad\\bar\{H\}\_\{s\}:=\\int\_\{0\}^\{1\}\\nabla^\{2\}\\mathcal\{L\}\_\{s,j\}\\\!\\left\(w^\{\(s\)\}\_\{\\setminus j\}\+t\(\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\(s\)\}\_\{\\setminus j\}\)\\right\)\\,dt\.Similarly, define

H¯A:=∫01∇2ℒA,j​\(θ∖j∗\+t​\(wA,∖j∗−θ∖j∗\)\)​𝑑t\.\\bar\{H\}\_\{A\}:=\\int\_\{0\}^\{1\}\\nabla^\{2\}\\mathcal\{L\}\_\{A,j\}\\\!\\left\(\\theta^\{\\ast\}\_\{\\setminus j\}\+t\(w^\{\\ast\}\_\{A,\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\)\\right\)\\,dt\.Then

0=∇ℒA,j​\(wA,∖j∗\)=∇ℒA,j​\(θ∖j∗\)\+H¯A​\(wA,∖j∗−θ∖j∗\)\\displaystyle 0=\\nabla\\mathcal\{L\}\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)=\\nabla\\mathcal\{L\}\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\+\\bar\{H\}\_\{A\}\(w^\{\\ast\}\_\{A,\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\)Combining the above and using∇ℒ0,j​\(θ∖j∗\)=0\\nabla\\mathcal\{L\}\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)=0gives

wA,∖j∗−θ∖j∗=−H¯A−1×∑s∈𝒜αs​H¯s​\(θ∖j∗−w∖j\(s\)\)\.w^\{\\ast\}\_\{A,\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}=\-\\bar\{H\}\_\{A\}^\{\-1\}\\times\\sum\_\{s\\in\\mathcal\{A\}\}\\alpha\_\{s\}\\,\\bar\{H\}\_\{s\}\\,\(\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\(s\)\}\_\{\\setminus j\}\)\.
Taking inducedℓ1\\ell\_\{1\}norms and using Assumption[3](https://arxiv.org/html/2607.03005#Thmassumption3)gives

‖wA,∖j∗−θ∖j∗‖1≤C​∑s∈𝒜αs​‖θ∖j∗−w∖j\(s\)‖1≤C​hj\.\\\|w^\{\\ast\}\_\{A,\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\leq C\\sum\_\{s\\in\\mathcal\{A\}\}\\alpha\_\{s\}\\,\\\|\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\(s\)\}\_\{\\setminus j\}\\\|\_\{1\}\\leq Ch\_\{j\}\.This completes the proof\. ∎

### S\.6\.2Proof of Lemma[2](https://arxiv.org/html/2607.03005#Thmlemma2)

###### Proof\.

For any coordinatesa,ba,b,

\[∇2ℓj\(r\)​\(u\)\]a​b=1nr​∑i=1nrf′′​\(zi\(r\)⊤​u\)​zi,a\(r\)​zi,b\(r\)\.\\big\[\\nabla^\{2\}\\ell^\{\(r\)\}\_\{j\}\(u\)\\big\]\_\{ab\}=\\frac\{1\}\{n\_\{r\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}f^\{\\prime\\prime\}\(z^\{\(r\)\\top\}\_\{i\}u\)\\,z^\{\(r\)\}\_\{i,a\}z^\{\(r\)\}\_\{i,b\}\.Because0≤f′′​\(t\)≤1/40\\leq f^\{\\prime\\prime\}\(t\)\\leq 1/4for allttand\|zi,a\(r\)​zi,b\(r\)\|≤4\|z^\{\(r\)\}\_\{i,a\}z^\{\(r\)\}\_\{i,b\}\|\\leq 4, we have\|\[∇2ℓj\(r\)​\(u\)\]a​b\|≤1\\big\|\\big\[\\nabla^\{2\}\\ell^\{\(r\)\}\_\{j\}\(u\)\\big\]\_\{ab\}\\big\|\\leq 1for alluu\. This gives‖∇2ℓj\(r\)​\(u\)‖max≤1\\\|\\nabla^\{2\}\\ell^\{\(r\)\}\_\{j\}\(u\)\\\|\_\{\\max\}\\leq 1\.

For the Lipschitz bound, use the integral form∇ℓj\(r\)​\(u\)−∇ℓj\(r\)​\(v\)=\{∫01∇2ℓj\(r\)​\(v\+t​\(u−v\)\)​𝑑t\}​\(u−v\)\\nabla\\ell^\{\(r\)\}\_\{j\}\(u\)\-\\nabla\\ell^\{\(r\)\}\_\{j\}\(v\)=\\\{\\int\_\{0\}^\{1\}\\nabla^\{2\}\\ell^\{\(r\)\}\_\{j\}\(v\+t\(u\-v\)\)\\,dt\\\}\(u\-v\), and then

‖∇ℓj\(r\)​\(u\)−∇ℓj\(r\)​\(v\)‖∞≤sup0≤t≤1‖∇2ℓj\(r\)​\(v\+t​\(u−v\)\)‖max​‖u−v‖1≤‖u−v‖1\.\\\|\\nabla\\ell^\{\(r\)\}\_\{j\}\(u\)\-\\nabla\\ell^\{\(r\)\}\_\{j\}\(v\)\\\|\_\{\\infty\}\\leq\\sup\_\{0\\leq t\\leq 1\}\\\|\\nabla^\{2\}\\ell^\{\(r\)\}\_\{j\}\(v\+t\(u\-v\)\)\\\|\_\{\\max\}\\,\\\|u\-v\\\|\_\{1\}\\leq\\\|u\-v\\\|\_\{1\}\.The population bounds follow by taking expectation of the entrywise bound\. This completes the proof\. ∎

### S\.6\.3Proof of Lemma[3](https://arxiv.org/html/2607.03005#Thmlemma3)

###### Proof\.

Fix a nodejj\. For the target sampleX\(0\)X^\{\(0\)\}, recall the signed design representation

Z∖j\(0\)=2​diag​\(x⋅j\(0\)\)​X∖j\(0\),uj,i​\(ϑ\)=\[Z∖j\(0\)\]i,⋅​ϑ,p^j,i​\(ϑ\)=σ​\(uj,i​\(ϑ\)\),\\displaystyle Z^\{\(0\)\}\_\{\\setminus j\}=2\\,\\mathrm\{diag\}\\\!\\big\(x^\{\(0\)\}\_\{\\cdot j\}\\big\)\\,X^\{\(0\)\}\_\{\\setminus j\},\\qquad u\_\{j,i\}\(\\vartheta\)=\\big\[Z^\{\(0\)\}\_\{\\setminus j\}\\big\]\_\{i,\\cdot\}\\,\\vartheta,\\qquad\\widehat\{p\}\_\{j,i\}\(\\vartheta\)=\\sigma\\\!\\big\(u\_\{j,i\}\(\\vartheta\)\\big\),whereσ​\(u\)=\(1\+e−u\)−1\\sigma\(u\)=\(1\+e^\{\-u\}\)^\{\-1\}\. Note thatxi​k∈\{−1,1\}x\_\{ik\}\\in\\\{\-1,1\\\}implies\|\[Z∖j\(0\)\]i​k\|≤2\|\[Z^\{\(0\)\}\_\{\\setminus j\}\]\_\{ik\}\|\\leq 2for alli,ki,k\.

