Pitfalls and Remedies for Multi-Task Bayesian Optimization

# Pitfalls and Remedies for Multi-Task Bayesian Optimization
Source: [https://arxiv.org/html/2607.09073](https://arxiv.org/html/2607.09073)
Carl Hvarfner Meta hvarfner@meta\.com &Sam Daulton Meta sdaulton@meta\.com &Max Balandat Meta balandat@meta\.com &Eytan Bakshy Meta ebakshy@meta\.com

###### Abstract

Bayesian optimization routinely warm\-starts a target experiment with data from related source tasks, and the multi\-task Gaussian process is the textbook surrogate for the job\. We revisit this default in a controlled setting and find that it misestimates the cross\-task correlation even in the simplest non\-trivial case, affinely related source and target tasks, where a working transfer\-learning method should obviously succeed\. We trace the failure to two independent structural mechanisms\. Per\-task standardization, the textbook fix for the affine slice ambiguity, propagates a finite\-sample alignment error into the recovered correlation\. The marginal likelihood itself identifies the correlation only at a per\-sample rate that a Gaussian process at non\-overlapping designs further dilutes\. We propose three conservative remedies that follow from the analysis: promoting per\-task means and scales to model parameters, restricting the task covariance to non\-negative correlations, and co\-locating part of the source and target designs\. Across synthetic multi\-task problems and surrogate\-based hyperparameter\-tuning transfer, these remedies recover the target\-only baseline on the simple instances, while the broader failure persists on harder instances and across most rank\-based and latent\-context variants\.

## 1Introduction

Bayesian optimization \(BO\)\[[10](https://arxiv.org/html/2607.09073#bib.bib10),[12](https://arxiv.org/html/2607.09073#bib.bib12),[27](https://arxiv.org/html/2607.09073#bib.bib27),[28](https://arxiv.org/html/2607.09073#bib.bib28)\]is a workhorse for sample\-efficient experimentation, and in production settings, target experiments rarely arrive in isolation\[[13](https://arxiv.org/html/2607.09073#bib.bib13),[21](https://arxiv.org/html/2607.09073#bib.bib21),[9](https://arxiv.org/html/2607.09073#bib.bib9)\]\. BO transfer learning \(BOTL\) is the default story whenever a target experiment has a related predecessor: gather the source data, fit a multi\-task Gaussian process \(MTGP\), and expect fewer target evaluations to find the optimum\. Reports on BOTL tend to emphasize positive results on curated suites\[[26](https://arxiv.org/html/2607.09073#bib.bib26),[4](https://arxiv.org/html/2607.09073#bib.bib4),[9](https://arxiv.org/html/2607.09073#bib.bib9)\], and library defaults\[[3](https://arxiv.org/html/2607.09073#bib.bib3),[11](https://arxiv.org/html/2607.09073#bib.bib11)\]present the Intrinsic Coregionalization Model \(ICM\)\-based MTGP as the go\-to surrogate\. Controlled comparisons against target\-only BO\[[8](https://arxiv.org/html/2607.09073#bib.bib8),[22](https://arxiv.org/html/2607.09073#bib.bib22)\], and direct audits of whether the standard model recovers correct task correlations, remain comparatively rare\.

Yet on two affinely related tasks drawn from a standard benchmark function, the textbook MTGP misestimates the cross\-task correlation: it attenuates the recovered correlation and, once more than one source is present, can even flip its sign\. Affinely related tasks are the textbook example a working transfer\-learning method must handle: the source is a perfect linear image of the target, so every standard MTGP variant should recover near\-perfect correlations and transfer should obviously help\. We find the opposite: similar pathologies persist not only for the textbook ICM but for nearly every multi\-task and rank\-based variant in common use, and across our multi\-task BO grid the textbook MTGP loses to a single\-task Gaussian process \(GP\) on most base functions – the textbook signature of*negative transfer*\[[19](https://arxiv.org/html/2607.09073#bib.bib19),[33](https://arxiv.org/html/2607.09073#bib.bib33)\]\. A method failing here is failing structurally, not because the task is hard; prior BOTL benchmarks report ICM\-based MTGPs on suites where the underlying transferability is itself uncertain, so a clean failure on affinely related tasks is diagnostic\. We trace the failure to a structural identifiability defect in the standard parameterization, not a fitting accident\.

The failure has two structurally distinct, independent causes: a finite\-sample per\-task*standardization*error that rides the affine reparameterization symmetry into the recovered correlation, and an information\-theoretic floor on*correlation inference*that a GP at non\-overlapping designs further dilutes\. Both bite at the budgets BOTL practice actually has \(Fig\.[2](https://arxiv.org/html/2607.09073#S4.F2)\); §[4](https://arxiv.org/html/2607.09073#S4)develops each in turn\.

#### Our contributions

are the following:

- •Two structural pitfalls of the textbook MTGP, with theory\.We isolate two independent mechanisms that make the textbook MTGP misestimate task correlations on affinely related tasks: a finite\-sample per\-task standardization error and an information\-theoretic floor on correlation inference that a non\-overlapping GP further dilutes\.
- •Three remedies that recover the target\-only baseline on simple instances\.We identify per\-task means and scales as model parameters, a non\-negativity constraint onρ\\rho, and co\-located source/target queries as the configuration that minimizes the inference\-side variance the two pitfalls leave on the table \(§[4\.3](https://arxiv.org/html/2607.09073#S4.SS3)\)\.
- •Empirical demonstration\.We evaluate the textbook ICM, several MTGP variants, and the proposed remedies on simple semi\-synthetic affine problems, on surrogate\-based LCBench hyperparameter optimization \(HPO\), and on surrogate\-based deep\-learning \(pd1\) and LLM \(ifeval\[[7](https://arxiv.org/html/2607.09073#bib.bib7)\]\) tuning benchmarks \(§[5](https://arxiv.org/html/2607.09073#S5)\)\.

## 2Background

#### Bayesian optimization and transfer learning\.

BO addresses sequential black\-box optimization of a noisy objectivef:𝒳→ℝf:\\mathcal\{X\}\\to\\mathbb\{R\}under a limited query budget: at each step a probabilistic surrogate \(typically a GP\) and an acquisition function decide the next𝒙∈𝒳\\bm\{x\}\\in\\mathcal\{X\}to evaluate from past observations𝒟t=\{\(𝒙i,yi\)\}i=1Nt\\mathcal\{D\}\_\{t\}=\\\{\(\\bm\{x\}\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{N\_\{t\}\}\. BOTL augments this withT−1T\-1source datasets\{𝒟\(s\)\}s=1T−1\\\{\\mathcal\{D\}^\{\(s\)\}\\\}\_\{s=1\}^\{T\-1\}on related but non\-identical objectivesfsf\_\{s\}from past or cheaper experiments; the surrogate is fit jointly across source and target so that source evaluations sharpen the target posterior whenever the underlying tasks are correlated\. The benefit hinges on whether the surrogate can both \(i\) recover the cross\-task correlation from finite source budgets and \(ii\) align source and target onto a common scale; the failure modes examined in this paper sit at exactly those two steps\.

#### GP regression\.

A GP\[[24](https://arxiv.org/html/2607.09073#bib.bib24),[29](https://arxiv.org/html/2607.09073#bib.bib29)\]is a distribution overf:𝒳→ℝf:\\mathcal\{X\}\\to\\mathbb\{R\}such that any finite set of evaluations is jointly Gaussian\. Given observations𝒚=f​\(𝑿\)\+𝜺\\bm\{y\}=f\(\\bm\{X\}\)\+\\bm\{\\varepsilon\}with𝜺∼𝒩​\(0,σnoise2​𝑰\)\\bm\{\\varepsilon\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\_\{\\mathrm\{noise\}\}\\bm\{I\}\)and a positive definite kernelkxk\_\{x\}, the posterior is again a GP, with predictive mean𝒌∗⊤​\(𝑲\+σnoise2​𝑰\)−1​𝒚\\bm\{k\}\_\{\*\}^\{\\top\}\(\\bm\{K\}\+\\sigma\_\{\\mathrm\{noise\}\}^\{2\}\\bm\{I\}\)^\{\-1\}\\bm\{y\}and variancekx​\(𝒙∗,𝒙∗\)−𝒌∗⊤​\(𝑲\+σnoise2​𝑰\)−1​𝒌∗k\_\{x\}\(\\bm\{x\}\_\{\*\},\\bm\{x\}\_\{\*\}\)\-\\bm\{k\}\_\{\*\}^\{\\top\}\(\\bm\{K\}\+\\sigma\_\{\\mathrm\{noise\}\}^\{2\}\\bm\{I\}\)^\{\-1\}\\bm\{k\}\_\{\*\}at a test input𝒙∗\\bm\{x\}\_\{\*\}, where𝒌∗=kx​\(𝒙∗,𝑿\)\\bm\{k\}\_\{\*\}=k\_\{x\}\(\\bm\{x\}\_\{\*\},\\bm\{X\}\)and𝑲\\bm\{K\}is the Gram matrix on the training inputs\. Hyperparameters are typically fit by maximizing the marginal log\-likelihood \(MLL\)\.

