MARCO: Click-Intent Decomposition for Calibrated Ads Conversion Prediction

# MARCO: Click-Intent Decomposition for Calibrated Ads Conversion Prediction
Source: [https://arxiv.org/html/2608.10562](https://arxiv.org/html/2608.10562)
Xiru HuangMeta AI, Menlo Park, California, USALiang LuoMeta AI, Menlo Park, California, USAJianbo SunMeta AI, Menlo Park, California, USAHe LyuMeta AI, Menlo Park, California, USAZihang FuMeta AI, Menlo Park, California, USAIvonne XuMeta AI, Menlo Park, California, USAZhizhuo LiMeta AI, Menlo Park, California, USAZhengyu ZhangMeta AI, Menlo Park, California, USAPei\-Ju SungMeta AI, Menlo Park, California, USAYunmiao WangMeta AI, Menlo Park, California, USAZixuan WangMeta AI, Menlo Park, California, USAZhengli ZhaoMeta AI, Menlo Park, California, USAQiang JinMeta AI, Menlo Park, California, USAMike JermannMeta AI, Menlo Park, California, USAMingda LiMeta AI, Menlo Park, California, USAYang XiaoMeta AI, Menlo Park, California, USABhavana ChallaMeta AI, Menlo Park, California, USABrooke BianMeta AI, Menlo Park, California, USAYang LiMeta AI, Menlo Park, California, USAAshish ChamoliMeta AI, Menlo Park, California, USABibek BhusalMeta AI, Menlo Park, California, USADanning DiMeta AI, Menlo Park, California, USAYuan JinMeta AI, Menlo Park, California, USAMeet RavalMeta AI, Menlo Park, California, USAZhiwen ChenMeta AI, Menlo Park, California, USABoyao SunMeta AI, Menlo Park, California, USAShuguang WangMeta AI, Menlo Park, California, USAYunlong HeMeta AI, Menlo Park, California, USAYantao YaoMeta AI, Menlo Park, California, USASagar ChordiaMeta AI, Menlo Park, California, USAWenlin ChenMeta AI, Menlo Park, California, USASantanu KolayMeta AI, Menlo Park, California, USAQin HuangMeta AI, Menlo Park, California, USAEllie WenMeta AI, Menlo Park, California, USA

###### Abstract

Not all clicks are equal\. Industrial ads ranking decouples conversion probability into click\-through rate \(CTR\) and post\-click conversion rate \(CVR\), yet treats every click as the same event\. In reality, users provide a free, self\-generated signal of intent through their physical UI interactions\. Different click types on the same ad exhibit a4×4\\timesdifference in actual conversion rates\. By conflating these signals, the standard CVR model systematically under\-predicts high\-intent clicks and over\-predicts low\-intent ones, which is a severe bias masked by near\-perfect aggregate calibration\.

We propose MARCO \(Multi\-intentAdsRankingCompositionOptimization\), a framework that resolves this bias by decomposing each click by intent\. Using the logged click type as a free behavioral label, MARCO trains per\-intent CVR heads on homogeneous populations, and at serving time composes their per\-intent CVR estimates under a predicted distribution over intents\. Theoretically, we prove that decomposition never raises population risk, give the exact headroom under squared loss and non\-negativity under the deployed loss, and show through a routing\-efficiency dial how much of it reaches serving\. Because the population\-optimal score is unchanged, any gain is a finite\-capacity estimation and calibration effect that we validated both offline and online\. For deployment at scale, we further cast multi\-impression, multi\-click attribution as credit assignment with a bias\-variance tradeoff analogous to RL return estimation, showing last\-impression, first\-click attribution is the low\-bias, low\-variance, deterministic choice under production constraints, and derive three consistency conditions enforced end\-to\-end at scale\.

Deployed at binary intent granularity, MARCO corrects per\-intent calibration to approximately100%100\\%, lifts conversions per click by\+2\.80%\+2\.80\\%, and drives\+0\.98%\+0\.98\\%cumulative improvement in topline metrics\.