#### \(a\) Target score atθ∖j∗\\theta^\{\\ast\}\_\{\\setminus j\}\.

For a fixed coordinatek≠jk\\neq j, thekk\-th component of the target score is

\[∇ℓ0,j​\(θ∖j∗\)\]k=1n0​∑i=1n0ξi​k,ξi​k:=\[Z∖j\(0\)\]i​k​\(p^j,i​\(θ∖j∗\)−1\)\.\\displaystyle\\big\[\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\big\]\_\{k\}=\\frac\{1\}\{n\_\{0\}\}\\sum\_\{i=1\}^\{n\_\{0\}\}\\xi\_\{ik\},\\qquad\\xi\_\{ik\}=\[Z^\{\(0\)\}\_\{\\setminus j\}\]\_\{ik\}\\Big\(\\widehat\{p\}\_\{j,i\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\-1\\Big\)\.We claim𝔼​\[ξi​k∣Xi,∖j\(0\)\]=0\\mathbb\{E\}\[\\xi\_\{ik\}\\mid X^\{\(0\)\}\_\{i,\\setminus j\}\]=0\. Indeed, setηi:=2​∑m≠jθj​m∗​xi​m\(0\)\\eta\_\{i\}:=2\\sum\_\{m\\neq j\}\\theta^\{\\ast\}\_\{jm\}x^\{\(0\)\}\_\{im\}\. Then

uj,i​\(θ∖j∗\)=2​xi​j\(0\)​∑m≠jθj​m∗​xi​m\(0\)=xi​j\(0\)​ηi\.u\_\{j,i\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\;=\\;2x^\{\(0\)\}\_\{ij\}\\sum\_\{m\\neq j\}\\theta^\{\\ast\}\_\{jm\}x^\{\(0\)\}\_\{im\}\\;=\\;x^\{\(0\)\}\_\{ij\}\\,\\eta\_\{i\}\.Writepi:=σ​\(ηi\)=ℙ​\{Xi​j\(0\)=1∣Xi,∖j\(0\)\}p\_\{i\}:=\\sigma\(\\eta\_\{i\}\)=\\mathbb\{P\}\\\{X^\{\(0\)\}\_\{ij\}=1\\mid X^\{\(0\)\}\_\{i,\\setminus j\}\\\}under the Ising conditional model\. Thenℙ​\{Xi​j\(0\)=−1∣Xi,∖j\(0\)\}=1−pi=σ​\(−ηi\)\\mathbb\{P\}\\\{X^\{\(0\)\}\_\{ij\}=\-1\\mid X^\{\(0\)\}\_\{i,\\setminus j\}\\\}=1\-p\_\{i\}=\\sigma\(\-\\eta\_\{i\}\)\. A short calculation gives

𝔼​\[xi​j\(0\)​\(σ​\(xi​j\(0\)​ηi\)−1\)\|Xi,∖j\(0\)\]=pi​\(pi−1\)\+\(1−pi\)​pi=0,\\mathbb\{E\}\\\!\\left\[x^\{\(0\)\}\_\{ij\}\\Big\(\\sigma\(x^\{\(0\)\}\_\{ij\}\\eta\_\{i\}\)\-1\\Big\)\\,\\Big\|\\,X^\{\(0\)\}\_\{i,\\setminus j\}\\right\]=p\_\{i\}\\,\(p\_\{i\}\-1\)\+\(1\-p\_\{i\}\)\\,p\_\{i\}=0,and multiplying by2​xi​k\(0\)2x^\{\(0\)\}\_\{ik\}yields𝔼​\[ξi​k∣Xi,∖j\(0\)\]=0\\mathbb\{E\}\[\\xi\_\{ik\}\\mid X^\{\(0\)\}\_\{i,\\setminus j\}\]=0\.

In addition,\|p^j,i​\(θ∖j∗\)−1\|≤1\|\\widehat\{p\}\_\{j,i\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\-1\|\\leq 1and\|\[Z∖j\(0\)\]i​k\|≤2\|\[Z^\{\(0\)\}\_\{\\setminus j\}\]\_\{ik\}\|\\leq 2, which implies\|ξi​k\|≤2\|\\xi\_\{ik\}\|\\leq 2almost surely\. By Hoeffding’s inequality, for anyt\>0t\>0,

ℙ​\(\|1n0​∑i=1n0ξi​k\|≥t\)≤2​exp⁡\(−n0​t28\)\.\\mathbb\{P\}\\\!\\left\(\\left\|\\frac\{1\}\{n\_\{0\}\}\\sum\_\{i=1\}^\{n\_\{0\}\}\\xi\_\{ik\}\\right\|\\geq t\\right\)\\leq 2\\exp\\\!\\left\(\-\\frac\{n\_\{0\}t^\{2\}\}\{8\}\\right\)\.Taking a union bound overk∈V∖\{j\}k\\in V\\setminus\\\{j\\\}and choosingt=C​log⁡p/n0t=C\\sqrt\{\\log p/n\_\{0\}\}gives

ℙ​\(‖∇ℓ0,j​\(θ∖j∗\)‖∞≥C​log⁡pn0\)≤2​p−c\\mathbb\{P\}\\\!\\left\(\\big\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\big\\\|\_\{\\infty\}\\geq C\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}\\right\)\\leq 2p^\{\-c\}for somec\>0c\>0\.

#### \(b\) Pooled score atwA,∖j∗w^\{\\ast\}\_\{A,\\setminus j\}\.

Recall the pooled objective

ℓA,j​\(u\)=∑r∈\{0\}∪𝒜αr​ℓj\(r\)​\(u\),\\ell\_\{A,j\}\(u\)=\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\alpha\_\{r\}\\,\\ell^\{\(r\)\}\_\{j\}\(u\),whereαr=nrN\\alpha\_\{r\}=\\frac\{n\_\{r\}\}\{N\}andN=∑r∈\{0\}∪𝒜nrN=\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}n\_\{r\}\.

This gives

∇ℓA,j​\(u\)=∑r∈\{0\}∪𝒜αr​∇ℓj\(r\)​\(u\),∇ℓj\(r\)​\(u\)=1nr​∑i=1nrψi\(r\)​\(u\),\\displaystyle\\nabla\\ell\_\{A,j\}\(u\)=\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\alpha\_\{r\}\\,\\nabla\\ell^\{\(r\)\}\_\{j\}\(u\),\\qquad\\nabla\\ell^\{\(r\)\}\_\{j\}\(u\)=\\frac\{1\}\{n\_\{r\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}\\psi^\{\(r\)\}\_\{i\}\(u\),whereψi\(r\)​\(u\)∈ℝp−1\\psi^\{\(r\)\}\_\{i\}\(u\)\\in\\mathbb\{R\}^\{p\-1\}is the single\-sample score contribution\. Atu=wA,∖j∗u=w^\{\\ast\}\_\{A,\\setminus j\}, population optimality gives∇ℒA,j​\(wA,∖j∗\)=0\\nabla\\mathcal\{L\}\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)=0, which implies

𝔼​\[∇ℓA,j​\(wA,∖j∗\)\]=0\.\\mathbb\{E\}\\big\[\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\]=0\.Fixk≠jk\\neq j\. The pooled score coordinate can be written as

\[∇ℓA,j​\(wA,∖j∗\)\]k=1N​∑r∈\{0\}∪𝒜∑i=1nrζi​k\(r\),\\big\[\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\]\_\{k\}=\\frac\{1\}\{N\}\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}\\zeta^\{\(r\)\}\_\{ik\},whereζi​k\(r\)\\zeta^\{\(r\)\}\_\{ik\}are independent across\(r,i\)\(r,i\)and uniformly bounded by a universal constant, sincex∈\{−1,1\}x\\in\\\{\-1,1\\\}\. Note thatζi​k\(r\)\\zeta^\{\(r\)\}\_\{ik\}need not be mean\-zero for eachrr, but the pooled mean equals zero:

𝔼​\[∇ℓA,j​\(wA,∖j∗\)\]=∇ℒA,j​\(wA,∖j∗\)=0\.\\mathbb\{E\}\\big\[\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\]=\\nabla\\mathcal\{L\}\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)=0\.Define the centered variables

ζ~i​k\(r\):=ζi​k\(r\)−𝔼​\[ζi​k\(r\)\],\\tilde\{\\zeta\}^\{\(r\)\}\_\{ik\}:=\\zeta^\{\(r\)\}\_\{ik\}\-\\mathbb\{E\}\[\\zeta^\{\(r\)\}\_\{ik\}\],Then𝔼​\[ζ~i​k\(r\)\]=0\\mathbb\{E\}\[\\tilde\{\\zeta\}^\{\(r\)\}\_\{ik\}\]=0and\|ζ~i​k\(r\)\|\|\\tilde\{\\zeta\}^\{\(r\)\}\_\{ik\}\|is still uniformly bounded\. Then

\[∇ℓA,j​\(wA,∖j∗\)\]k=1N​∑r∈\{0\}∪𝒜∑i=1nrζ~i​k\(r\),\\big\[\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\]\_\{k\}=\\frac\{1\}\{N\}\\sum\_\{r\\in\\\{0\\\}\\cup\\mathcal\{A\}\}\\sum\_\{i=1\}^\{n\_\{r\}\}\\tilde\{\\zeta\}^\{\(r\)\}\_\{ik\},and Hoeffding’s inequality yields

ℙ​\(\|\[∇ℓA,j​\(wA,∖j∗\)\]k\|≥t\)≤2​exp⁡\(−c​N​t2\),\\mathbb\{P\}\\\!\\left\(\\left\|\\big\[\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\]\_\{k\}\\right\|\\geq t\\right\)\\leq 2\\exp\\\!\\left\(\-cNt^\{2\}\\right\),for a universalc\>0c\>0\. A union bound overkkgives

ℙ​\(‖∇ℓA,j​\(wA,∖j∗\)‖∞≥C​log⁡pN\)≤2​p−c′\\mathbb\{P\}\\\!\\left\(\\big\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\\\|\_\{\\infty\}\\geq C\\sqrt\{\\frac\{\\log p\}\{N\}\}\\right\)\\leq 2p^\{\-c^\{\\prime\}\}for somec′\>0c^\{\\prime\}\>0\.