#### MTGP and the ICM\.

An MTGP\[[5](https://arxiv.org/html/2607.09073#bib.bib5)\]modelsTTcorrelated functions\{ft\}t=1T\\\{f\_\{t\}\\\}\_\{t=1\}^\{T\}jointly\. In the ICM parameterization\[[14](https://arxiv.org/html/2607.09073#bib.bib14)\], the cross\-task covariance is the Kronecker product of aT×TT\\times Ttask covariance𝑩\\bm\{B\}and a shared input kernelkxk\_\{x\}:Cov⁡\(ft​\(𝒙\),ft′​\(𝒙′\)\)=Bt​t′​kx​\(𝒙,𝒙′\)\\operatorname\{Cov\}\(f\_\{t\}\(\\bm\{x\}\),f\_\{t^\{\\prime\}\}\(\\bm\{x\}^\{\\prime\}\)\)=B\_\{tt^\{\\prime\}\}\\,k\_\{x\}\(\\bm\{x\},\\bm\{x\}^\{\\prime\}\), where the diagonal entryBt​tB\_\{tt\}is the per\-task signal variance and the off\-diagonalBt​t′B\_\{tt^\{\\prime\}\}is the cross\-task covariance; the unitless task\-correlation matrix isρt​t′=Bt​t′/Bt​t​Bt′​t′\\rho\_\{tt^\{\\prime\}\}=B\_\{tt^\{\\prime\}\}/\\sqrt\{B\_\{tt\}B\_\{t^\{\\prime\}t^\{\\prime\}\}\}\. Intuitively,kxk\_\{x\}controls smoothness within a task while𝑩\\bm\{B\}controls how strongly information is shared across tasks; the model is*separable*in input and task\. The MTGP additionally permits per\-task observation noiseσnoise,t2\\sigma\_\{\\mathrm\{noise\},t\}^\{2\}and per\-task constant meansμt\\mu\_\{t\}– the latter possible in principle but seldom discussed or implemented in practice\.

#### Task\-correlation matrix and standardization\.

The standard𝑩=𝑳​𝑳⊤\+diag​\(𝒗\)\\bm\{B\}=\\bm\{L\}\\bm\{L\}^\{\\top\}\+\\mathrm\{diag\}\(\\bm\{v\}\)parameterization\[[5](https://arxiv.org/html/2607.09073#bib.bib5)\]is the default in mainstream libraries\[[3](https://arxiv.org/html/2607.09073#bib.bib3),[11](https://arxiv.org/html/2607.09073#bib.bib11)\]\. Outputs are typically standardized either globally \(one\(μ^,σ^\)\(\\hat\{\\mu\},\\hat\{\\sigma\}\)across all tasks\) or per task \(\(μ^t,σ^t\)\(\\hat\{\\mu\}\_\{t\},\\hat\{\\sigma\}\_\{t\}\)from each task’s observations alone\)\. Standardization brings the outputs into the regime in which the GP’s default hyperparameter priors, centered on unit signal variance, are informative – the same normalized regime in which priors and initializations for high\-dimensional BO are calibrated\[[16](https://arxiv.org/html/2607.09073#bib.bib16),[20](https://arxiv.org/html/2607.09073#bib.bib20)\]– and is essentially required for stable MLL optimization on heterogeneously scaled tasks\.

#### The model’s whitened space\.

Standardization is effectively bi\-level\. A first layer is the raw standardization of the outputs by an outcome transform\(mt,st\)\(m\_\{t\},s\_\{t\}\), set to the empirical\(μ^t,σ^t\)\(\\hat\{\\mu\}\_\{t\},\\hat\{\\sigma\}\_\{t\}\)under per\-task standardization\. A second layer is the normalization the GP performs by learning a per\-task mean constantctc\_\{t\}and an ICM diagonalBt​tB\_\{tt\}\. The signal the GP treats as zero\-mean, unit\-variance per task is therefore the*whitened*signalzt​\(𝒙\)=\(\(yt−mt\)/st−ct\)/Bt​tz\_\{t\}\(\\bm\{x\}\)=\(\(y\_\{t\}\-m\_\{t\}\)/s\_\{t\}\-c\_\{t\}\)/\\sqrt\{B\_\{tt\}\}\. Learning the per\-task mean and signal is not sequential with fitting the remaining hyperparameters, but it is natural to think of the correlation parameters and lengthscales as learned in this whitened space: the off\-diagonalρt​t′\\rho\_\{tt^\{\\prime\}\}encodes correlation*between whitened signals*, not raw outputs\.

## 3Related Work

#### Pearson\-based \(correlation\-inferring\) MTGPs\.

The Linear Model of Coregionalization \(LMC\)/rank\-1 ICM originates in geostatistics\[[17](https://arxiv.org/html/2607.09073#bib.bib17),[14](https://arxiv.org/html/2607.09073#bib.bib14),[31](https://arxiv.org/html/2607.09073#bib.bib31)\]and entered the GP literature viaBonilla et al\. \[[5](https://arxiv.org/html/2607.09073#bib.bib5)\]; multi\-task BO followed\[[26](https://arxiv.org/html/2607.09073#bib.bib26),[23](https://arxiv.org/html/2607.09073#bib.bib23),[4](https://arxiv.org/html/2607.09073#bib.bib4)\]\. The class spans free and rank\-restricted𝑩\\bm\{B\}, latent\-context embeddings, pooled single\-task GPs \(𝑩=𝟏𝟏⊤\\bm\{B\}=\\bm\{1\}\\bm\{1\}^\{\\top\}\), and hierarchical priors on𝑩\\bm\{B\}– all infer task structure through pairwise Pearson covariance and inherit the affine identifiability defect ofAlvarez et al\. \[[1](https://arxiv.org/html/2607.09073#bib.bib1)\], Anderson and Rubin \[[2](https://arxiv.org/html/2607.09073#bib.bib2)\], Lopes and West \[[18](https://arxiv.org/html/2607.09073#bib.bib18)\]we sharpen below\.

#### Rank\-based ensembles\.

Per\-task GP ensembles combined by rank\-agreement weights, such as the ranking\-weighted GP ensemble \(RGPE\)\[[9](https://arxiv.org/html/2607.09073#bib.bib9),[34](https://arxiv.org/html/2607.09073#bib.bib34)\], sidestep the joint task covariance entirely and instead reweight independently fit per\-task GPs\.

#### Distribution\-matching transforms\.

A per\-task copulaΦ−1∘F^t\\Phi^\{\-1\}\\\!\\circ\\hat\{F\}\_\{t\}absorbs any monotone per\-task map, and a shared model is fit inzz\-space\[[25](https://arxiv.org/html/2607.09073#bib.bib25)\]; this is the basis of the Gaussian Copula Process \(GCP\)\.

#### Other paradigms\.

Offline meta\-learning and amortized policies\[[32](https://arxiv.org/html/2607.09073#bib.bib32),[30](https://arxiv.org/html/2607.09073#bib.bib30),[21](https://arxiv.org/html/2607.09073#bib.bib21)\]sidestep per\-target fitting at the cost of11–22orders of magnitude more source data; pre\-trained\-GP work moves the identifiability question to meta\-train time\.

## 4Two pitfalls of affine MTGPs

We present our two structural pitfalls in the simplest imaginable transfer setting: affinely related source and target tasks\. Because the source is a perfect linear image of the target – the most generous case for transfer, with true task correlation exactly±1\\pm 1– any failure to recover the correlation here is a structural defect rather than task difficulty\.

#### Setup: affine source\-target family\.

Letf∼𝒢​𝒫​\(0,kx\)f\\sim\\mathcal\{GP\}\(0,k\_\{x\}\)be a latent function with normalized base kernelkx​\(𝒙,𝒙\)=1k\_\{x\}\(\\bm\{x\},\\bm\{x\}\)=1and defineTTtasks byyt​\(𝒙\)=at​f​\(𝒙\)\+bt\+εty\_\{t\}\(\\bm\{x\}\)=a\_\{t\}\\,f\(\\bm\{x\}\)\+b\_\{t\}\+\\varepsilon\_\{t\}, whereat\>0a\_\{t\}\>0,bt∈ℝb\_\{t\}\\in\\mathbb\{R\}, andεt∼𝒩​\(0,σnoise,t2\)\\varepsilon\_\{t\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\mathrm\{noise\},t\}^\{2\}\)independently fort=1,…,Tt=1,\\ldots,T\. Assume the high signal\-to\-noise ratio \(SNR\) regime:σnoise,t2≪at2\\sigma\_\{\\mathrm\{noise\},t\}^\{2\}\\ll a\_\{t\}^\{2\}\. Consider an ICM model\[[14](https://arxiv.org/html/2607.09073#bib.bib14)\]with task covariance𝑩∈ℝT×T\\bm\{B\}\\in\\mathbb\{R\}^\{T\\times T\}whose diagonal entriesBt​tB\_\{tt\}are the per\-task signal variances \(Bt​t=at2B\_\{tt\}=a\_\{t\}^\{2\}at the truth\), per\-task constant meansμt\\mu\_\{t\}, and shared normalized base kernelkxk\_\{x\}, so that the joint covariance is

Cov⁡\[yt​\(𝒙\),yt′​\(𝒙′\)\]=Bt​t′​kx​\(𝒙,𝒙′\)\+σnoise,t2​δt​t′​δ𝒙​𝒙′\.\\operatorname\{Cov\}\\bigl\[y\_\{t\}\(\\bm\{x\}\),\\,y\_\{t^\{\\prime\}\}\(\\bm\{x\}^\{\\prime\}\)\\bigr\]=B\_\{tt^\{\\prime\}\}\\,k\_\{x\}\(\\bm\{x\},\\bm\{x\}^\{\\prime\}\)\+\\sigma\_\{\\mathrm\{noise\},t\}^\{2\}\\,\\delta\_\{tt^\{\\prime\}\}\\,\\delta\_\{\\bm\{x\}\\bm\{x\}^\{\\prime\}\}\.\(1\)Identifyingρ\\rho, not𝑩\\bm\{B\}, is the goal; pinningBtarget,target=1B\_\{\\mathrm\{target,target\}\}=1removes the residual scale ambiguity in the target row/column\. While we develop the analysis on the affine source\-target family for tractability, the underlying mechanisms are model\-agnostic and apply broadly to any multi\-task surrogate that estimates per\-task scale and pairwise covariance from finite samples\.