*K*eywordscomputational advertising, conversion rate prediction, click\-through rate prediction, model calibration, click\-intent decomposition, prediction composition, credit attribution

## 1Introduction

Industrial ads ranking systems estimate impression conversion probability by factorizing it into a click\-through rate \(CTR\) and a post\-click conversion rate \(CVR\) as an exact marginal factorization\[[1](https://arxiv.org/html/2608.10562#bib.bib1),[2](https://arxiv.org/html/2608.10562#bib.bib2)\]:

Φstd​\(x\)=λ​\(x\)⋅μ​\(x\),\\Phi\_\{\\text\{std\}\}\(x\)=\\lambda\(x\)\\cdot\\mu\(x\),\(1\)wherexxdenotes impression\-time features,λ​\(x\):=P​\(click∣imp,x\)\\lambda\(x\):=P\(\\text\{click\}\\mid\\text\{imp\},x\)is the CTR,μ​\(x\):=P​\(conv∣click,x\)\\mu\(x\):=P\(\\text\{conv\}\\mid\\text\{click\},x\)is the CVR, andΦstd​\(x\):=P​\(conv∣imp,x\)\\Phi\_\{\\mathrm\{std\}\}\(x\):=P\(\\text\{conv\}\\mid\\text\{imp\},x\)is the conversion score used for auction ranking\. Each component is estimated by a dedicated machine learning model—a compositional approach that has served as the industry standard for over a decade\.

The primary operational challenge lies in CVR estimation: predicting conversion rates across a mixed click population is difficult due to substantial variance across interaction types\. High\-intent actions convert at elevated rates, whereas low\-intent actions convert far less frequently\. Compressing this heterogeneous click mixture into a single scalar prediction introduces group\-conditional bias that no finite\-capacity model can resolve\. Conversely, partitioning clicks into homogeneous intent strata improves estimation efficiency, leveraging foundational principles of post\-stratified estimation\[[3](https://arxiv.org/html/2608.10562#bib.bib3),[4](https://arxiv.org/html/2608.10562#bib.bib4)\]\. Unlike classical post\-stratification, the intent stratum in ad ranking is a post\-impression outcome unobserved during scoring\. We address this by predicting the intent distribution at inference time and routing per\-stratum estimates through the predicted distribution\.

Standard interaction logs already contain the signal required to decouple user intent\. Modern social ads feature distinct UI surfaces: call\-to\-action \(CTA\) taps navigate off\-platform, whereas social interactions \(e\.g\., likes, comments\) reflect lightweight on\-platform engagement\. Conventional architectures collapse these behaviors into a single click label, leaving a zero\-cost supervision signal unexploited\. MARCO instead leverages logged click types as free supervision for intent\. Empirically, high\-intent CTA taps convert at nearly4×4\\timesthe rate of social actions—a gap single\-head models cannot capture\.

Ignoring this signal forces a single CVR head to fit distinct conversion funnels, causing systematic subgroup*miscalibration*: under\-predicting high\-intent traffic while over\-predicting low\-intent traffic\. This directly degrades ranking, as inflated low\-intent scores displace higher\-converting candidates in the auction\. Because these opposing errors cancel in aggregate, standard monitoring tools report healthy overall calibration while masking severe internal distortion\. We term this silent failure mode*click\-intent heterogeneity*and formalize its observability limit in Section[2](https://arxiv.org/html/2608.10562#S2)\.

While prior literature addresses adjacent modeling challenges, no existing framework restructures the CTR\-CVR composition formula to isolate heterogeneous click populations\. Vertical funnel methods\[[1](https://arxiv.org/html/2608.10562#bib.bib1),[2](https://arxiv.org/html/2608.10562#bib.bib2),[5](https://arxiv.org/html/2608.10562#bib.bib5),[6](https://arxiv.org/html/2608.10562#bib.bib6),[7](https://arxiv.org/html/2608.10562#bib.bib7)\]decompose the user journey after the click\. Multi\-task learning and mixture\-of\-experts \(MoE\) architectures\[[8](https://arxiv.org/html/2608.10562#bib.bib8),[9](https://arxiv.org/html/2608.10562#bib.bib9),[10](https://arxiv.org/html/2608.10562#bib.bib10)\]optimize routing across shared experts, yet ultimately blend representations into a single prediction over the mixed click population\. Expert gating modifies architectural capacity rather than the supervised target, leaving the underlying label distribution unaltered for each task\. Post\-hoc calibration techniques\[[11](https://arxiv.org/html/2608.10562#bib.bib11),[12](https://arxiv.org/html/2608.10562#bib.bib12),[13](https://arxiv.org/html/2608.10562#bib.bib13),[14](https://arxiv.org/html/2608.10562#bib.bib14)\]cannot condition on click intent, as the specific interaction type is a post\-impression event unobservable at inference time\. While intent models\[[15](https://arxiv.org/html/2608.10562#bib.bib15)\]enrich feature representations, they leave the underlying prediction composition unchanged\.