#### \(c\) Pooled score atθ∖j∗\\theta^\{\\ast\}\_\{\\setminus j\}\.

Decompose

∇ℓA,j​\(θ∖j∗\)=\(∇ℓA,j​\(θ∖j∗\)−𝔼​\[∇ℓA,j​\(θ∖j∗\)\]\)\+𝔼​\[∇ℓA,j​\(θ∖j∗\)\]\.\\displaystyle\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)=\\Big\(\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\-\\mathbb\{E\}\[\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\]\\Big\)\+\\mathbb\{E\}\[\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\]\.The fluctuation term is an average ofNNindependent bounded summands, which gives

‖∇ℓA,j​\(θ∖j∗\)−𝔼​\[∇ℓA,j​\(θ∖j∗\)\]‖∞≲log⁡pNw\.h\.p\.\\big\\\|\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\-\\mathbb\{E\}\[\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\]\\big\\\|\_\{\\infty\}\\lesssim\\sqrt\{\\frac\{\\log p\}\{N\}\}\\quad\\text\{w\.h\.p\.\}
For the bias term, note that𝔼​\[∇ℓA,j​\(u\)\]=∇ℒA,j​\(u\)\\mathbb\{E\}\[\\nabla\\ell\_\{A,j\}\(u\)\]=\\nabla\\mathcal\{L\}\_\{A,j\}\(u\)and∇ℒA,j​\(wA,∖j∗\)=0\\nabla\\mathcal\{L\}\_\{A,j\}\(w^\{\\ast\}\_\{A,\\setminus j\}\)=0\. By the integral form of Taylor’s theorem,

∇ℒA,j​\(θ∖j∗\)=\{∫01∇2ℒA,j​\(wA,∖j∗\+t​\(θ∖j∗−wA,∖j∗\)\)​𝑑t\}​\(θ∖j∗−wA,∖j∗\)\\nabla\\mathcal\{L\}\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)=\\left\\\{\\int\_\{0\}^\{1\}\\nabla^\{2\}\\mathcal\{L\}\_\{A,j\}\\big\(w^\{\\ast\}\_\{A,\\setminus j\}\+t\(\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\)\\big\)\\,dt\\right\\\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\)and Lemma[2](https://arxiv.org/html/2607.03005#Thmlemma2)bounds the integrated Hessian in max norm by11, which gives

‖𝔼​\[∇ℓA,j​\(θ∖j∗\)\]‖∞=‖∇ℒA,j​\(θ∖j∗\)‖∞≤‖θ∖j∗−wA,∖j∗‖1=‖δ∖j∗‖1≲hj,\\displaystyle\\big\\\|\\mathbb\{E\}\[\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\]\\big\\\|\_\{\\infty\}=\\\|\\nabla\\mathcal\{L\}\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\\leq\\\|\\theta^\{\\ast\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\\\|\_\{1\}=\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim h\_\{j\},where the last step uses Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1)\. Combining both parts yields

‖∇ℓA,j​\(θ∖j∗\)‖∞≲log⁡pN\+hjw\.h\.p\.\\big\\\|\\nabla\\ell\_\{A,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\big\\\|\_\{\\infty\}\\ \\lesssim\\ \\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\\quad\\text\{w\.h\.p\.\}
This completes the proof\. ∎

### S\.6\.4Proof of Lemma[4](https://arxiv.org/html/2607.03005#Thmlemma4)

###### Proof\.

Fix a nodejjand abbreviate

w^:=w^∖jA,w∗:=wA,∖j∗,Δ:=w^−w∗\.\\hat\{w\}:=\\hat\{w\}^\{A\}\_\{\\setminus j\},\\qquad w^\{\\ast\}:=w^\{\\ast\}\_\{A,\\setminus j\},\\qquad\\Delta:=\\hat\{w\}\-w^\{\\ast\}\.LetS:=Sj=supp​\(θ∖j∗\)S:=S\_\{j\}=\\mathrm\{supp\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)ands:=\|S\|s:=\|S\|\. Note thatw∗w^\{\\ast\}need not be supported onSS, but by Lemma[1](https://arxiv.org/html/2607.03005#Thmlemma1),

‖wSc∗‖1≤‖w∗−θ∖j∗‖1=‖δ∖j∗‖1≲hj\.\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\\leq\\\|w^\{\\ast\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}=\\\|\\delta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim h\_\{j\}\.\(22\)

#### Step 1: Basic inequality\.

By optimality ofw^\\hat\{w\}for \([17](https://arxiv.org/html/2607.03005#S4.E17)\),

ℓA,j​\(w^\)\+λw​‖w^‖1≤ℓA,j​\(w∗\)\+λw​‖w∗‖1\.\\ell\_\{A,j\}\(\\hat\{w\}\)\+\\lambda\_\{w\}\\\|\\hat\{w\}\\\|\_\{1\}\\leq\\ell\_\{A,j\}\(w^\{\\ast\}\)\+\\lambda\_\{w\}\\\|w^\{\\ast\}\\\|\_\{1\}\.Rearrange:

ℓA,j​\(w^\)−ℓA,j​\(w∗\)≤λw​\(‖w∗‖1−‖w^‖1\)\.\\ell\_\{A,j\}\(\\hat\{w\}\)\-\\ell\_\{A,j\}\(w^\{\\ast\}\)\\leq\\lambda\_\{w\}\(\\\|w^\{\\ast\}\\\|\_\{1\}\-\\\|\\hat\{w\}\\\|\_\{1\}\)\.\(23\)Also, by convexity ofℓA,j\\ell\_\{A,j\},

ℓA,j​\(w^\)−ℓA,j​\(w∗\)≥⟨∇ℓA,j​\(w∗\),Δ⟩\.\\ell\_\{A,j\}\(\\hat\{w\}\)\-\\ell\_\{A,j\}\(w^\{\\ast\}\)\\ \\geq\\ \\langle\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\),\\Delta\\rangle\.\(24\)Combining \([23](https://arxiv.org/html/2607.03005#A6.E23)\)–\([24](https://arxiv.org/html/2607.03005#A6.E24)\) gives

⟨∇ℓA,j​\(w∗\),Δ⟩≤λw​\(‖w∗‖1−‖w∗\+Δ‖1\)\.\\langle\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\),\\Delta\\rangle\\leq\\lambda\_\{w\}\(\\\|w^\{\\ast\}\\\|\_\{1\}\-\\\|w^\{\\ast\}\+\\Delta\\\|\_\{1\}\)\.\(25\)

#### Step 2: Control of theℓ1\\ell\_\{1\}\-difference\.

Using decomposability of theℓ1\\ell\_\{1\}norm,

‖w∗‖1−‖w∗\+Δ‖1=‖wS∗‖1\+‖wSc∗‖1−‖wS∗\+ΔS‖1−‖wSc∗\+ΔSc‖1≤‖ΔS‖1−‖ΔSc‖1\+2​‖wSc∗‖1\.\\displaystyle\\\|w^\{\\ast\}\\\|\_\{1\}\-\\\|w^\{\\ast\}\+\\Delta\\\|\_\{1\}=\\\|w^\{\\ast\}\_\{S\}\\\|\_\{1\}\+\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\-\\\|w^\{\\ast\}\_\{S\}\+\\Delta\_\{S\}\\\|\_\{1\}\-\\\|w^\{\\ast\}\_\{S^\{c\}\}\+\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\\leq\\\|\\Delta\_\{S\}\\\|\_\{1\}\-\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\+2\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\.\(26\)On the other hand,

⟨∇ℓA,j​\(w∗\),Δ⟩≥−‖∇ℓA,j​\(w∗\)‖∞​‖Δ‖1\.\\langle\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\),\\Delta\\rangle\\geq\-\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\)\\\|\_\{\\infty\}\\,\\\|\\Delta\\\|\_\{1\}\.Plugging this and \([26](https://arxiv.org/html/2607.03005#A6.E26)\) into \([25](https://arxiv.org/html/2607.03005#A6.E25)\) yields