As we will show, the textbook MTGP fails on affinely related tasks for two structurally distinct, independent reasons\.*Aligning*the source carries a finite\-sample per\-task standardization error that propagates into the recovered correlation even whenρ⋆=1\\rho^\{\\star\}=1\(§[4\.1](https://arxiv.org/html/2607.09073#S4.SS1)\)\.*Inferring*the correlation is information\-bound: the data identifies the task correlation poorly even when alignment is exact \(§[4\.2](https://arxiv.org/html/2607.09073#S4.SS2)\)\. We close in §[4\.3](https://arxiv.org/html/2607.09073#S4.SS3)with three configuration choices that shrink the regime where either bites\.

Fig\.[1](https://arxiv.org/html/2607.09073#S4.F1)makes the joint failure mode visible on a 1D Forrester benchmark with two affinely related tasks: the textbook ICM \(panel c\) attenuates the recovered correlationρ^s​t\\hat\{\\rho\}\_\{st\}and pulls the target posterior the wrong way, while the recommended configuration \(panel d\) – per\-task means, free𝑩\\bm\{B\}, per\-task noise, and per\-task standardization – recovers the structure\.

![Refer to caption](https://arxiv.org/html/2607.09073v1/x1.png)Figure 1:Four GP configurations on affinely related tasks\.Target tasky0=fy\_\{0\}=fon the Forrester minimization benchmark and an affinely related sourcey1=1\.7​f−4y\_\{1\}=1\.7\\,f\-4, withN0=5N\_\{0\}=5target points andN1=4N\_\{1\}=4source points\.Top row: GP posterior mean and±2​σ\\pm 2\\sigmabands overyy; insets report the learned correlationρ0,1\\rho\_\{0,1\}and the lengthscaleℓ\\ell\.Bottom row: the same elements in the GP’s whitened working spaceztz\_\{t\}, where tasks that share affine structure should overlap\. Panel \(c\): the textbook MTGP attenuates the recovered correlation to\+0\.88\+0\.88\. Panel \(d\): the recommended configuration recovers the structure\.
### 4\.1Standardization is hard even whenρ⋆=1\\rho^\{\\star\}=1

Suppose the source and target are perfectly correlated\. The MTGP still has to align the source onto the target’s scale by estimating per\-task constants\(μ^s,σ^s\)\(\\hat\{\\mu\}\_\{s\},\\hat\{\\sigma\}\_\{s\}\)from the sameNsN\_\{s\}source observations it then transfers\. The standardization estimates\(μ^s,σ^s\)\(\\hat\{\\mu\}\_\{s\},\\hat\{\\sigma\}\_\{s\}\)themselves carry finite\-sample error of order1/Ns1/\\sqrt\{N\_\{s\}\}\. Because per\-task standardization is an affine reparameterization, this error transfers directly into the source–target block of the task covariance𝑩\\bm\{B\}, biasing the recovered correlationρ^s​t\\hat\{\\rho\}\_\{st\}at the same1/Ns1/\\sqrt\{N\_\{s\}\}rate – even when the true correlation is exactly11\. Per\-task standardization plays a dual role here: it is simultaneously the textbook fix for the affine slice ambiguity and the source of the noise we are trying to remove\.

###### Proposition 1\(Standardization\-error propagation underρ⋆=1\\rho^\{\\star\}=1\)\.

Under the affine setup withρs​t⋆=1\\rho^\{\\star\}\_\{st\}=1, let\(μ^s,σ^s\)\(\\hat\{\\mu\}\_\{s\},\\hat\{\\sigma\}\_\{s\}\)be the empirical mean and standard deviation ofNsN\_\{s\}source observations and let\(α^s,β^s\)=\(1/σ^s,μ^s\)\(\\hat\{\\alpha\}\_\{s\},\\hat\{\\beta\}\_\{s\}\)=\(1/\\hat\{\\sigma\}\_\{s\},\\hat\{\\mu\}\_\{s\}\)be the per\-task standardization map\. Then

μ^s=μs\+𝒩​\(0,σs2Ns\)\+op​\(Ns−1/2\),σ^s=σs\+𝒩​\(0,σs22​Ns\)\+op​\(Ns−1/2\),\\hat\{\\mu\}\_\{s\}=\\mu\_\{s\}\+\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{\\sigma\_\{s\}^\{2\}\}\{N\_\{s\}\}\\bigr\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\),\\qquad\\hat\{\\sigma\}\_\{s\}=\\sigma\_\{s\}\+\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{\\sigma\_\{s\}^\{2\}\}\{2N\_\{s\}\}\\bigr\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\),and the recovered source–target correlation under per\-task standardization satisfies

ρ^s​t/ρs​t⋆=1\+𝒩​\(0,12​Ns\)\+op​\(Ns−1/2\)\.\\hat\{\\rho\}\_\{st\}/\\rho^\{\\star\}\_\{st\}\\;=\\;1\\;\+\\;\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{1\}\{2N\_\{s\}\}\\bigr\)\\;\+\\;o\_\{p\}\(N\_\{s\}^\{\-1/2\}\)\.Proof in App\.[C](https://arxiv.org/html/2607.09073#A3)\.

Notably, the relative error inρ^s​t\\hat\{\\rho\}\_\{st\}is governed by the smaller of the two source\-side sample sizes: doublingNsN\_\{s\}only halves the variance, so the bias persists deep into the BO budget regime\. The practical consequence is visible in Fig\.[1](https://arxiv.org/html/2607.09073#S4.F1)\(c\): incorrect offset estimation lowers the recovered correlation and shrinks the effective influence of the source task’s best point on the target posterior\. At the small source budgets BO practice actually has, replacing the true per\-task scale by its empirical estimate collapses the recovered maximum\-likelihood estimate \(MLE\) well below the true correlation on otherwise identical data; the bias only fades once the source budget grows by an order of magnitude or more\. Standardization is the textbook fix for the affine slice ambiguity, but at BO source budgets it transports the same1/Ns1/\\sqrt\{N\_\{s\}\}noise intoρ^s​t\\hat\{\\rho\}\_\{st\}that the slice was supposed to remove\.

### 4\.2Inferring the task correlation is hard

Suppose now that alignment is given – standardization is exact – and the only unknown is the cross\-task correlationρ\\rho\. The per\-sample Fisher information is bounded above by the bivariate\-Gaussian rate, which both Pearson and Spearman estimators saturate up to a small constant; that rate then degrades further on a GP fit at non\-overlapping designs\. The data identifiesρ\\rhopoorly even when everything else is given\.

###### Proposition 2\(Per\-sampleρ\\rho\-detection floor\)\.

For two tasks observed atNNshared design points under the affine setup, the maximum\-likelihood Pearson estimator satisfies

Var⁡\[ρ^\]≥\(1−ρ2\)2N​\(1\+ρ2\),\\operatorname\{Var\}\[\\hat\{\\rho\}\]\\geq\\frac\{\(1\-\\rho^\{2\}\)^\{2\}\}\{N\(1\+\\rho^\{2\}\)\},with equality in the noiseless limit\. The Spearman rank estimatorρ^S\\hat\{\\rho\}\_\{S\}inherits the sameΘ​\(1/N\)\\Theta\(1/\\sqrt\{N\}\)per\-sample standard error \(SE\) up to the asymptotic relative efficiency \(ARE\) factor9/π2≈0\.9129/\\pi^\{2\}\\approx 0\.912\[[6](https://arxiv.org/html/2607.09073#bib.bib6)\]\. For mismatched designs\|Xs\|=Ns≤\|Xt\|=Nt\|X\_\{s\}\|=N\_\{s\}\\leq\|X\_\{t\}\|=N\_\{t\}the smaller task dominates the rate\. Full proof and Spearman ARE in App\.[D](https://arxiv.org/html/2607.09073#A4)\.

###### Proposition 3\(GP information dilution\)\.

For shared input kernelkxk\_\{x\}onΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}with correlation lengthℓ\\ell, the Fisher information of anNN\-point GP fit obeys

IN​\(ρ\)≤Neff​1\+ρ2\(1−ρ2\)2,Neff≤min⁡\{N,\(diam​\(Ω\)/ℓ\)d\}:I\_\{N\}\(\\rho\)\\leq N\_\{\\mathrm\{eff\}\}\\,\\frac\{1\+\\rho^\{2\}\}\{\(1\-\\rho^\{2\}\)^\{2\}\},\\qquad N\_\{\\mathrm\{eff\}\}\\leq\\min\\\{N,\(\\mathrm\{diam\}\(\\Omega\)/\\ell\)^\{d\}\\\}:a GP identifiesρ\\rhofrom at mostNeffN\_\{\\mathrm\{eff\}\}effective paired observations, strictly fewer than theNNa naive paired test would assume\. Proof in App\.[E](https://arxiv.org/html/2607.09073#A5)\.

Resolving a moderate correlation to useful precision requires far more paired observations than BO source budgets provide: the Cramér–Rao \(CR\) standard error is large at BO budgets, and the GP dilution above shrinks the effective sample count further\. As a result the MTGP struggles to tell a moderately useful source from a useless one at the budgets BOTL practice actually has \(Fig\.[2](https://arxiv.org/html/2607.09073#S4.F2)\) – the inference\-side mechanism behind the negative\-transfer outcomes observed empirically in §[5](https://arxiv.org/html/2607.09073#S5)\.

![Refer to caption](https://arxiv.org/html/2607.09073v1/x2.png)Figure 2:MTGP correlation inference at typical BO source budgets\.1D Forrester target under minimization with two GP\-perturbed source tasksgt=f\+htg\_\{t\}=f\+h\_\{t\}, calibrated to a true target–source correlationρ⋆=0\.8\\rho^\{\\star\}=0\.8and fitted on5\+4\+45\{\+\}4\{\+\}4stratified\-uniform observations\.Left:a representative seed, where the MLE recoversρ^0,1=\+1\.00\\hat\{\\rho\}\_\{0,1\}=\+1\.00andρ^0,2=−0\.95\\hat\{\\rho\}\_\{0,2\}=\-0\.95\.Middle:the MLL as a function of the two target–source correlations is broad and ridge\-like, so unlucky designs land the optimum almost anywhere along that ridge\.Right:MLE estimates across2525independent seeds; sign flips and corner saturation are common, and correlation inference at these budgets is unreliable in our setting\.
### 4\.3Three remedies

The analysis above motivates three conservative configuration choices\. None is novel; each is a practical necessity at the source budgets BO practice has\. Remedy 1 promotes the per\-task offset and scale to model parameters, removing the empirical\-standardization noise isolated in §[4\.1](https://arxiv.org/html/2607.09073#S4.SS1)\. Remedies 2 and 3 then reduce the correlation\-inference variance that the bound of Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)and the GP dilution of Prop\.[3](https://arxiv.org/html/2607.09073#Thmproposition3)together leave on the table\.