We proposeMARCO\(Multi\-intentAdsRankingCompositionOptimization\), a framework that resolves click\-intent heterogeneity through structural intent decomposition\. MARCO partitions click events into intent\-differentiated sub\-categories, estimating dedicated CTR and CVR predictions over more homogeneous sub\-populations\. At training time, MARCO uses the logged click type as a free supervision label; at serving time, it predicts the latent intent distribution across categories\. By reformulating the prediction composition, MARCO eliminates group\-conditional miscalibration while remaining orthogonal to and fully composable with existing CVR architectures\. Our contributions are:

1. 1\.Problem Formulation & Observability Limit\.We identify*click\-intent heterogeneity*as a root cause of systematic miscalibration that conventional architectures cannot eliminate, and prove that a single impression\-time prediction is inherently incapable of achieving per\-intent calibration \(§[2](https://arxiv.org/html/2608.10562#S2)\)\.
2. 2\.Free Supervision & Mechanism Isolation\.We leverage logged click types as zero\-cost behavioral supervision labels\. Through controlled ablations, we demonstrate that single\-output architectures consistently fail to eliminate per\-intent miscalibration, even when augmented with auxiliary intent tasks, historical features, or Mixture\-of\-Experts \(MoE\) gating\. Conversely, exposing per\-intent outputs via output\-layer decomposition eliminates over99%99\\%of calibration error with zero annotation cost and negligible parameter overhead \(§[6](https://arxiv.org/html/2608.10562#S6)\)\.
3. 3\.Production System & Industrial Impact\.We articulate the end\-to\-end system design, including a credit\-attribution design and the cross\-pipeline consistency conditions the system must satisfy \(§[5](https://arxiv.org/html/2608.10562#S5)\)\. Deployed at scale across Meta’s ad platforms under binary intent granularity, MARCO increases conversions per click by\+2\.80%\+2\.80\\%in a production holdback and delivers a\+0\.98%\+0\.98\\%cumulative topline metric lift across successive launches \(§[6](https://arxiv.org/html/2608.10562#S6)\)\.
4. 4\.Theoretical Foundation & Routing Efficiency\.We formalize intent decomposition as a population\-level class enlargement, proving weak dominance over pooled models and establishing generic strictly positive headroomΔℱ\\Delta\_\{\\mathcal\{F\}\}\. Furthermore, we introduce the routing\-efficiency parameterη\\etato quantify the fraction of theoretical headroom realized at inference \(§[3](https://arxiv.org/html/2608.10562#S3)\)\.

The remainder of this paper is organized as follows\. Section[2](https://arxiv.org/html/2608.10562#S2)formalizes the click\-intent heterogeneity problem\. Section[3](https://arxiv.org/html/2608.10562#S3)establishes the theoretical framing and headroom analysis for MARCO\. Section[4](https://arxiv.org/html/2608.10562#S4)details the proposed MARCO model, while Section[5](https://arxiv.org/html/2608.10562#S5)describes the credit attribution design and end\-to\-end system architecture\. Section[6](https://arxiv.org/html/2608.10562#S6)presents our offline evaluations and online A/B testing results\. All mathematical proofs are deferred to Appendix[A](https://arxiv.org/html/2608.10562#A1)\.

![Refer to caption](https://arxiv.org/html/2608.10562v1/x1.png)Figure 1:The MARCO ads\-ranking system\. Gray marks the unchanged baseline and teal marks what MARCO adds\.
## 2Click\-Intent Heterogeneity in Ads Ranking

Recall from Eq\.[1](https://arxiv.org/html/2608.10562#S1.E1)that conventional ads ranking scores an impression viaΦstd​\(x\)=λ​\(x\)⋅μ​\(x\)\\Phi\_\{\\mathrm\{std\}\}\(x\)=\\lambda\(x\)\\cdot\\mu\(x\), relying on a single pooled conversion headμ​\(x\)\\mu\(x\)across all click types\. To capture this intent variation, we partition the click space intoKKmutually exclusive and collectively exhaustive intent categories covering all click types, indexed byk∈\{1,…,K\}k\\in\\\{1,\\dots,K\\\}, grouped by historical click\-type conversion likelihood\. Intent routing relies strictly on logged interaction types rather than conversion outcomes, eliminating target leakage\.