−‖∇ℓA,j​\(w∗\)‖∞​\(‖ΔS‖1\+‖ΔSc‖1\)≤λw​\(‖ΔS‖1−‖ΔSc‖1\+2​‖wSc∗‖1\)\.\\displaystyle\-\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\)\\\|\_\{\\infty\}\(\\\|\\Delta\_\{S\}\\\|\_\{1\}\+\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\)\\leq\\lambda\_\{w\}\\big\(\\\|\\Delta\_\{S\}\\\|\_\{1\}\-\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\+2\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\\big\)\.Chooseλw≥2​‖∇ℓA,j​\(w∗\)‖∞\\lambda\_\{w\}\\geq 2\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\)\\\|\_\{\\infty\}\(w\.h\.p\. by Lemma[3](https://arxiv.org/html/2607.03005#Thmlemma3)whenλw=Cw​log⁡p/N\\lambda\_\{w\}=C\_\{w\}\\sqrt\{\\log p/N\}andCw\>0C\_\{w\}\>0is sufficiently large\)\. Then

λw2​‖ΔSc‖1≤3​λw2​‖ΔS‖1\+2​λw​‖wSc∗‖1,\\frac\{\\lambda\_\{w\}\}\{2\}\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\\leq\\frac\{3\\lambda\_\{w\}\}\{2\}\\\|\\Delta\_\{S\}\\\|\_\{1\}\+2\\lambda\_\{w\}\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\},i\.e\. the shifted cone condition

‖ΔSc‖1≤3​‖ΔS‖1\+4​‖wSc∗‖1\.\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\\leq 3\\\|\\Delta\_\{S\}\\\|\_\{1\}\+4\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\.\(27\)In particular, using \([22](https://arxiv.org/html/2607.03005#A6.E22)\),

‖ΔSc‖1≤3​‖ΔS‖1\+C​hjw\.h\.p\.\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\\leq 3\\\|\\Delta\_\{S\}\\\|\_\{1\}\+Ch\_\{j\}\\qquad\\text\{w\.h\.p\.\}\(28\)
Condition[1](https://arxiv.org/html/2607.03005#Thmcondition1)includes the required containment of the segment betweenw^\\hat\{w\}andw∗w^\{\\ast\}in the local RSC neighborhood\.

#### Step 3: Apply pooled RSC\.

By Condition[1](https://arxiv.org/html/2607.03005#Thmcondition1), forϑ\\varthetaon the segment betweenw^\\hat\{w\}andw∗w^\{\\ast\},

ℓA,j​\(w^\)−ℓA,j​\(w∗\)−⟨∇ℓA,j​\(w∗\),Δ⟩≥κA2​‖Δ‖22−τA​log⁡pN​‖Δ‖12\.\\displaystyle\\ell\_\{A,j\}\(\\hat\{w\}\)\-\\ell\_\{A,j\}\(w^\{\\ast\}\)\-\\langle\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\),\\Delta\\rangle\\geq\\frac\{\\kappa\_\{A\}\}\{2\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-\\tau\_\{A\}\\frac\{\\log p\}\{N\}\\\|\\Delta\\\|\_\{1\}^\{2\}\.Combine this with \([23](https://arxiv.org/html/2607.03005#A6.E23)\):

κA2​‖Δ‖22−τA​log⁡pN​‖Δ‖12≤λw​\(‖w∗‖1−‖w^‖1\)−⟨∇ℓA,j​\(w∗\),Δ⟩\\displaystyle\\frac\{\\kappa\_\{A\}\}\{2\}\\\|\\Delta\\\|\_\{2\}^\{2\}\-\\tau\_\{A\}\\frac\{\\log p\}\{N\}\\\|\\Delta\\\|\_\{1\}^\{2\}\\leq\\lambda\_\{w\}\(\\\|w^\{\\ast\}\\\|\_\{1\}\-\\\|\\hat\{w\}\\\|\_\{1\}\)\-\\langle\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\),\\Delta\\rangle≤λw​\(‖ΔS‖1−‖ΔSc‖1\+2​‖wSc∗‖1\)\+‖∇ℓA,j​\(w∗\)‖∞​‖Δ‖1\.\\displaystyle\\leq\\lambda\_\{w\}\\big\(\\\|\\Delta\_\{S\}\\\|\_\{1\}\-\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\+2\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\\big\)\+\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\)\\\|\_\{\\infty\}\\\|\\Delta\\\|\_\{1\}\.Using againλw≥2​‖∇ℓA,j​\(w∗\)‖∞\\lambda\_\{w\}\\geq 2\\\|\\nabla\\ell\_\{A,j\}\(w^\{\\ast\}\)\\\|\_\{\\infty\}and \([27](https://arxiv.org/html/2607.03005#A6.E27)\), the right\-hand side is bounded by

3​λw2​‖ΔS‖1\+2​λw​‖wSc∗‖1≤3​λw2​s​‖Δ‖2\+2​λw​‖wSc∗‖1\.\\frac\{3\\lambda\_\{w\}\}\{2\}\\\|\\Delta\_\{S\}\\\|\_\{1\}\+2\\lambda\_\{w\}\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\\leq\\frac\{3\\lambda\_\{w\}\}\{2\}\\sqrt\{s\}\\,\\\|\\Delta\\\|\_\{2\}\+2\\lambda\_\{w\}\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\.Also from \([27](https://arxiv.org/html/2607.03005#A6.E27)\),

‖Δ‖1=‖ΔS‖1\+‖ΔSc‖1≤4​‖ΔS‖1\+4​‖wSc∗‖1≤4​s​‖Δ‖2\+4​‖wSc∗‖1\.\\displaystyle\\\|\\Delta\\\|\_\{1\}=\\\|\\Delta\_\{S\}\\\|\_\{1\}\+\\\|\\Delta\_\{S^\{c\}\}\\\|\_\{1\}\\leq 4\\\|\\Delta\_\{S\}\\\|\_\{1\}\+4\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\\leq 4\\sqrt\{s\}\\\|\\Delta\\\|\_\{2\}\+4\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\.

#### Step 4: Conclude the rates\.

Using the previous bounds and absorbing the RSC tolerance term unders​log⁡p/N\+\(log⁡p/N\)​hj2=o​\(1\)s\\log p/N\+\(\\log p/N\)h\_\{j\}^\{2\}=o\(1\), we arrive at an inequality of the form

‖Δ‖22≲λw​s​‖Δ‖2\+λw​‖wSc∗‖1\.\\\|\\Delta\\\|\_\{2\}^\{2\}\\ \\lesssim\\ \\lambda\_\{w\}\\sqrt\{s\}\\,\\\|\\Delta\\\|\_\{2\}\\;\+\\;\\lambda\_\{w\}\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\.Solving this quadratic inequality first yields

‖Δ‖2≲λw​s\+λw​‖wSc∗‖1\.\\\|\\Delta\\\|\_\{2\}\\ \\lesssim\\ \\lambda\_\{w\}\\sqrt\{s\}\\;\+\\;\\sqrt\{\\lambda\_\{w\}\\,\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\}\.LetB:=‖wSc∗‖1B:=\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\. IfB≥λwB\\geq\\lambda\_\{w\}, thenλw​B≤B\\sqrt\{\\lambda\_\{w\}B\}\\leq B\. IfB<λwB<\\lambda\_\{w\}ands≥1s\\geq 1, thenλw​B≤λw≤λw​s\\sqrt\{\\lambda\_\{w\}B\}\\leq\\lambda\_\{w\}\\leq\\lambda\_\{w\}\\sqrt\{s\}, and this term is absorbed into the leading term\. Ifs=0s=0, \([27](https://arxiv.org/html/2607.03005#A6.E27)\) gives‖Δ‖1≤4​B\\\|\\Delta\\\|\_\{1\}\\leq 4B, which implies‖Δ‖2≤4​B\\\|\\Delta\\\|\_\{2\}\\leq 4B\. Combining these cases,

‖Δ‖2≲λw​s\+\(λw​B∧B\)\.\\\|\\Delta\\\|\_\{2\}\\ \\lesssim\\ \\lambda\_\{w\}\\sqrt\{s\}\\;\+\\;\\Big\(\\sqrt\{\\lambda\_\{w\}B\}\\ \\wedge\\ B\\Big\)\.In addition, the shifted cone bound gives

‖Δ‖1≲s​λw\+‖wSc∗‖1\.\\\|\\Delta\\\|\_\{1\}\\ \\lesssim\\ s\\,\\lambda\_\{w\}\+\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\.Withλw=Cw​\(log⁡p\)/N\\lambda\_\{w\}=C\_\{w\}\\sqrt\{\(\\log p\)/N\}and‖wSc∗‖1≲hj\\\|w^\{\\ast\}\_\{S^\{c\}\}\\\|\_\{1\}\\lesssim h\_\{j\}, this yields