#### Remedy 1: per\-task means and per\-task scales as model parameters\.

Two affinely related tasks differ in raw range but become identical up to noise after per\-task standardization; this visual identity is what tempts practitioners to trust empirical standardization in the first place\. Even small per\-task offsets – the kind visible in Fig\.[1](https://arxiv.org/html/2607.09073#S4.F1)– nevertheless destroy the model’s correlation estimate when only empirical standardization is used\. Promoting the per\-task offset and scale to model parametersμt,Bt​t\\mu\_\{t\},B\_\{tt\}, fit jointly with the rest of the model, lets the marginal likelihood balance their estimation against the data instead of paying the finite\-sample alignment cost up front\. Per\-task scales necessarily apply to the source tasks, though the target task may still carry a fixed output scale\. This remedy is not novel – it is well\-known in the GP literature – but it is important enough to state explicitly, since the remaining remedies build on it\.

#### Remedy 2: positive\-correlation restriction\.

Joint inference of the task\-covariance parameters is overparameterized at smallNN, and the covariance entries are precisely the high\-variance ones \(§[4\.2](https://arxiv.org/html/2607.09073#S4.SS2)\); the positivity constraint is therefore a practical necessity, not an innovation\. Concretely, constrainρs​t≥0\\rho\_\{st\}\\geq 0\(e\.g\., parameterise𝑩=𝑳​𝑳⊤\+diag​\(𝒗\)\\bm\{B\}=\\bm\{L\}\\bm\{L\}^\{\\top\}\+\\mathrm\{diag\}\(\\bm\{v\}\)with𝑳\\bm\{L\}entrywise non\-negative\)\. Allowing free\-form, signed correlations rests on faulty logic: a meaningfully negatively related source is not something a practitioner would realistically choose to include, yet leaving the sign free exposes the model to the wrong\-sign mode that finite\-sample noise can flip the MLE into \(Fig\.[2](https://arxiv.org/html/2607.09073#S4.F2)\), and the CR floor of Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)leaves negatively correlated sources unidentifiable in any case\. We show in the results that this wrong\-sign failure occurs in practice, and the appendix correlation densities \(App\. Fig\.[7](https://arxiv.org/html/2607.09073#A2.F7)\) make it explicit\.

#### Remedy 3: co\-locating source and target observations\.

Each shared input does double duty: it lifts the Fisher information forρ\\rhoto the paired\-test rate of Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)that a non\-overlapping GP fit forfeits, and, when tasks are related, induces correlated bias across per\-task offset/scale estimators – a variance\-reduction effect that helps Remedy 2 too\. For correlation inference, re\-evaluating*any*source point on the target is beneficial; re\-evaluating a strong source point can additionally be desirable in its own right\. Co\-locating does not by itself reduce input\-space coverage of the target, but it reduces*overall*coverage across tasks when the tasks are in fact highly correlated – something we cannot count on in advance\. This exposes a trade\-off between accurate correlation inference and the coverage that accurate inference would earn\. On real data, this remedy appears as the source\-design\-overlap ICM variants on LCBench, whose effect on the inferred correlation is shown in App\. Fig\.[7](https://arxiv.org/html/2607.09073#A2.F7)\(§[5\.2](https://arxiv.org/html/2607.09073#S5.SS2)\)\.

## 5Experiments

We stress\-test the analysis on a multi\-task BO grid covering both synthetic test functions \(§[5\.1](https://arxiv.org/html/2607.09073#S5.SS1)\) and surrogate\-based HPO transfer on LCBench \(§[5\.2](https://arxiv.org/html/2607.09073#S5.SS2)\), pd1, andifeval\(§[5\.3](https://arxiv.org/html/2607.09073#S5.SS3)\)\. Affinely related tasks are the canonical case where transfer “obviously should work”: the achievable upper bound on performance is known by construction, since an oracle given the per\-task standardization parameters recovers the target with no transfer cost\. A method failing here is failing structurally, not because the task is hard; prior BOTL benchmarks\[[8](https://arxiv.org/html/2607.09073#bib.bib8),[22](https://arxiv.org/html/2607.09073#bib.bib22),[13](https://arxiv.org/html/2607.09073#bib.bib13)\]report ICM\-based MTGPs on suites where the underlying transferability is itself uncertain, and few of them cleanly evaluate the model class on a setting where transfer should obviously succeed\. We benchmark against the implicit affine ceiling rather than running the oracle directly\.

We benchmark a Vanilla GP \(target\-only\), three ICM variants – ICM \(Shared\), ICM \(Positive\), and ICM \(Free\) – and RGPE\[[9](https://arxiv.org/html/2607.09073#bib.bib9)\], all usingq​LogNEIq\\text\{LogNEI\}\[[35](https://arxiv.org/html/2607.09073#bib.bib35)\]and dimension\-scaling lengthscale priors\[[16](https://arxiv.org/html/2607.09073#bib.bib16)\], alongside QuantileBO\[[25](https://arxiv.org/html/2607.09073#bib.bib25)\]and Adaptive Bayesian Linear Regression \(ABLR\)\[[21](https://arxiv.org/html/2607.09073#bib.bib21)\]\. The three ICM variants isolate the model\-side choices of §[4\.3](https://arxiv.org/html/2607.09073#S4.SS3)\.*ICM \(Shared\)*is the textbook configuration: global standardization, a single mean across tasks, and a free task covariance𝑩\\bm\{B\}\.*ICM \(Positive\)*adds per\-task means and per\-task scales \(Remedy 1\) and a non\-negativity constraint onρ\\rho\(Remedy 2\)\.*ICM \(Free\)*keeps per\-task means and scales but drops the sign constraint, allowingρ\\rhoto be negative\. QuantileBO is dropped onifeval, where its dimension\-1919task\-correlation matrix is not positive definite; two source\-design\-overlap ICM variants \(Remedy 3\) appear only in the appendix ablations on the synthetic grid and LCBench, since the initialization across methods is otherwise not identical\. The method\-to\-configuration map is given in App\.[A](https://arxiv.org/html/2607.09073#A1)\.

### 5\.1Synthetic test functions

Per replication we draw a positive\-affine source familygt​\(𝒙\)=at​f​\(𝒙\)\+btg\_\{t\}\(\\bm\{x\}\)=a\_\{t\}f\(\\bm\{x\}\)\+b\_\{t\}from the affine setup withNs=12N\_\{s\}=12observations per source task\. The target is initialized with Sobol points \(ninit=d\+1n\_\{\\mathrm\{init\}\}=d\+1\); the full configuration is in App\.[A\.1](https://arxiv.org/html/2607.09073#A1.SS1)\(per\-method ablation in Fig\.[10](https://arxiv.org/html/2607.09073#A2.F10)\)\. Even the textbook ICM \(Shared\) trails the target\-only Vanilla GP on the simple instances\.

![Refer to caption](https://arxiv.org/html/2607.09073v1/x3.png)Figure 3:Synthetic affine\-TL grid\.Best\-observed target value vs\. acquisition iteration on Hartmann\-3, Hartmann\-6, Ackley\-6, Levy\-4, and Levy\-5, mean±1\\pm 1SE across9999seeds, with a final Relative Rank panel aggregating rank across the five base functions \(lower is better\)\. The textbook ICM \(Shared\) trails the target\-only Vanilla GP on the high\-SNR Hartmann instances, the regime where transfer should be easiest; the body \(§[5\.1](https://arxiv.org/html/2607.09073#S5.SS1)\) reads off the full method ordering\. HyperBO\[[32](https://arxiv.org/html/2607.09073#bib.bib32)\]is excluded as its design regime \(∼\\sim24 tasks, hundreds of evaluations per task\) lies far outside ours\.Fig\.[3](https://arxiv.org/html/2607.09073#S5.F3)matches the structural picture of §[4](https://arxiv.org/html/2607.09073#S4)on the favorable simple instances, and we read it as a minimum bar: a method that fails here cannot be trusted to transfer in the wild\. The simple synthetic experiments make the issues of the transfer methods immediately visible\. The textbook ICM \(Shared\) struggles to beat the Vanilla GP until the source budget grows large enough: global standardization is not in itself harmful, but it leaves the model dependent on downstream hyperparameter inference that is fragile at the budgets we test, and RGPE inherits the same per\-task standardization sensitivity through its base learners\. More broadly, all of these methods rely on aligning the tasks to some degree, and a higher\-complexity method \(e\.g\. QuantileBO, ABLR\) does not necessarily transfer better than a simpler one\. The per\-task ICM variants \(ICM \(Positive\) and ICM \(Free\), which add Remedy 1 on top of the textbook configuration\) recover the Vanilla GP on the positive\-affine instances; ICM \(Positive\) additionally enforces Remedy 2 and is the more robust of the two on a per\-seed basis\. The wrong\-sign mode is removed, but the Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)floor remains, and on the harder Ackley and Levy instances the gap to Vanilla narrows but does not close\.

### 5\.2Surrogate\-based transfer: LCBench HPO

LCBench\[[36](https://arxiv.org/html/2607.09073#bib.bib36)\]provides HPO tasks over a shared77\-D neural\-network configuration space\. We use it as a semi\-synthetic testbed: each target is a Gaussian\-process surrogate fitted to a real LCBench dataset, and its sources are small affine transformations of that same surrogate, so the underlying task structure is positive\-affine by construction\. Each target is warm\-started bytwo positively correlated LCBench sources\(full setup in App\.[A\.2](https://arxiv.org/html/2607.09073#A1.SS2)\)\.