For impression featuresxx, we define the per\-intent CTRλk​\(x\):=P​\(clickk∣imp,x\)\\lambda\_\{k\}\(x\):=P\(\\text\{click\}\_\{k\}\\mid\\text\{imp\},x\), per\-intent CVRμk​\(x\):=P​\(conv∣clickk,x\)\\mu\_\{k\}\(x\):=P\(\\text\{conv\}\\mid\\text\{click\}\_\{k\},x\), and intent distributionπk​\(x\):=P​\(k∣click,x\)\\pi\_\{k\}\(x\):=P\(k\\mid\\text\{click\},x\)\. By the law of total probability, the overall impression conversion probability estimated by MARCO is:

ΦMARCO​\(x\)=∑k=1Kλk​\(x\)​μk​\(x\)\.\\Phi\_\{\\mathrm\{MARCO\}\}\(x\)=\\sum\_\{k=1\}^\{K\}\\lambda\_\{k\}\(x\)\\mu\_\{k\}\(x\)\.\(2\)WhileΦMARCO​\(x\)\\Phi\_\{\\mathrm\{MARCO\}\}\(x\)mathematically generalizes to arbitraryKK, our evaluation and deployment focus on the binary instantiation \(K=2K=2\), capturing the dominant contrast between off\-platform CTA taps and on\-platform social engagements\.*Throughout the remainder of this paper, we drop the explicit feature argument\(x\)\(x\)for brevity whenever feature conditioning is clear from context\.*

We evaluate prediction quality using the*calibration ratio*\(predicted over observed conversion rate\)\. Because a pooled headμ\\muoutputs a single scalar across heterogeneous clicks, opposing errors from under\-predicting high\-intent traffic and over\-predicting low\-intent traffic cancel in aggregate, masking severe subgroup bias behind a deceptively healthy overall ratio\.

Existing ads ranking architectures \(the gray baseline path in Figure[1](https://arxiv.org/html/2608.10562#S1.F1)\) are structurally blind to per\-intent bias across every ranking stage:

1. 1\.Attribution System:Clicks and conversions are joined without distinguishing interaction types, generating pooled training targets\.
2. 2\.Model Training:CTR and CVR heads optimize a single loss over mixed click traffic, ignoring intent differentiation\.
3. 3\.Calibration Service:Calibration relies strictly on impression\-time features, leaving unobserved post\-impression intent miscalibrated beneath aggregate cancellation\.
4. 4\.Prediction Composition:Pooled estimates are composed via Eq\.[1](https://arxiv.org/html/2608.10562#S1.E1), propagating single\-click bias into auction scores\.

This limitation represents a structural observability barrier rather than a model capacity constraint\. As long as a ranking system outputs a single scalar predictionμ\\muprior to interaction observation, no added capacity, architectural gating, or post\-hoc recalibration can achieve simultaneous group\-conditional calibration\. We formalize this fundamental limit in Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)\.

###### Proposition 1\(Per\-intent calibration is unattainable by a single output before the click occurs\)

Before the click intent is observed, under a proper scoring rule and a fixed impression\-time feature set, no model emitting a single CVR valueμ\\mucan be simultaneously calibrated for all intent categories whenever the per\-intent CVRsμk\\mu\_\{k\}differ\. It equals at most oneμk\\mu\_\{k\}\.

Section[3](https://arxiv.org/html/2608.10562#S3)formalizes this mechanism as a class\-enlargement headroom, proving it is non\-negative and generically strictly positive\. Section[6](https://arxiv.org/html/2608.10562#S6)directly quantifies this headroom on held\-out benchmarks\.

## 3Theoretical Analysis

We frame intent decomposition as a population\-level*class enlargement*\(ℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}\) and establish three key theoretical guarantees:

1. \(1\)Weak Dominance:The intent\-decomposed class never yields higher risk than the pooled class at the population optimum \(Theorem[1](https://arxiv.org/html/2608.10562#Thmtheorem1)\(i\)\)\.
2. \(2\)Genericity:Strict improvement over pooled models occurs almost surely across parameter space \(Proposition[4](https://arxiv.org/html/2608.10562#Thmproposition4)\)\.
3. \(3\)Routing Monotonicity:Expanding CTR model capacity provably realizes a larger fraction of the theoretical headroom \(Proposition[7](https://arxiv.org/html/2608.10562#Thmproposition7)\)\.