‖Δ‖2≲s​log⁡pN\+\(\[\(log⁡pN\)1/4​hj1/2\]∧hj\),‖Δ‖1≲s​log⁡pN\+hj\.\\displaystyle\\\|\\Delta\\\|\_\{2\}\\lesssim\\sqrt\{\\frac\{s\\log p\}\{N\}\}\+\\Big\(\\big\[\(\\tfrac\{\\log p\}\{N\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\big\]\\wedge h\_\{j\}\\Big\),\\qquad\\\|\\Delta\\\|\_\{1\}\\lesssim s\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\.This completes the proof\. ∎

### S\.6\.5Proof of Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)

###### Proof\.

Fix a nodejj\. Write

δ^:=δ^∖jA,δ∗:=δ∖j∗,v:=δ^−δ∗,θ^:=w^∖jA\+δ^\.\\hat\{\\delta\}:=\\hat\{\\delta\}^\{A\}\_\{\\setminus j\},\\qquad\\delta^\{\\ast\}:=\\delta^\{\\ast\}\_\{\\setminus j\},\\qquad v:=\\hat\{\\delta\}\-\\delta^\{\\ast\},\\qquad\\hat\{\\theta\}:=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}\.Recall the Step 2 objective

Q​\(δ\):=ℓ0,j​\(w^∖jA\+δ\)\+λδ​‖δ‖1\+∑k≠jPλ​\(w^j​kA\+δj​k\),Q\(\\delta\):=\\ell\_\{0,j\}\(\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\delta\)\+\\lambda\_\{\\delta\}\\\|\\delta\\\|\_\{1\}\+\\sum\_\{k\\neq j\}P\_\{\\lambda\}\(\\hat\{w\}^\{A\}\_\{jk\}\+\\delta\_\{jk\}\),wherePλP\_\{\\lambda\}is the SCAD penalty with parametera\>2a\>2and levelλ\\lambda\. Let

θ†:=w^∖jA\+δ∗\.\\theta^\{\\dagger\}:=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\delta^\{\\ast\}\.Note thatθ†=θ∖j∗\+Δw\\theta^\{\\dagger\}=\\theta^\{\\ast\}\_\{\\setminus j\}\+\\Delta\_\{w\}, whereΔw:=w^∖jA−wA,∖j∗\\Delta\_\{w\}:=\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}is the Step 1 error\.

By Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3)\(ii\),δ^=δ∗\+v\\hat\{\\delta\}=\\delta^\{\\ast\}\+vsatisfies‖δ^−δ∗‖1≤rδ\\\|\\hat\{\\delta\}\-\\delta^\{\\ast\}\\\|\_\{1\}\\leq r\_\{\\delta\}and is locally optimal on theℓ1\\ell\_\{1\}\-ball aroundδ∗\\delta^\{\\ast\}\. Takingδ=δ∗\\delta=\\delta^\{\\ast\}yields

Q​\(δ∗\+v\)≤Q​\(δ∗\)\.Q\(\\delta^\{\\ast\}\+v\)\\ \\leq\\ Q\(\\delta^\{\\ast\}\)\.\(29\)

#### Step 1: Basic inequality and Taylor expansion\.

Expanding \([29](https://arxiv.org/html/2607.03005#A6.E29)\) yields

ℓj\(0\)​\(θ†\+v\)−ℓj\(0\)​\(θ†\)≤λδ​\(‖δ∗‖1−‖δ∗\+v‖1\)\+∑k≠j\(Pλ​\(θk†\)−Pλ​\(θk†\+vk\)\)\.\\displaystyle\\ell^\{\(0\)\}\_\{j\}\(\\theta^\{\\dagger\}\+v\)\-\\ell^\{\(0\)\}\_\{j\}\(\\theta^\{\\dagger\}\)\\leq\\lambda\_\{\\delta\}\\big\(\\\|\\delta^\{\\ast\}\\\|\_\{1\}\-\\\|\\delta^\{\\ast\}\+v\\\|\_\{1\}\\big\)\+\\sum\_\{k\\neq j\}\\Big\(P\_\{\\lambda\}\(\\theta^\{\\dagger\}\_\{k\}\)\-P\_\{\\lambda\}\(\\theta^\{\\dagger\}\_\{k\}\+v\_\{k\}\)\\Big\)\.\(30\)Letθ~\\tilde\{\\theta\}be a point on the segment betweenθ†\\theta^\{\\dagger\}andθ†\+v\\theta^\{\\dagger\}\+v\. Then

ℓj\(0\)​\(θ†\+v\)−ℓj\(0\)​\(θ†\)=⟨∇ℓj\(0\)​\(θ†\),v⟩\+12​v⊤​∇2ℓj\(0\)​\(θ~\)​v\.\\ell^\{\(0\)\}\_\{j\}\(\\theta^\{\\dagger\}\+v\)\-\\ell^\{\(0\)\}\_\{j\}\(\\theta^\{\\dagger\}\)=\\langle\\nabla\\ell^\{\(0\)\}\_\{j\}\(\\theta^\{\\dagger\}\),v\\rangle\+\\frac\{1\}\{2\}\\,v^\{\\top\}\\nabla^\{2\}\\ell^\{\(0\)\}\_\{j\}\(\\tilde\{\\theta\}\)\\,v\.\(31\)Combining \([30](https://arxiv.org/html/2607.03005#A6.E30)\) and \([31](https://arxiv.org/html/2607.03005#A6.E31)\) gives

⟨∇ℓj\(0\)​\(θ†\),v⟩\+12​v⊤​∇2ℓj\(0\)​\(θ~\)​v≤λδ​\(‖δ∗‖1−‖δ∗\+v‖1\)\+∑k≠j\(Pλ​\(θk†\)−Pλ​\(θk†\+vk\)\)\.\\displaystyle\\langle\\nabla\\ell^\{\(0\)\}\_\{j\}\(\\theta^\{\\dagger\}\),v\\rangle\+\\frac\{1\}\{2\}\\,v^\{\\top\}\\nabla^\{2\}\\ell^\{\(0\)\}\_\{j\}\(\\tilde\{\\theta\}\)\\,v\\leq\\lambda\_\{\\delta\}\\big\(\\\|\\delta^\{\\ast\}\\\|\_\{1\}\-\\\|\\delta^\{\\ast\}\+v\\\|\_\{1\}\\big\)\+\\sum\_\{k\\neq j\}\\Big\(P\_\{\\lambda\}\(\\theta^\{\\dagger\}\_\{k\}\)\-P\_\{\\lambda\}\(\\theta^\{\\dagger\}\_\{k\}\+v\_\{k\}\)\\Big\)\.\(32\)
For SCAD, the subgradient is uniformly bounded: for alltt, anyg∈∂Pλ​\(t\)g\\in\\partial P\_\{\\lambda\}\(t\)satisfies\|g\|≤λ\|g\|\\leq\\lambda\. Equivalently,Pλ​\(⋅\)P\_\{\\lambda\}\(\\cdot\)isλ\\lambda\-Lipschitz\. This implies that, for anyu,b∈ℝu,b\\in\\mathbb\{R\},

Pλ​\(u\)−Pλ​\(u\+b\)≤λ​\|b\|\.P\_\{\\lambda\}\(u\)\-P\_\{\\lambda\}\(u\+b\)\\ \\leq\\ \\lambda\|b\|\.Applying this coordinate\-wise yields the clean bound

∑k≠j\(Pλ​\(θk†\)−Pλ​\(θk†\+vk\)\)≤λ​‖v‖1\.\\sum\_\{k\\neq j\}\\Big\(P\_\{\\lambda\}\(\\theta^\{\\dagger\}\_\{k\}\)\-P\_\{\\lambda\}\(\\theta^\{\\dagger\}\_\{k\}\+v\_\{k\}\)\\Big\)\\ \\leq\\ \\lambda\\\|v\\\|\_\{1\}\.\(33\)

#### Step 2: Score control atθ†\\theta^\{\\dagger\}\.

Recallθ†=θ∖j∗\+Δw\\theta^\{\\dagger\}=\\theta^\{\\ast\}\_\{\\setminus j\}\+\\Delta\_\{w\}whereΔw=w^∖jA−wA,∖j∗\\Delta\_\{w\}=\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}\. By Lemma[2](https://arxiv.org/html/2607.03005#Thmlemma2),