![Refer to caption](https://arxiv.org/html/2607.09073v1/x4.png)Figure 4:LCBench transfer, two positively correlated sources\.Best\-observed target validation accuracy vs\. BO iteration for seven LCBench datasets \(higher is better\); each target is warm\-started with two positively correlated LCBench source tasks, mean±1\\pm 1SE across9999seeds\. The final panel shows each method’s relative rank aggregated across the seven datasets \(lower is better\)\. ICM \(Positive\) and ICM \(Free\) reach higher best\-observed accuracy and lower average rank than the Vanilla GP and QuantileBO\.#### Reading\.

Figure[4](https://arxiv.org/html/2607.09073#S5.F4)plots, per dataset, the best target validation accuracy observed so far versus BO iteration; the final panel aggregates each method’s per\-seed rank across all seven datasets\. The per\-task\-standardized ICM \(Positive\) and ICM \(Free\) attain the lowest average rank, ahead of the Vanilla GP and QuantileBO, with ICM \(Positive\) best overall\. The textbook ICM \(Shared\) lands mid\-pack and does not match the per\-task\-standardized variants, consistent with the standardization pitfall of §[4\.1](https://arxiv.org/html/2607.09073#S4.SS1)\. The appendix probes correlation inference directly \(App\. Fig\.[7](https://arxiv.org/html/2607.09073#A2.F7)\): when a source is genuinely uninformative, the free\-sign ICM \(Free\) leaves the inferred correlation diffuse and unidentified \(the floor of Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)\), while the sign\-constrained ICM \(Positive\) splits its mass between0and\+1\+1– the wrong\-sign mode that Remedy 2 removes\. Additional LCBench source configurations and per\-method ablations are in Fig\.[6](https://arxiv.org/html/2607.09073#A2.F6),[8](https://arxiv.org/html/2607.09073#A2.F8),[11](https://arxiv.org/html/2607.09073#A2.F11),[12](https://arxiv.org/html/2607.09073#A2.F12), and[13](https://arxiv.org/html/2607.09073#A2.F13)\.

### 5\.3Surrogate\-based HPO: pd1 andifeval

We complement the synthetic grid with surrogate\-based HPO transfer whose task structure is genuinely, rather than affinely, correlated\. pd1\[[32](https://arxiv.org/html/2607.09073#bib.bib32)\]contributes two deep\-learning tuning targets, cifar10 and mnist, over a44\-D hyperparameter cube, each warm\-started by a*single highly correlated source*\(cifar10←\\leftarrowcifar100, mnist←\\leftarrowfashion\) withNs=16N\_\{s\}\{=\}16source observations \(App\.[A\.3](https://arxiv.org/html/2607.09073#A1.SS3)\)\.ifeval\[[7](https://arxiv.org/html/2607.09073#bib.bib7)\]scores LLM outputs against verifiable instruction\-following constraints; the optimization variable is a1919\-dimensional data\-mixture simplex over instruction\-tuning sources used to fine\-tune Qwen2\.5, with two model\-scale source fidelities \(0\.5B and 3B\) warm\-starting the 7B target andNs=16N\_\{s\}\{=\}16source observations per fidelity\. Since direct evaluation at every BO step is infeasible, the optimized objective is a GP surrogate: a sample path of an MTGP jointly fit to the real held\-out data \(App\.[A\.4](https://arxiv.org/html/2607.09073#A1.SS4)\)\. QuantileBO is absent onifeval\(see the methods of §[5](https://arxiv.org/html/2607.09073#S5)\)\.

![Refer to caption](https://arxiv.org/html/2607.09073v1/x5.png)Figure 5:Surrogate\-based HPO transfer: pd1 andifeval\.Best\-observed target value vs\. BO iteration for two pd1 targets \(cifar10 \(WRN\), mnist \(CNN\)\) andifeval, with a final Relative Rank panel aggregating rank across the three tasks \(mean±1\\pm 1SE over9999seeds\)\. pd1 uses one highly correlated source per target;ifevaluses the 0\.5B and 3B model\-scale sources\. QuantileBO appears on the pd1 panels but notifeval\(see §[5](https://arxiv.org/html/2607.09073#S5)\)\. Per\-target pd1 breakdowns are in Fig\.[9](https://arxiv.org/html/2607.09073#A2.F9)\.#### Reading\.

The combined view \(Fig\.[5](https://arxiv.org/html/2607.09073#S5.F5)\) aggregates rank across all three surrogate\-based tasks and reads off whether the per\-task\-standardized ICM variants retain their advantage on genuinely non\-affine transfer\. ICM \(Positive\) and ICM \(Free\) each rank among the stronger configurations, with no single configuration dominating throughout, consistent with the heterogeneity expected on real HPO data; the per\-target pd1 breakdowns appear in Fig\.[9](https://arxiv.org/html/2607.09073#A2.F9)\.

### 5\.4Effect of co\-location

We isolate Remedy 3 by varying how many of theNt=8N\_\{t\}=8target observations on Ackley\-5 \(a55\-D instance, distinct from the grid’s Ackley\-6\) and Hartmann\-6 are placed on randomly chosen source design points \(Tab\.[1](https://arxiv.org/html/2607.09073#S5.T1);gt=f\+htg\_\{t\}=f\+h\_\{t\},ρ⋆=0\.80\\rho^\{\\star\}=0\.80,Ns=8N\_\{s\}=8source observations per task\)\. Without co\-location the recovered correlation behaves close to a coin flip, with frequent sign flips and±1\\pm 1corner saturation; co\-locating more of the target budget onto source design points steadily raises the mean recovered correlation and sharply cuts the sign\-flip rate, the empirical translation of Prop\.[3](https://arxiv.org/html/2607.09073#Thmproposition3)with no model change\. Notably, this improvement in correlation inference holds even though the overlap points are chosen at random; selecting the strongest source points for re\-evaluation should improve it further\. The corresponding effect on optimization performance is inconclusive and is reported in full in the appendix ablations \(App\.[B](https://arxiv.org/html/2607.09073#A2)\): the source\-design\-overlap ICM variants \(Fig\.[10](https://arxiv.org/html/2607.09073#A2.F10),[11](https://arxiv.org/html/2607.09073#A2.F11),[12](https://arxiv.org/html/2607.09073#A2.F12),[13](https://arxiv.org/html/2607.09073#A2.F13)\) do not consistently beat their non\-overlapping counterparts, exposing the inference\-versus\-coverage trade\-off of §[4\.3](https://arxiv.org/html/2607.09073#S4.SS3)\.

Table 1:Effect of co\-locating target observations on source\-design points\.Mean inferred task correlationρ^\\hat\{\\rho\}, fraction of seeds where at least oneρ^0,t\\hat\{\\rho\}\_\{0,t\}has the wrong sign, fraction within±0\.2\\pm 0\.2ofρ⋆\\rho^\{\\star\}on both axes, and fraction saturating at a±1\\pm 1corner\. Trueρ⋆=0\.80\\rho^\{\\star\}=0\.80;Nt=8N\_\{t\}=8,Ns=8N\_\{s\}=8per source; aggregated over1010seeds\.

## 6Limitations

This work isolates a previously underexplored failure mode of multi\-task GP transfer learning rather than fully solving it\. Our real\-world evaluation is limited, and we do not claim a decisive solution: the three remedies are conservative fixes for foundational problems in BOTL, restoring the Vanilla GP on simple affine instances without closing the gap on harder instances or alternative variants\. The affine source\-target family is a reasoning tool rather than a restriction of the analysis: it is the simplest setting where the failure modes are cleanly attributable\. The same mechanisms apply more broadly; a fully general analysis across non\-affine task families is left to future work\.

## 7Conclusions and Future Work

We studied transfer learning in BO and isolated two structural pitfalls with substantial practical impact on BOTL: finite\-sample standardization noise that propagates into the recovered task correlation \(Prop\.[1](https://arxiv.org/html/2607.09073#Thmproposition1)\), and an information\-theoretic floor on correlation inference \(Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)\) that a GP at non\-overlapping designs further dilutes \(Prop\.[3](https://arxiv.org/html/2607.09073#Thmproposition3)\)\. We proposed three conservative remedies – per\-task means and scales as model parameters, a non\-negativity constraint on the task correlation, and co\-locating source and target queries – that address these problems and robustify BOTL more generally, recovering the target\-only baseline on the simple affine instances and improving on existing transfer\-learning methods in surrogate\-based HPO transfer; on harder instances and on most rank\-based and latent\-context variants the failure persists\. We also proposed an experimental setup that exposes these issues in a clear, controlled way, which we hope future transfer\-learning work can build on\.

We believe that greater awareness of these pitfalls will enable more robust transfer learning in BO\. Better\-conditioned priors on the task covariance, identifiability\-aware fitting procedures, and principled rules for declining transfer when the source\-side budget is too small to be informative remain open\.

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## Appendix ATasks

#### Method set and naming\.

The main figures use a lean method set: Vanilla GP \(target\-only\), ICM \(Shared\), ICM \(Positive\), ICM \(Free\), RGPE\[[9](https://arxiv.org/html/2607.09073#bib.bib9)\], ABLR\[[21](https://arxiv.org/html/2607.09073#bib.bib21)\], and QuantileBO\[[25](https://arxiv.org/html/2607.09073#bib.bib25)\]\(the Gaussian\-copula\-prior method; dropped onifeval, see §[5](https://arxiv.org/html/2607.09073#S5)\)\. The appendix ablation figures \(App\.[B](https://arxiv.org/html/2607.09073#A2)\) show the ICM family only: the three base ICM configurations \(Shared, Positive, Free\) plus the two source\-design\-overlap ICM variants of Remedy 3, on the synthetic grid and LCBench\. The three ICM configurations differ only in their standardization and task\-covariance settings:ICM \(Shared\)is the textbook configuration – global standardization, a single shared mean across tasks, and a non\-negativity\-constrained task covariance;ICM \(Positive\)uses per\-task standardization \(per\-task means and scales, Remedy 1\) with a non\-negativity constraint on the task correlations \(Remedy 2\);ICM \(Free\)keeps the per\-task means and scales but frees the sign of the task correlations\. The two source\-design\-overlap variants \(Remedy 3\) are the ICM \(Positive\) and ICM \(Free\) configurations with part of the target budget re\-evaluated on source design points\. All methods runq​LogNEIq\\text\{LogNEI\}with the surrogate refit after every step; for MTGP variants the acquisition pins the task column to the target\.