These guarantees characterize the best achievable risk within each hypothesis class\. Because the pooled and decomposed scores coincide at the true Bayes optimum \(ΦMARCO=Φstd\\Phi\_\{\\mathrm\{MARCO\}\}=\\Phi\_\{\\mathrm\{std\}\}, Lemma[1](https://arxiv.org/html/2608.10562#Thmlemma1)\), any realized gain is strictly a finite\-capacity estimation and calibration effect\.

##### Class Enlargement and Headroom \(Δℱ\\Delta\_\{\\mathcal\{F\}\}\)\.

Letℱ\\mathcal\{F\}denote the hypothesis class of a single pooled conversion head, and let𝒢=\{g=∑k=1Kπk​fk:fk∈ℱ\}\\mathcal\{G\}=\\\{g=\\sum\_\{k=1\}^\{K\}\\pi\_\{k\}f\_\{k\}:f\_\{k\}\\in\\mathcal\{F\}\\\}denote the intent\-decomposed class\. We define the theoretical headroom opened by decomposition as the risk gap between the best\-in\-class pooled model and the best\-in\-class decomposed model:

Δℱ:=inff∈ℱℛ​\(f\)−infg∈𝒢ℛ​\(g\)\\Delta\_\{\\mathcal\{F\}\}:=\\inf\_\{f\\in\\mathcal\{F\}\}\\mathcal\{R\}\(f\)\-\\inf\_\{g\\in\\mathcal\{G\}\}\\mathcal\{R\}\(g\)\(3\)Becauseℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}, weak dominance \(Δℱ≥0\\Delta\_\{\\mathcal\{F\}\}\\geq 0\) holds universally regardless of backbone architecture, feature representations, or loss function \(Theorem[1](https://arxiv.org/html/2608.10562#Thmtheorem1)\(i\)\)\. Under squared loss,Δℱ\\Delta\_\{\\mathcal\{F\}\}resolves to an exactL2L^\{2\}projection energy‖P𝒢​μ−Pℱ​μ‖2≥0\\\|P\_\{\\mathcal\{G\}\}\\mu\-P\_\{\\mathcal\{F\}\}\\mu\\\|^\{2\}\\geq 0\(Proposition[3](https://arxiv.org/html/2608.10562#Thmproposition3)\)\. Under the deployed Normalized Entropy metric, while non\-linearity prevents a closed\-form expression, weak dominance \(Δℱ≥0\\Delta\_\{\\mathcal\{F\}\}\\geq 0\) holds unconditionally\.

Furthermore, Proposition[4](https://arxiv.org/html/2608.10562#Thmproposition4)establishes that parameter configurations yielding zero headroom \(Δℱ=0\\Delta\_\{\\mathcal\{F\}\}=0\) form a set of Lebesgue measure zero\. Thus, strict headroom \(Δℱ\>0\\Delta\_\{\\mathcal\{F\}\}\>0\) is generic, i\.e\., decomposition strictly improves population risk whenever per\-intent conversion rates differ\. In Section[6](https://arxiv.org/html/2608.10562#S6), we directly measure this headroom on held\-out production traffic atK=2K=2\.

##### Routing Efficiency and CTR Capacity\.

The theoretical headroomΔℱ\\Delta\_\{\\mathcal\{F\}\}is a CVR\-side quantity\. How much of this potential reaches inference depends on the intent distributionπ^k=λ^k/λ^\\hat\{\\pi\}\_\{k\}=\\hat\{\\lambda\}\_\{k\}/\\hat\{\\lambda\}predicted by the CTR model\. We formalize this transfer via the*routing efficiency*:

η^:=1−εrouteΔℱ\\hat\{\\eta\}:=1\-\\frac\{\\varepsilon\_\{\\mathrm\{route\}\}\}\{\\Delta\_\{\\mathcal\{F\}\}\}\(4\)whereεroute\\varepsilon\_\{\\mathrm\{route\}\}represents the heterogeneity\-weighted routing cost incurred by imperfect intent predictions \(Lemma[11](https://arxiv.org/html/2608.10562#Thmlemma11)\)\. The routing cost obeysεroute≤𝔼​\[‖π^−π‖2​s2\]\\varepsilon\_\{\\mathrm\{route\}\}\\leq\\mathbb\{E\}\[\\\|\\hat\{\\pi\}\-\\pi\\\|^\{2\}s^\{2\}\], wheres2=∑k\(μk−μ\)2s^\{2\}=\\sum\_\{k\}\(\\mu\_\{k\}\-\\mu\)^\{2\}measures intent variance\. This cost vanishes where intent is homogeneous \(s=0s=0\) and concentrates strictly where per\-intent conversion rates diverge\.

In Proposition[7](https://arxiv.org/html/2608.10562#Thmproposition7), we prove that expanding CTR model capacity expands the realizable routing classΠ\(c\)\\Pi^\{\(c\)\}, monotonically reducing routing errorεroute∗\\varepsilon\_\{\\mathrm\{route\}\}^\{\*\}and increasing serving efficiencyη^\\hat\{\\eta\}\. Consequently, larger CTR models unlock a higher fraction of the CVR headroom—a theoretical property directly validated by our offline capacity sweeps in Section[6](https://arxiv.org/html/2608.10562#S6)\.

## 4The MARCO Model

Guided by our theoretical framing, MARCO introduces a minimal architectural modification to capture per\-intent estimation gains while remaining strictly constrained in parameter and latency budgets\. Rather than predicting a single scalar CVR, MARCO estimates per\-intent CTR and CVR vectors and composes the total conversion probabilityΦMARCO\\Phi\_\{\\mathrm\{MARCO\}\}via the law of total probability as formulated in Eq\.[2](https://arxiv.org/html/2608.10562#S2.E2)\. Estimating eachμk\\mu\_\{k\}on a homogeneous click sub\-population eliminates the label conflation that causes systematic group\-conditional bias\. In our production platform, we instantiate MARCO atK=2K=2\.

![Refer to caption](https://arxiv.org/html/2608.10562v1/x2.png)Figure 2:Model architecture: Standard vs\. MARCO\.##### Free Behavioral Supervision\.

MARCO leverages logged user interaction types as zero\-cost supervision targets\. Rather than requiring manual annotation, the intent label for each converted click is derived directly from physical UI behavior, serving as a behavioral pretext signal for latent intent\. At training time, each per\-intent CVR headμk\\mu\_\{k\}is trained exclusively on clicks belonging to intent categorykk\(attributed via Section[5](https://arxiv.org/html/2608.10562#S5)\)\. At serving time, because the true click intent is unobserved prior to user interaction, MARCO evaluates all per\-intent CVR heads and weights them by the predicted intent distributionπ^k=λ^k/λ^\\hat\{\\pi\}\_\{k\}=\\hat\{\\lambda\}\_\{k\}/\\hat\{\\lambda\}, computing the expected CVR over predicted latent intents\.

##### Vector\-Head Architecture\.

As shown in Figure[2](https://arxiv.org/html/2608.10562#S4.F2), MARCO implements this decomposition via multi\-output task heads without altering the underlying feature representations or model backbones\. Shared embedding layers and deep neural network \(DNN\) representations remain intact for both CTR and CVR models\. Alongside the baseline scalar projection, each model appends aKK\-dimensional vector head \(ℝd→ℝK\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{K\}, wheredddenotes the final backbone representation dimension\)\. The CTR head emits the complete per\-intent click rate vector\(λ1,…,λK\)\(\\lambda\_\{1\},\\dots,\\lambda\_\{K\}\), while the CVR head emits the corresponding conversion rate vector\(μ1,…,μK\)\(\\mu\_\{1\},\\dots,\\mu\_\{K\}\)\. The original scalar output is retained strictly as an operational baseline for automated fallback \(Section[5\.3](https://arxiv.org/html/2608.10562#S5.SS3)\)\.

##### Parameter Efficiency & Operational Overhead\.

ConstructingKKseparate dedicated models would significantly inflate parameter footprint, memory consumption, and serving latency, causing severe model fragmentation\[[16](https://arxiv.org/html/2608.10562#bib.bib16)\]\. By expanding scalar projection heads intoKK\-vector heads within existing backbones, MARCO adds fewer than10−7%10^\{\-7\}\\%parameters relative to total model capacity\. Because representation layers and embedding lookups are shared, serving latency remains entirely unaffected while successfully restoring per\-intent calibration\.