‖∇ℓ0,j​\(θ†\)−∇ℓ0,j​\(θ∖j∗\)‖∞≤‖θ†−θ∖j∗‖1=‖Δw‖1\.\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\dagger\}\)\-\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\\leq\\\|\\theta^\{\\dagger\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}=\\\|\\Delta\_\{w\}\\\|\_\{1\}\.This gives

‖∇ℓ0,j​\(θ†\)‖∞≤‖∇ℓ0,j​\(θ∖j∗\)‖∞\+‖Δw‖1\.\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\dagger\}\)\\\|\_\{\\infty\}\\leq\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\+\\\|\\Delta\_\{w\}\\\|\_\{1\}\.This implies

\|⟨∇ℓ0,j​\(θ†\),v⟩\|≤\(‖∇ℓ0,j​\(θ∖j∗\)‖∞\+‖Δw‖1\)​‖v‖1\.\|\\langle\\nabla\\ell\_\{0,j\}\(\\theta^\{\\dagger\}\),v\\rangle\|\\leq\\Big\(\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\+\\\|\\Delta\_\{w\}\\\|\_\{1\}\\Big\)\\,\\\|v\\\|\_\{1\}\.By Lemma[3](https://arxiv.org/html/2607.03005#Thmlemma3),‖∇ℓ0,j​\(θ∖j∗\)‖∞≲log⁡p/n0\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\\lesssim\\sqrt\{\\log p/n\_\{0\}\}w\.h\.p\. The Step 1 hypothesis of this lemma gives‖Δw‖1≲λδ\\\|\\Delta\_\{w\}\\\|\_\{1\}\\lesssim\\lambda\_\{\\delta\}w\.h\.p\. Becauseλδ≍log⁡p/n0\\lambda\_\{\\delta\}\\asymp\\sqrt\{\\log p/n\_\{0\}\}, this gives

\|⟨∇ℓ0,j​\(θ†\),v⟩\|≲λδ​‖v‖1​w\.h\.p\.\|\\langle\\nabla\\ell\_\{0,j\}\(\\theta^\{\\dagger\}\),v\\rangle\|\\ \\lesssim\\ \\lambda\_\{\\delta\}\\,\\\|v\\\|\_\{1\}\\ \\text\{w\.h\.p\.\}\(34\)

#### Step 3:ℓ1\\ell\_\{1\}term via effective sparsity ofδ∗\\delta^\{\\ast\}\.

The vectorδ∗\\delta^\{\\ast\}is not assumed sparse, but satisfies‖δ∗‖1≤Ch​hj\\\|\\delta^\{\\ast\}\\\|\_\{1\}\\leq C\_\{h\}h\_\{j\}\. Define the “large” index set at levelλδ\\lambda\_\{\\delta\}:

T:=T​\(λδ\):=\{k≠j:\|δk∗\|\>λδ\},t:=\|T\|\.T:=T\(\\lambda\_\{\\delta\}\):=\\\{k\\neq j:\\ \|\\delta^\{\\ast\}\_\{k\}\|\>\\lambda\_\{\\delta\}\\\},\\qquad t:=\|T\|\.Thent≤‖δ∗‖1/λδ≤Ch​hj/λδt\\leq\\\|\\delta^\{\\ast\}\\\|\_\{1\}/\\lambda\_\{\\delta\}\\leq C\_\{h\}h\_\{j\}/\\lambda\_\{\\delta\}\. By decomposability,

‖δ∗‖1−‖δ∗\+v‖1≤‖vT‖1−‖vTc‖1\+2​‖δTc∗‖1≤‖vT‖1−‖vTc‖1\+2​Ch​hj\.\\displaystyle\\\|\\delta^\{\\ast\}\\\|\_\{1\}\-\\\|\\delta^\{\\ast\}\+v\\\|\_\{1\}\\leq\\\|v\_\{T\}\\\|\_\{1\}\-\\\|v\_\{T^\{c\}\}\\\|\_\{1\}\+2\\\|\\delta^\{\\ast\}\_\{T^\{c\}\}\\\|\_\{1\}\\leq\\\|v\_\{T\}\\\|\_\{1\}\-\\\|v\_\{T^\{c\}\}\\\|\_\{1\}\+2C\_\{h\}h\_\{j\}\.\(35\)

#### Step 4: Cone condition and mixed rate\.

Condition[4](https://arxiv.org/html/2607.03005#Thmcondition4)gives

‖vTc‖1≤3​‖vT‖1\+3​Ch​hj\.\\\|v\_\{T^\{c\}\}\\\|\_\{1\}\\leq 3\\\|v\_\{T\}\\\|\_\{1\}\+3C\_\{h\}h\_\{j\}\.Together with Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3), this permits the use of Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2)in the Taylor expansion\. Combine \([32](https://arxiv.org/html/2607.03005#A6.E32)\), \([33](https://arxiv.org/html/2607.03005#A6.E33)\), \([34](https://arxiv.org/html/2607.03005#A6.E34)\), \([35](https://arxiv.org/html/2607.03005#A6.E35)\), and Condition[2](https://arxiv.org/html/2607.03005#Thmcondition2)\. With high probability,

κ02​‖v‖22−τ0​log⁡pn0​‖v‖12≲λδ​‖v‖1\+λ​‖v‖1\+λδ​\(‖vT‖1−‖vTc‖1\)\+λδ​hj\.\\displaystyle\\frac\{\\kappa\_\{0\}\}\{2\}\\\|v\\\|\_\{2\}^\{2\}\-\\tau\_\{0\}\\frac\{\\log p\}\{n\_\{0\}\}\\\|v\\\|\_\{1\}^\{2\}\\lesssim\\lambda\_\{\\delta\}\\\|v\\\|\_\{1\}\+\\lambda\\\|v\\\|\_\{1\}\+\\lambda\_\{\\delta\}\(\\\|v\_\{T\}\\\|\_\{1\}\-\\\|v\_\{T^\{c\}\}\\\|\_\{1\}\)\+\\lambda\_\{\\delta\}h\_\{j\}\.\(36\)Chooseλ≍λδ≍log⁡p/n0\\lambda\\asymp\\lambda\_\{\\delta\}\\asymp\\sqrt\{\\log p/n\_\{0\}\}\. By Condition[4](https://arxiv.org/html/2607.03005#Thmcondition4),

‖v‖1=‖vT‖1\+‖vTc‖1≤4​‖vT‖1\+3​Ch​hj≲t​‖v‖2\+hj\.\\\|v\\\|\_\{1\}=\\\|v\_\{T\}\\\|\_\{1\}\+\\\|v\_\{T^\{c\}\}\\\|\_\{1\}\\leq 4\\\|v\_\{T\}\\\|\_\{1\}\+3C\_\{h\}h\_\{j\}\\lesssim\\sqrt\{t\}\\,\\\|v\\\|\_\{2\}\+h\_\{j\}\.Substituting this bound into \([36](https://arxiv.org/html/2607.03005#A6.E36)\) reduces the bound to an inequality in‖v‖2\\\|v\\\|\_\{2\}:

‖v‖22≲λδ​t​‖v‖2\+λδ​hj\+log⁡pn0​\(t​‖v‖2\+hj\)2\.\\\|v\\\|\_\{2\}^\{2\}\\ \\lesssim\\ \\lambda\_\{\\delta\}\\sqrt\{t\}\\,\\\|v\\\|\_\{2\}\+\\lambda\_\{\\delta\}h\_\{j\}\+\\frac\{\\log p\}\{n\_\{0\}\}\\big\(\\sqrt\{t\}\\,\\\|v\\\|\_\{2\}\+h\_\{j\}\\big\)^\{2\}\.Usingt≲hj/λδt\\lesssim h\_\{j\}/\\lambda\_\{\\delta\}andλδ≍log⁡p/n0\\lambda\_\{\\delta\}\\asymp\\sqrt\{\\log p/n\_\{0\}\}, putan:=λδa\_\{n\}:=\\lambda\_\{\\delta\}andx:=‖v‖2x:=\\\|v\\\|\_\{2\}\. The preceding inequality gives

x2≲an​t​x\+an​hj\+an2​\(t​x\+hj\)2\.x^\{2\}\\lesssim a\_\{n\}\\sqrt\{t\}\\,x\+a\_\{n\}h\_\{j\}\+a\_\{n\}^\{2\}\(\\sqrt\{t\}\\,x\+h\_\{j\}\)^\{2\}\.Becauset≲hj/ant\\lesssim h\_\{j\}/a\_\{n\},hj≲anh\_\{j\}\\lesssim a\_\{n\}, andan2=log⁡p/n0=o​\(1\)a\_\{n\}^\{2\}=\\log p/n\_\{0\}=o\(1\), the coefficient ofx2x^\{2\}inan2​t​x2a\_\{n\}^\{2\}tx^\{2\}isO​\(an​hj\)=o​\(1\)O\(a\_\{n\}h\_\{j\}\)=o\(1\)and can be absorbed into the left\-hand side\. The remaining terms inan2​\(t​x\+hj\)2a\_\{n\}^\{2\}\(\\sqrt\{t\}\\,x\+h\_\{j\}\)^\{2\}are bounded by a constant multiple of\(an​hj\)1/2​x\+an​hj\(a\_\{n\}h\_\{j\}\)^\{1/2\}x\+a\_\{n\}h\_\{j\}\. This givesx2≲\(an​hj\)1/2​x\+an​hj\.x^\{2\}\\lesssim\(a\_\{n\}h\_\{j\}\)^\{1/2\}x\+a\_\{n\}h\_\{j\}\.Solving this scalar inequality gives