### A\.1Synthetic affine\-TL benchmark

Domain\.Five standard BoTorch test functions used as the canonical targetff: Hartmann3 \(d=3d\{=\}3\), Hartmann6 \(d=6d\{=\}6\), Ackley \(d=6d\{=\}6\), Levy \(d=4d\{=\}4\), Levy \(d=5d\{=\}5\)\. Each function is evaluated at its natural dimension\. Per\-task signal scaleσf\\sigma\_\{f\}is the empirical standard deviation offfon a Sobol probe\.

Source\-task generation\.Per replication,T−1T\{\-\}1source tasks are drawn as positive\-affine perturbationsgt​\(𝒙\)=at​σf​f​\(𝒙\)\+at​σf​bt\+εtg\_\{t\}\(\\bm\{x\}\)=a\_\{t\}\\,\\sigma\_\{f\}\\,f\(\\bm\{x\}\)\+a\_\{t\}\\,\\sigma\_\{f\}\\,b\_\{t\}\+\\varepsilon\_\{t\}withat∼LogNormal​\(μ=0\.25,σ=0\.5\)a\_\{t\}\\sim\\mathrm\{LogNormal\}\(\\mu\{=\}0\.25,\\sigma\{=\}0\.5\)\(strictly positive\),bt∼𝒩​\(0,1\)b\_\{t\}\\sim\\mathcal\{N\}\(0,1\), and observation noiseεt∼𝒩​\(0,σnoise2\)\\varepsilon\_\{t\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\mathrm\{noise\}\}^\{2\}\)\. The target is the canonicalff\(a=1,b=0a\{=\}1,b\{=\}0\)\. This is the affine family of §[4](https://arxiv.org/html/2607.09073#S4)and Eq\. \([1](https://arxiv.org/html/2607.09073#S4.E1)\) rescaled by the per\-task signal scaleσf\\sigma\_\{f\}, so slopes and offsets are expressed inσf\\sigma\_\{f\}units\. Source design points are drawn from independent Sobol streams seeded byseed\+1000\+t\\mathrm\{seed\}\+1000\+t; the configuration is summarized in Tab\.[2](https://arxiv.org/html/2607.09073#A1.T2)\.

Table 2:Synthetic affine\-TL benchmark configuration\.
### A\.2LCBench

LCBench\[[36](https://arxiv.org/html/2607.09073#bib.bib36)\]is used as a semi\-synthetic benchmark: each target is a GP surrogate fitted to a real classification dataset over a common77\-D search space, and its sources are small affine transformations of that surrogate\. The main figure \(Fig\.[4](https://arxiv.org/html/2607.09073#S5.F4)\) uses thetwo\-positive\-sourcebaseline: two sources,1212points per source,dim=7\\dim\{=\}7\. Targets are thesevendatasets Car, Fashion\-MNIST, Higgs, Segment, Adult, Christine, and Phoneme \(Numerai28\.6 dropped\)\. Budget is88init\+\+1717BO==2525total, over9999seeds\. Appendix\-only variants use the same budget/seeds:one positive\+\+one uncorrelated source\(the source configuration for the density panel of Fig\.[7](https://arxiv.org/html/2607.09073#A2.F7)\),one anti\-correlated source, and the per\-method ablations of App\.[B](https://arxiv.org/html/2607.09073#A2)\.

### A\.3pd1

Domain\.pd1\[[32](https://arxiv.org/html/2607.09073#bib.bib32)\]is a deep\-learning hyperparameter\-tuning benchmark \(non\-affine\) released as part of the HyperBO line of work; we optimize a GP surrogate fit to its tabular data\. We use two targets, \{cifar10\_wrn, mnist\_simple\_cnn\}, over a44\-dimensional hyperparameter cube \(learning rate, momentum, weight decay, label\-smoothing\)\. Each target is warm\-started by asingle highly correlated source\(cifar10←\\leftarrowcifar100, mnist←\\leftarrowfashion\); see Tab\.[3](https://arxiv.org/html/2607.09073#A1.T3)\.

Table 3:pd1 transfer benchmark configuration\.
### A\.4ifeval, evaluated on a GP surrogate

Domain\.19\-d data\-mix simplex; the objective is theifevalcomposite score for a Qwen2\.5 model finetuned on the chosen mixture\. Sources: Qwen2\.5 0\.5B and 3B; target: 7B \(Tab\.[4](https://arxiv.org/html/2607.09073#A1.T4)\)\.

Surrogate\.An MTGP is fit jointly on the real held\-out data across all three fidelities\. A single multi\-output posterior sample is drawn at512512Sobol anchors \(fixed seed\) and interpolated per fidelity by a single\-task GP, yielding the GP surrogate that BO optimizes\. The interpolation step works around missing kernel feature generation on the product kernel; cross\-fidelity correlation is exact at the anchors\.

Source training data\.For each source fidelity,Ns=16N\_\{s\}\{=\}16rows are uniformly randomly subsampled from the raw observations \(deterministic per seed\)\. SourceX,YX,Yare real raw rows; only the BO target is a GP surrogate sample path\. QuantileBO is dropped on this benchmark \(see §[5](https://arxiv.org/html/2607.09073#S5)\), leaving six methods\.

Table 4:ifevalGP\-surrogate benchmark configuration\.#### Source construction for pd1 andifeval\.

For pd1 andifevalthe optimized objective is a GP surrogate fit to*real held\-out task data*: the objective surface is a sample path of a jointly fit MTGP over the tabular data \(a standard continuous\-surrogate\-over\-tabular construction\), not a synthetic perturbation\. LCBench, by contrast, fits a GP surrogate to each real dataset and derives its sources as affine transformations of that surrogate, making it semi\-synthetic\. The uncorrelated\- and anti\-correlated\-source configurations are used only on LCBench and the synthetic grid; pd1 andifevaluse genuinely correlated tasks\.

#### Reproducibility\.

Code and instructions for reproducing all experiments will be released as a public GitHub repository upon acceptance\.

## Appendix BAdditional transfer\-learning results

The main text references these LCBench variant and ablation figures \(§[5\.2](https://arxiv.org/html/2607.09073#S5.SS2)\) and the standalone pd1 figures \(§[5\.3](https://arxiv.org/html/2607.09073#S5.SS3)\)\. All share the LCBench/HPO budgets and9999seeds of App\.[A](https://arxiv.org/html/2607.09073#A1)\.

![Refer to caption](https://arxiv.org/html/2607.09073v1/x6.png)Figure 6:LCBench transfer, one positive\+\+one uncorrelated source\.Best\-observed target validation accuracy vs\. BO iteration for the seven LCBench datasets \(higher is better\), with a final relative\-rank panel \(lower is better\); mean±1\\pm 1SE across9999seeds\. Each target is warm\-started by one genuinely correlated and one uncorrelated source\. The per\-task\-standardized ICM \(Positive\) and ICM \(Free\) retain their rank advantage despite the uninformative second source; this is the source configuration from which the correlation\-density diagnostic of Fig\.[7](https://arxiv.org/html/2607.09073#A2.F7)is drawn\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x7.png)Figure 7:Inferred task correlation, correlated vs\. uncorrelated source\.Post\-initialization density of the inferred target–source correlation for ICM \(Free\) and ICM \(Positive\) \(columns\), with overlapping and non\-overlapping source designs \(rows\), for the one\-correlated\-plus\-one\-uncorrelated source configuration; the crimson density is the correlated source and the teal density the uncorrelated source\. Against the uncorrelated source, ICM \(Free\) is*diffuse and unidentified*while the sign\-constrained ICM \(Positive\) is*bimodal*, splitting mass between0and\+1\+1; both keep the genuinely correlated source concentrated near\+1\+1\. This is the direct evidence for the correlation\-inference floor of Prop\.[2](https://arxiv.org/html/2607.09073#Thmproposition2)and the wrong\-sign mode that Remedy 2 removes \(see the §[5\.2](https://arxiv.org/html/2607.09073#S5.SS2)*Reading*text\)\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x8.png)Figure 8:LCBench transfer, one anti\-correlated source\.Best\-observed target validation accuracy vs\. BO iteration for the seven LCBench datasets \(higher is better\), with a final relative\-rank panel \(lower is better\); mean±1\\pm 1SE across9999seeds\. Each target is warm\-started by a single anti\-correlated source – the negative\-transfer stress test for Remedy 3 \(§[4\.3](https://arxiv.org/html/2607.09073#S4.SS3)\)\. The sign\-constrained ICM \(Positive\) is the most robust to the misleading source, while free\-sign methods are more exposed to negative transfer\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x9.png)Figure 9:Standalone pd1 per\-target breakdowns\.Best\-observed target value vs\. BO iteration for the cifar10 and mnist targets separately, each warm\-started by one highly correlated source; mean±1\\pm 1SE over9999seeds\. The per\-target view resolves the aggregate of Fig\.[5](https://arxiv.org/html/2607.09073#S5.F5): the per\-task\-standardized ICM \(Positive\) and ICM \(Free\) are competitive on both targets, with no single method dominating across the two\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x10.png)Figure 10:Synthetic grid, per\-method ablation\.Best\-observed target value vs\. acquisition iteration on the five synthetic base functions, mean±1\\pm 1SE across seeds, with a final relative\-rank panel \(lower is better\)\. This ablates the headline grid \(Fig\.[3](https://arxiv.org/html/2607.09073#S5.F3)\) over the ICM family only, adding the two source\-design\-overlap variants \(Remedy 3\) omitted from the main figure\. The overlap variants do not consistently beat their non\-overlapping counterparts, exposing the inference\-versus\-coverage trade\-off of §[4\.3](https://arxiv.org/html/2607.09073#S4.SS3)\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x11.png)Figure 11:LCBench two\-positive\-source baseline, per\-method ablation\.Best\-observed target validation accuracy vs\. BO iteration across the seven LCBench datasets, with a final relative\-rank panel; mean±1\\pm 1SE over9999seeds\. This ablates the baseline of Fig\.[4](https://arxiv.org/html/2607.09073#S5.F4)over the ICM family, adding the source\-design\-overlap variants \(Remedy 3\), which track their non\-overlapping counterparts closely\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x12.png)Figure 12:LCBench one\-positive\-plus\-one\-uncorrelated\-source, per\-method ablation\.Best\-observed target validation accuracy vs\. BO iteration across the seven LCBench datasets, with a final relative\-rank panel; mean±1\\pm 1SE over9999seeds\. This ablates Fig\.[6](https://arxiv.org/html/2607.09073#A2.F6)over the ICM family, adding the source\-design\-overlap variants \(Remedy 3\); co\-location does not consistently improve optimization performance despite sharpening correlation inference\.![Refer to caption](https://arxiv.org/html/2607.09073v1/x13.png)Figure 13:LCBench one\-anti\-correlated\-source, per\-method ablation\.Best\-observed target validation accuracy vs\. BO iteration across the seven LCBench datasets, with a final relative\-rank panel; mean±1\\pm 1SE over9999seeds\. This ablates Fig\.[8](https://arxiv.org/html/2607.09073#A2.F8)over the ICM family, adding the source\-design\-overlap variants \(Remedy 3\); the sign\-constrained ICM \(Positive\) remains the most robust to the anti\-correlated source\.
## Appendix CFull proof of Proposition[1](https://arxiv.org/html/2607.09073#Thmproposition1)

Lety1,…,yNsy\_\{1\},\\dots,y\_\{N\_\{s\}\}be the source observations with empirical meanμ^s\\hat\{\\mu\}\_\{s\}and empirical varianceσ^s2=Ns−1​∑i\(yi−μ^s\)2\\hat\{\\sigma\}\_\{s\}^\{2\}=N\_\{s\}^\{\-1\}\\sum\_\{i\}\(y\_\{i\}\-\\hat\{\\mu\}\_\{s\}\)^\{2\}\. We work underρs​t⋆=1\\rho^\{\\star\}\_\{st\}=1and treat the target divisorσ^t\\hat\{\\sigma\}\_\{t\}as fixed at its truth, since the target is data\-rich withNt≫NsN\_\{t\}\\gg N\_\{s\}\.

#### Step 1: sampling error of the standardization estimates\.

For Gaussian, or more generally finite\-fourth\-moment, observations the central limit theorem gives

μ^s\\displaystyle\\hat\{\\mu\}\_\{s\}=μs\+𝒩​\(0,σs2Ns\)\+op​\(Ns−1/2\),\\displaystyle=\\mu\_\{s\}\+\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{\\sigma\_\{s\}^\{2\}\}\{N\_\{s\}\}\\bigr\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\),\(2\)σ^s2\\displaystyle\\hat\{\\sigma\}\_\{s\}^\{2\}=σs2\+𝒩​\(0,2​σs4Ns\)\+op​\(Ns−1/2\)\.\\displaystyle=\\sigma\_\{s\}^\{2\}\+\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{2\\sigma\_\{s\}^\{4\}\}\{N\_\{s\}\}\\bigr\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\)\.\(3\)The delta method withg​\(u\)=ug\(u\)=\\sqrt\{u\}andg′​\(σs2\)=1/\(2​σs\)g^\{\\prime\}\(\\sigma\_\{s\}^\{2\}\)=1/\(2\\sigma\_\{s\}\)then gives

σ^s=σs\+𝒩​\(0,σs22​Ns\)\+op​\(Ns−1/2\)\.\\hat\{\\sigma\}\_\{s\}=\\sigma\_\{s\}\+\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{\\sigma\_\{s\}^\{2\}\}\{2N\_\{s\}\}\\bigr\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\)\.\(4\)

#### Step 2: propagation into the task covariance\.

Per\-task standardization divides the source byσ^s\\hat\{\\sigma\}\_\{s\}instead ofσs\\sigma\_\{s\}, scaling its row and column of the task covariance byσs/σ^s\\sigma\_\{s\}/\\hat\{\\sigma\}\_\{s\}\. Underρs​t⋆=1\\rho^\{\\star\}\_\{st\}=1the true blocks areBs​t⋆=σs​σtB^\{\\star\}\_\{st\}=\\sigma\_\{s\}\\sigma\_\{t\},Bs​s⋆=σs2B^\{\\star\}\_\{ss\}=\\sigma\_\{s\}^\{2\}, andBt​t⋆=σt2B^\{\\star\}\_\{tt\}=\\sigma\_\{t\}^\{2\}, so the standardized blocks are

Bs​teff=σsσ^s​Bs​t⋆,Bs​seff=σs2σ^s2​Bs​s⋆,Bt​teff=Bt​t⋆\.B^\{\\mathrm\{eff\}\}\_\{st\}=\\frac\{\\sigma\_\{s\}\}\{\\hat\{\\sigma\}\_\{s\}\}\\,B^\{\\star\}\_\{st\},\\qquad B^\{\\mathrm\{eff\}\}\_\{ss\}=\\frac\{\\sigma\_\{s\}^\{2\}\}\{\\hat\{\\sigma\}\_\{s\}^\{2\}\}\\,B^\{\\star\}\_\{ss\},\\qquad B^\{\\mathrm\{eff\}\}\_\{tt\}=B^\{\\star\}\_\{tt\}\.\(5\)

#### Step 3: residual variance inρ^s​t\\hat\{\\rho\}\_\{st\}\.

Writeηs:=\(σ^s−σs\)/σs\\eta\_\{s\}:=\(\\hat\{\\sigma\}\_\{s\}\-\\sigma\_\{s\}\)/\\sigma\_\{s\}, so that by Step 1ηs=𝒩​\(0,12​Ns\)\+op​\(Ns−1/2\)\\eta\_\{s\}=\\mathcal\{N\}\(0,\\tfrac\{1\}\{2N\_\{s\}\}\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\)\. The recovered correlation is

ρ^s​t=Bs​teffBs​seff​Bt​teff=\(1\+ηs\)−1​Bs​t⋆\(1\+ηs\)−2​σs2​σt2=ρs​t⋆,\\hat\{\\rho\}\_\{st\}=\\frac\{B^\{\\mathrm\{eff\}\}\_\{st\}\}\{\\sqrt\{B^\{\\mathrm\{eff\}\}\_\{ss\}\\,B^\{\\mathrm\{eff\}\}\_\{tt\}\}\}=\\frac\{\(1\+\\eta\_\{s\}\)^\{\-1\}B^\{\\star\}\_\{st\}\}\{\\sqrt\{\(1\+\\eta\_\{s\}\)^\{\-2\}\\sigma\_\{s\}^\{2\}\\;\\sigma\_\{t\}^\{2\}\}\}=\\rho^\{\\star\}\_\{st\},\(6\)so the scale factor cancels to first order and the bias is second order\. The surviving first\-order stochastic term is the standardization errorηs\\eta\_\{s\}itself, propagated jointly with the mean estimateμ^s\\hat\{\\mu\}\_\{s\}that entersσ^s2\\hat\{\\sigma\}\_\{s\}^\{2\}through the sameNsN\_\{s\}observations, giving

ρ^s​tρs​t⋆=1\+𝒩​\(0,12​Ns\)\+op​\(Ns−1/2\)\.\\frac\{\\hat\{\\rho\}\_\{st\}\}\{\\rho^\{\\star\}\_\{st\}\}=1\+\\mathcal\{N\}\\\!\\bigl\(0,\\tfrac\{1\}\{2N\_\{s\}\}\\bigr\)\+o\_\{p\}\(N\_\{s\}^\{\-1/2\}\)\.\(7\)The standard error isΘ​\(1/Ns\)\\Theta\(1/\\sqrt\{N\_\{s\}\}\)regardless ofNtN\_\{t\}: the source\-side sample size dominates\. AtNs=1N\_\{s\}=1the conventionσ^s=1\\hat\{\\sigma\}\_\{s\}=1makes the divisor pure noise; because the target posterior weights the source byρ^s​t\\hat\{\\rho\}\_\{st\}, this noise enters target predictions directly and can drive the multi\-task posterior strictly below the single\-task baseline\. ∎

## Appendix DFull proof of Proposition[2](https://arxiv.org/html/2607.09073#Thmproposition2)

#### Step 1: eigenmode decomposition\.

Consider the two\-task ICM model under the affine setup with shared designX=\{x1,…,xN\}X=\\\{x\_\{1\},\\ldots,x\_\{N\}\\\}and observationsyt=ft​\(X\)\+εty\_\{t\}=f\_\{t\}\(X\)\+\\varepsilon\_\{t\},εt∼𝒩​\(0,σ2​𝑰N\)\\varepsilon\_\{t\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\\bm\{I\}\_\{N\}\), and suppose all hyperparameters exceptρ:=B12/B11​B22\\rho:=B\_\{12\}/\\sqrt\{B\_\{11\}B\_\{22\}\}are known\. Stacking𝒚=\(𝒚1,𝒚2\)∈ℝ2​N\\bm\{y\}=\(\\bm\{y\}\_\{1\},\\bm\{y\}\_\{2\}\)\\in\\mathbb\{R\}^\{2N\}gives𝚺=𝑩⊗𝑲x\+σ2​𝑰2​N\\bm\{\\Sigma\}=\\bm\{B\}\\otimes\\bm\{K\}\_\{x\}\+\\sigma^\{2\}\\bm\{I\}\_\{2N\}\. Diagonalizing𝑲x=𝑼​𝚲​𝑼⊤\\bm\{K\}\_\{x\}=\\bm\{U\}\\bm\{\\Lambda\}\\bm\{U\}^\{\\top\}and applying𝑰2⊗𝑼⊤\\bm\{I\}\_\{2\}\\otimes\\bm\{U\}^\{\\top\}splits the joint law intoNNindependent bivariate Gaussians, modeiihaving covariance𝚺~i=λi​𝑩\+σ2​𝑰2\\widetilde\{\\bm\{\\Sigma\}\}\_\{i\}=\\lambda\_\{i\}\\bm\{B\}\+\\sigma^\{2\}\\bm\{I\}\_\{2\}\.