‖v‖2≲\(an​hj\)1/2=\(log⁡pn0\)1/4​hj1/2\.\\\|v\\\|\_\{2\}\\lesssim\(a\_\{n\}h\_\{j\}\)^\{1/2\}=\\left\(\\frac\{\\log p\}\{n\_\{0\}\}\\right\)^\{1/4\}h\_\{j\}^\{1/2\}\.In addition, the cone bound gives

‖v‖1≲t​‖v‖2\+hj≲hj,\\\|v\\\|\_\{1\}\\lesssim\\sqrt\{t\}\\\|v\\\|\_\{2\}\+h\_\{j\}\\lesssim h\_\{j\},which implies‖v‖2≤‖v‖1≲hj\\\|v\\\|\_\{2\}\\leq\\\|v\\\|\_\{1\}\\lesssim h\_\{j\}\. Combining the two bounds,

‖v‖2≲\(\[\(log⁡pn0\)1/4​hj1/2\]∧hj\),‖v‖1≲hj,\\\|v\\\|\_\{2\}\\ \\lesssim\\ \\Big\(\\big\[\(\\tfrac\{\\log p\}\{n\_\{0\}\}\)^\{1/4\}h\_\{j\}^\{1/2\}\\big\]\\wedge h\_\{j\}\\Big\),\\qquad\\\|v\\\|\_\{1\}\\ \\lesssim\\ h\_\{j\},as claimed\. This completes the proof\. ∎

### S\.6\.6Proof of Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)

###### Proof\.

Fix a nodejj\. Recall

θ^∖j=w^∖jA\+δ^∖jA,θ∖j∗=wA,∖j∗\+δ∖j∗\.\\hat\{\\theta\}\_\{\\setminus j\}=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\},\\qquad\\theta^\{\\ast\}\_\{\\setminus j\}=w^\{\\ast\}\_\{A,\\setminus j\}\+\\delta^\{\\ast\}\_\{\\setminus j\}\.Define the Step 1 errorΔw:=w^∖jA−wA,∖j∗\\Delta\_\{w\}:=\\hat\{w\}^\{A\}\_\{\\setminus j\}\-w^\{\\ast\}\_\{A,\\setminus j\}and the Step 2 errorv:=δ^∖jA−δ∖j∗v:=\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\-\\delta^\{\\ast\}\_\{\\setminus j\}\. We condition on the intersection of the event on which Conditions[1](https://arxiv.org/html/2607.03005#Thmcondition1)–[4](https://arxiv.org/html/2607.03005#Thmcondition4)hold, the score\-concentration event in Lemma[3](https://arxiv.org/html/2607.03005#Thmlemma3), and the high\-probability events in Lemmas[4](https://arxiv.org/html/2607.03005#Thmlemma4)and[5](https://arxiv.org/html/2607.03005#Thmlemma5)\. This intersection has probability at least1−C0​p−c01\-C\_\{0\}p^\{\-c\_\{0\}\}after changing constants\. Lemma[4](https://arxiv.org/html/2607.03005#Thmlemma4)and the rate conditionsj​log⁡p/N\+hj≲λδs\_\{j\}\\sqrt\{\\log p/N\}\+h\_\{j\}\\lesssim\\lambda\_\{\\delta\}give

‖Δw‖1≲sj​log⁡pN\+hj≲λδ\.\\\|\\Delta\_\{w\}\\\|\_\{1\}\\lesssim s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\\lesssim\\lambda\_\{\\delta\}\.This verifies the Step 1 condition in Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)\. Then

θ^∖j−θ∖j∗=Δw\+v,\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}=\\Delta\_\{w\}\+v,and the triangle inequality gives

‖θ^∖j−θ∖j∗‖2≤‖Δw‖2\+‖v‖2\.\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{2\}\\leq\\\|\\Delta\_\{w\}\\\|\_\{2\}\+\\\|v\\\|\_\{2\}\.Apply Lemma[4](https://arxiv.org/html/2607.03005#Thmlemma4)to bound‖Δw‖2\\\|\\Delta\_\{w\}\\\|\_\{2\}and Lemma[5](https://arxiv.org/html/2607.03005#Thmlemma5)to bound‖v‖2\\\|v\\\|\_\{2\}\. Combining the two bounds yields the claimedℓ2\\ell\_\{2\}rate\. The same triangle inequality, together with theℓ1\\ell\_\{1\}bounds in Lemmas[4](https://arxiv.org/html/2607.03005#Thmlemma4)and[5](https://arxiv.org/html/2607.03005#Thmlemma5), gives

‖θ^∖j−θ∖j∗‖1≲sj​log⁡pN\+hj\.\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\\lesssim s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\.This completes the proof\. ∎

### S\.6\.7Proof of Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2)

###### Proof\.

Fix a nodej∈Vj\\in Vand letS:=Sj=\{k≠j:θj​k∗≠0\}S:=S\_\{j\}=\\\{k\\neq j:\\theta^\{\\ast\}\_\{jk\}\\neq 0\\\}\. Recallθ^∖j=w^∖jA\+δ^∖jA\\hat\{\\theta\}\_\{\\setminus j\}=\\hat\{w\}^\{A\}\_\{\\setminus j\}\+\\hat\{\\delta\}^\{A\}\_\{\\setminus j\}\. By Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1), with probability tending to one,

‖θ^∖j−θ∖j∗‖∞≤‖θ^∖j−θ∖j∗‖2≤rn,j\.\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{\\infty\}\\leq\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{2\}\\leq r\_\{n,j\}\.On this event, we prove the following two claims\.

#### \(a\) Sign Consistency and Vanishing Bias onSS\.

For anyk∈Sk\\in S, under the beta\-min condition in Theorem[2](https://arxiv.org/html/2607.03005#Thmtheorem2),

\|θ^j​k−θj​k∗\|≤rn,j<\|θj​k∗\|,\|\\hat\{\\theta\}\_\{jk\}\-\\theta^\{\\ast\}\_\{jk\}\|\\leq r\_\{n,j\}<\|\\theta^\{\\ast\}\_\{jk\}\|,which implies thatsign⁡\(θ^j​k\)=sign⁡\(θj​k∗\)\\operatorname\{sign\}\(\\hat\{\\theta\}\_\{jk\}\)=\\operatorname\{sign\}\(\\theta^\{\\ast\}\_\{jk\}\)\. In addition,

\|θ^j​k\|≥\|θj​k∗\|−‖θ^∖j−θ∖j∗‖∞≥\(a​λ\+rn,j\)−rn,j=a​λ\.\|\\hat\{\\theta\}\_\{jk\}\|\\geq\|\\theta^\{\\ast\}\_\{jk\}\|\-\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{\\infty\}\\geq\(a\\lambda\+r\_\{n,j\}\)\-r\_\{n,j\}=a\\lambda\.This has two implications:

1. 1\.Since\|θ^j​k\|\>0\|\\hat\{\\theta\}\_\{jk\}\|\>0, we haveS^j⊇S\\hat\{S\}\_\{j\}\\supseteq S\(No false negatives\)\.
2. 2\.Since\|θ^j​k\|≥a​λ\|\\hat\{\\theta\}\_\{jk\}\|\\geq a\\lambda, we are in the flat region of the SCAD penalty\. The subgradient of the SCAD penalty is zero: Pλ′​\(\|θ^j​k\|\)=0∀k∈S\.P^\{\\prime\}\_\{\\lambda\}\(\|\\hat\{\\theta\}\_\{jk\}\|\)=0\\quad\\forall k\\in S\.