#### Step 2: per\-mode Fisher information\.

Define the effective SNR of modeii,

ui:=λi2​B11​B22\(λi​B11\+σ2\)​\(λi​B22\+σ2\)∈\[0,1\]\.u\_\{i\}:=\\frac\{\\lambda\_\{i\}^\{2\}B\_\{11\}B\_\{22\}\}\{\(\\lambda\_\{i\}B\_\{11\}\+\\sigma^\{2\}\)\(\\lambda\_\{i\}B\_\{22\}\+\\sigma^\{2\}\)\}\\in\[0,1\]\.\(8\)Direct computation of12​tr​\(\(𝚺~i−1​∂ρ𝚺~i\)2\)\\tfrac\{1\}\{2\}\\mathrm\{tr\}\\bigl\(\(\\widetilde\{\\bm\{\\Sigma\}\}\_\{i\}^\{\-1\}\\partial\_\{\\rho\}\\widetilde\{\\bm\{\\Sigma\}\}\_\{i\}\)^\{2\}\\bigr\)gives

Ii​\(ρ\)=ui​\(1\+ui​ρ2\)\(1−ui​ρ2\)2,I\_\{i\}\(\\rho\)=\\frac\{u\_\{i\}\(1\+u\_\{i\}\\rho^\{2\}\)\}\{\(1\-u\_\{i\}\\rho^\{2\}\)^\{2\}\},\(9\)which increases inuiu\_\{i\}, withui=1u\_\{i\}=1exactly in the noiseless limitσ2=0\\sigma^\{2\}=0\.

#### Step 3: aggregate bound\.

Summing the per\-mode informations,

IN​\(ρ\)=∑iIi​\(ρ\)≤N​1\+ρ2\(1−ρ2\)2,I\_\{N\}\(\\rho\)=\\sum\_\{i\}I\_\{i\}\(\\rho\)\\leq N\\,\\frac\{1\+\\rho^\{2\}\}\{\(1\-\\rho^\{2\}\)^\{2\}\},\(10\)and the Cramér–Rao bound yieldsVar⁡\[ρ^\]≥\(1−ρ2\)2/\[N​\(1\+ρ2\)\]\\operatorname\{Var\}\[\\hat\{\\rho\}\]\\geq\(1\-\\rho^\{2\}\)^\{2\}/\[N\(1\+\\rho^\{2\}\)\], with equality in the noiseless limit\. The rate staysΘ​\(1/N\)\\Theta\(1/N\)at any fixed noise level; only the constant degrades, controlled by the smallest eigenmode of𝑲x\\bm\{K\}\_\{x\}\. For mismatched designs with\|Xs\|=Ns≤\|Xt\|=Nt\|X\_\{s\}\|=N\_\{s\}\\leq\|X\_\{t\}\|=N\_\{t\}, a kernel\-overlap argument givesI​\(ρ\)≤c​Ns​\(1\+ρ2\)/\(1−ρ2\)2I\(\\rho\)\\leq c\\,N\_\{s\}\(1\+\\rho^\{2\}\)/\(1\-\\rho^\{2\}\)^\{2\}, so the smaller task dominates the rate\.

#### Step 4: Spearman asymptotic relative efficiency\.

At the bivariate normal the Pearson MLE attains the Cramér–Rao boundVar⁡\[ρ^\]=\(1−ρ2\)2/N\\operatorname\{Var\}\[\\hat\{\\rho\}\]=\(1\-\\rho^\{2\}\)^\{2\}/Nin the noiseless limit\. The Spearman rank estimatorρ^S\\hat\{\\rho\}\_\{S\}has the asymptotic variance derived byHoeffding \[[15](https://arxiv.org/html/2607.09073#bib.bib15)\]and tabulated byBorkowf \[[6](https://arxiv.org/html/2607.09073#bib.bib6)\], whose ratio to the Pearson variance is

Var⁡\[ρ^S\]Var⁡\[ρ^\]=π29≈1\.097,i\.e\. an asymptotic relative efficiency \(ARE\) of​9π2≈0\.912\.\\frac\{\\operatorname\{Var\}\[\\hat\{\\rho\}\_\{S\}\]\}\{\\operatorname\{Var\}\[\\hat\{\\rho\}\]\}=\\frac\{\\pi^\{2\}\}\{9\}\\approx 1\.097,\\qquad\\text\{i\.e\.\\ an asymptotic relative efficiency \(ARE\) of \}\\frac\{9\}\{\\pi^\{2\}\}\\approx 0\.912\.\(11\)The Spearman standard error therefore inherits the sameΘ​\(1/N\)\\Theta\(1/\\sqrt\{N\}\)rate, inflated by at most10%10\\%\. The rank weights of RGPE and the copula kernel of QuantileBO are smooth functionals of the empirical rank vector and inherit this rate through the functional delta method\.

## Appendix EFull proof of Proposition[3](https://arxiv.org/html/2607.09073#Thmproposition3)

#### Step 1: packing the design\.

Letkxk\_\{x\}be a stationary correlation kernel onΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}with characteristic lengthℓ\\ell, sokx​\(x,x′\)≤exp⁡\(−‖x−x′‖/ℓ\)k\_\{x\}\(x,x^\{\\prime\}\)\\leq\\exp\(\-\\\|x\-x^\{\\prime\}\\\|/\\ell\)up to a kernel\-dependent constant\. LetX=\{x1,…,xN\}⊂ΩX=\\\{x\_\{1\},\\dots,x\_\{N\}\\\}\\subset\\Omegabe the design andXℓ⊆XX\_\{\\ell\}\\subseteq Xa maximal subset with pairwise distance≥ℓ\\geq\\ell\. Volume comparison gives\|Xℓ\|≤\(diam​\(Ω\)/ℓ\)d\|X\_\{\\ell\}\|\\leq\(\\mathrm\{diam\}\(\\Omega\)/\\ell\)^\{d\}\.

#### Step 2: effective\-rank bound\.

Points inside a common packing cell satisfy\|kx​\(xi,xj\)\|≥exp⁡\(−1\)\|k\_\{x\}\(x\_\{i\},x\_\{j\}\)\|\\geq\\exp\(\-1\), so𝑲x\\bm\{K\}\_\{x\}is withinO​\(1\)O\(1\)of a block\-constant matrix with\|Xℓ\|\|X\_\{\\ell\}\|blocks\. Up to a fixed constant,𝑲x\\bm\{K\}\_\{x\}therefore has effective rankNeff:=\|Xℓ\|≤min⁡\{N,\(diam​\(Ω\)/ℓ\)d\}N\_\{\\mathrm\{eff\}\}:=\|X\_\{\\ell\}\|\\leq\\min\\\{N,\\,\(\\mathrm\{diam\}\(\\Omega\)/\\ell\)^\{d\}\\\}, and only the topNeffN\_\{\\mathrm\{eff\}\}eigenmodes carry signal\.

#### Step 3: Fisher information\.

Reusing the eigenmode decomposition of App\.[D](https://arxiv.org/html/2607.09073#A4), the Fisher information forρ\\rhoisIN​\(ρ\)=∑iIi​\(ρ\)I\_\{N\}\(\\rho\)=\\sum\_\{i\}I\_\{i\}\(\\rho\)withIiI\_\{i\}proportional to the SNRuiu\_\{i\}of modeii\. Modes outside the packing haveui=O​\(exp⁡\(−2\)\)u\_\{i\}=O\(\\exp\(\-2\)\)and contributeO​\(1\)O\(1\)in total, so the sum is dominated by theNeffN\_\{\\mathrm\{eff\}\}in\-packing modes:

IN​\(ρ\)≤Neff​1\+ρ2\(1−ρ2\)2≤min⁡\{N,\(diam​\(Ω\)/ℓ\)d\}​1\+ρ2\(1−ρ2\)2\.I\_\{N\}\(\\rho\)\\;\\leq\\;N\_\{\\mathrm\{eff\}\}\\,\\frac\{1\+\\rho^\{2\}\}\{\(1\-\\rho^\{2\}\)^\{2\}\}\\;\\leq\\;\\min\\\!\\bigl\\\{N,\\,\(\\mathrm\{diam\}\(\\Omega\)/\\ell\)^\{d\}\\bigr\\\}\\,\\frac\{1\+\\rho^\{2\}\}\{\(1\-\\rho^\{2\}\)^\{2\}\}\.\(12\)The Cramér–Rao bound then givesVar⁡\[ρ^\]≥\(1−ρ2\)2/\[Neff​\(1\+ρ2\)\]\\operatorname\{Var\}\[\\hat\{\\rho\}\]\\geq\(1\-\\rho^\{2\}\)^\{2\}/\[N\_\{\\mathrm\{eff\}\}\(1\+\\rho^\{2\}\)\]: the GP identifiesρ\\rhofrom at mostNeffN\_\{\\mathrm\{eff\}\}effective paired observations\. ∎