This property, namely zero penalty derivative for large coefficients, is specific to folded\-concave penalties such as SCAD and distinguishes this proof from Lasso\. For Lasso, the penalty derivative would beλ⋅sign⁡\(θ^j​k\)\\lambda\\cdot\\operatorname\{sign\}\(\\hat\{\\theta\}\_\{jk\}\), introducing a bias that requires the Irrepresentable Condition to manage\.

#### \(b\) No False Positives \(Selection Consistency\)\.

We showθ^j​k=0\\hat\{\\theta\}\_\{jk\}=0for allk∈Sck\\in S^\{c\}\. Fork∈Sck\\in S^\{c\}, suppose for contradiction thatθ^j​k≠0\\hat\{\\theta\}\_\{jk\}\\neq 0\. On this event and underrn,j≤λr\_\{n,j\}\\leq\\lambda, we have0<\|θ^j​k\|≤λ0<\|\\hat\{\\theta\}\_\{jk\}\|\\leq\\lambda\. The SCAD subgradient with respect toθ^j​k\\hat\{\\theta\}\_\{jk\}has absolute valueλ\\lambda\. By Condition[3](https://arxiv.org/html/2607.03005#Thmcondition3), there existgj​k∈∂\|δ^j​kA\|g\_\{jk\}\\in\\partial\|\\hat\{\\delta\}^\{A\}\_\{jk\}\|andhj​k∈∂Pλ​\(θ^j​k\)h\_\{jk\}\\in\\partial P\_\{\\lambda\}\(\\hat\{\\theta\}\_\{jk\}\)such that

\|\[∇ℓ0,j​\(θ^∖j\)\]k\+λδ​gj​k\+hj​k\|≤εn,j\.\\left\|\\Big\[\\nabla\\ell\_\{0,j\}\(\\hat\{\\theta\}\_\{\\setminus j\}\)\\Big\]\_\{k\}\+\\lambda\_\{\\delta\}g\_\{jk\}\+h\_\{jk\}\\right\|\\leq\\varepsilon\_\{n,j\}\.Because\|gj​k\|≤1\|g\_\{jk\}\|\\leq 1and\|hj​k\|=λ\|h\_\{jk\}\|=\\lambda, this implies

\|\[∇ℓ0,j​\(θ^∖j\)\]k\|≥λ−λδ−εn,j\.\\left\|\\Big\[\\nabla\\ell\_\{0,j\}\(\\hat\{\\theta\}\_\{\\setminus j\}\)\\Big\]\_\{k\}\\right\|\\geq\\lambda\-\\lambda\_\{\\delta\}\-\\varepsilon\_\{n,j\}\.The integral form of Taylor’s theorem and Lemma[2](https://arxiv.org/html/2607.03005#Thmlemma2)give

\|\[∇ℓ0,j​\(θ^∖j\)\]k\|≤‖∇ℓ0,j​\(θ∖j∗\)‖∞\+‖θ^∖j−θ∖j∗‖1\.\\left\|\\Big\[\\nabla\\ell\_\{0,j\}\(\\hat\{\\theta\}\_\{\\setminus j\}\)\\Big\]\_\{k\}\\right\|\\leq\\\|\\nabla\\ell\_\{0,j\}\(\\theta^\{\\ast\}\_\{\\setminus j\}\)\\\|\_\{\\infty\}\+\\\|\\hat\{\\theta\}\_\{\\setminus j\}\-\\theta^\{\\ast\}\_\{\\setminus j\}\\\|\_\{1\}\.Lemma[3](https://arxiv.org/html/2607.03005#Thmlemma3)and Theorem[1](https://arxiv.org/html/2607.03005#Thmtheorem1)imply

\|\[∇ℓ0,j​\(θ^∖j\)\]k\|≲log⁡pn0\+sj​log⁡pN\+hj\.\\left\|\\Big\[\\nabla\\ell\_\{0,j\}\(\\hat\{\\theta\}\_\{\\setminus j\}\)\\Big\]\_\{k\}\\right\|\\lesssim\\sqrt\{\\frac\{\\log p\}\{n\_\{0\}\}\}\+s\_\{j\}\\sqrt\{\\frac\{\\log p\}\{N\}\}\+h\_\{j\}\.This contradiction implies thatθ^j​k=0\\hat\{\\theta\}\_\{jk\}=0for allk∈Sck\\in S^\{c\}\.

We have shown thatS^j⊇S\\hat\{S\}\_\{j\}\\supseteq Sfrom beta\-min and flat penalty region andS^j⊆S\\hat\{S\}\_\{j\}\\subseteq Sfrom noise control dominating the gradient\. This givesS^j=S\\hat\{S\}\_\{j\}=S\. The result holds without imposing the mutual incoherence condition on the Hessian, because the SCAD derivative equals zero on the true support\. ∎

### S\.6\.8Proof of Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3)

In this section, we prove Theorem[3](https://arxiv.org/html/2607.03005#Thmtheorem3)from the graph\-level separation and threshold condition\.

###### Proof\.

Recall that the compatibility score for sourcessis the difference between the averaged empirical cross\-validation losses on the target domain\. Define

ℒ¯0:=12​∑r=12ℒ^0\(r\),ℒ¯s:=12​∑r=12ℒ^s\(r\)\.\\bar\{\\mathcal\{L\}\}\_\{0\}:=\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\hat\{\\mathcal\{L\}\}^\{\(r\)\}\_\{0\},\\qquad\\bar\{\\mathcal\{L\}\}\_\{s\}:=\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\hat\{\\mathcal\{L\}\}^\{\(r\)\}\_\{s\}\.Then

Δs=ℒ¯s−ℒ¯0=12​∑r=12\(ℒ^s\(r\)−ℒ^0\(r\)\)\.\\Delta\_\{s\}=\\bar\{\\mathcal\{L\}\}\_\{s\}\-\\bar\{\\mathcal\{L\}\}\_\{0\}=\\frac\{1\}\{2\}\\sum\_\{r=1\}^\{2\}\\left\(\\hat\{\\mathcal\{L\}\}^\{\(r\)\}\_\{s\}\-\\hat\{\\mathcal\{L\}\}^\{\(r\)\}\_\{0\}\\right\)\.On the event in Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5), we have

max1≤s≤S⁡\|Δs−ℰ​\(s\)\|≤C​ϵn\\max\_\{1\\leq s\\leq S\}\|\\Delta\_\{s\}\-\\mathcal\{E\}\(s\)\|\\leq C\\epsilon\_\{n\}and

cgap​νn\+C​ϵn<Cτ​σ^<\(1\+cgap\)​νn−C​ϵn\.c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\+C\\epsilon\_\{n\}<C\_\{\\tau\}\\hat\{\\sigma\}<\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\-C\\epsilon\_\{n\}\.Ifs∈𝒜hs\\in\\mathcal\{A\}\_\{h\}, thenℰ​\(s\)≤cgap​νn\\mathcal\{E\}\(s\)\\leq c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}, which implies

Δs≤ℰ​\(s\)\+C​ϵn≤cgap​νn\+C​ϵn<Cτ​σ^\.\\Delta\_\{s\}\\leq\\mathcal\{E\}\(s\)\+C\\epsilon\_\{n\}\\leq c\_\{\\mathrm\{gap\}\}\\nu\_\{n\}\+C\\epsilon\_\{n\}<C\_\{\\tau\}\\hat\{\\sigma\}\.This givess∈𝒜^s\\in\\hat\{\\mathcal\{A\}\}\. Ifs∉𝒜hs\\notin\\mathcal\{A\}\_\{h\}, thenℰ​\(s\)≥\(1\+cgap\)​νn\\mathcal\{E\}\(s\)\\geq\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}, which implies

Δs≥ℰ​\(s\)−C​ϵn≥\(1\+cgap\)​νn−C​ϵn\>Cτ​σ^\.\\Delta\_\{s\}\\geq\\mathcal\{E\}\(s\)\-C\\epsilon\_\{n\}\\geq\(1\+c\_\{\\mathrm\{gap\}\}\)\\nu\_\{n\}\-C\\epsilon\_\{n\}\>C\_\{\\tau\}\\hat\{\\sigma\}\.This givess∉𝒜^s\\notin\\hat\{\\mathcal\{A\}\}\. The event in Condition[5](https://arxiv.org/html/2607.03005#Thmcondition5)has probability tending to one, which provesℙ​\(𝒜^=𝒜h\)→1\\mathbb\{P\}\(\\hat\{\\mathcal\{A\}\}=\\mathcal\{A\}\_\{h\}\)\\to 1\. This completes the proof\. ∎