##### System Integration\.

MARCO operates within an end\-to\-end industrial ranking architecture \(Figure[1](https://arxiv.org/html/2608.10562#S1.F1)\), where data logging, credit attribution, model training, online serving, and real\-time calibration are managed by independent infrastructure components\. For the composite score in Eq\.[2](https://arxiv.org/html/2608.10562#S2.E2)to remain semantically coherent, all pipeline stages must maintain strict agreement on intent category assignments—a system requirement we formalize and enforce in Section[5](https://arxiv.org/html/2608.10562#S5)\.

## 5Attribution and System Design

Deploying the decomposed composition Eq\.[2](https://arxiv.org/html/2608.10562#S2.E2)in a distributed industrial ranking environment requires all logging, training, serving, and calibration pipelines to maintain strict agreement on which interaction defines the credited click intent for every impression\. This section formalizes the credit attribution rationale \(§[5\.1](https://arxiv.org/html/2608.10562#S5.SS1)\), establishes the cross\-pipeline consistency invariants \(§[5\.2](https://arxiv.org/html/2608.10562#S5.SS2)\), and details the production infrastructure built to enforce them at scale \(§[5\.3](https://arxiv.org/html/2608.10562#S5.SS3)\)\.

### 5\.1Credit Assignment Framework

In conversion\-optimized advertising, a user’s pre\-conversion journey spans multiple impressions and multi\-touch interaction sequences\. We model this trajectory as an ordered historyHHofnnimpressions, where each impressionimpi\\text\{imp\}\_\{i\}logged at render timeτi\\tau\_\{i\}contains a multisetCiC\_\{i\}ofmi≥0m\_\{i\}\\geq 0within\-impression clicks logged at timesti,jt\_\{i,j\}:

H=\[\(impi,τi,Ci\)\]i=1n⟶conv,where​Ci=\{\(clicki,j,ti,j\)\}j=1miH=\\big\[\(\\text\{imp\}\_\{i\},\\tau\_\{i\},C\_\{i\}\)\\big\]\_\{i=1\}^\{n\}\\longrightarrow\\text\{conv\},\\quad\\text\{where \}C\_\{i\}=\\big\\\{\(\\text\{click\}\_\{i,j\},t\_\{i,j\}\)\\big\\\}\_\{j=1\}^\{m\_\{i\}\}\(5\)withτ1<τ2<⋯<τn\\tau\_\{1\}<\\tau\_\{2\}<\\dots<\\tau\_\{n\}andτi≤ti,1≤ti,2≤⋯≤ti,mi\\tau\_\{i\}\\leq t\_\{i,1\}\\leq t\_\{i,2\}\\leq\\dots\\leq t\_\{i,m\_\{i\}\}\. Because a single rendered ad unit can generate a sequence of heterogeneous interactions \(e\.g\., a social like atti,1t\_\{i,1\}followed by a CTA tap atti,3t\_\{i,3\}\), credit attribution must assign two non\-negative weight vectors: a cross\-impression weightwiw\_\{i\}and a within\-impression click weightvi,jv\_\{i,j\}\. The credited conversion outcome is distributed across training tuples via the joint weightwi⋅vi,jw\_\{i\}\\cdot v\_\{i,j\}, yielding the per\-impression training target decomposition:

Φ¯​\(H\)=∑i=1nwi​∑j=1mivi,j⋅P​\(conv∣impi,clicki,j\)\\bar\{\\Phi\}\(H\)=\\sum\_\{i=1\}^\{n\}w\_\{i\}\\sum\_\{j=1\}^\{m\_\{i\}\}v\_\{i,j\}\\cdot P\(\\text\{conv\}\\mid\\text\{imp\}\_\{i\},\\text\{click\}\_\{i,j\}\)\(6\)enforcing∑iwi=1\\sum\_\{i\}w\_\{i\}=1and∑jvi,j=1\\sum\_\{j\}v\_\{i,j\}=1to guarantee total conversion mass\.

##### Bias\-Variance Tradeoff & RL Analog\.

Selecting\(wi,vi,j\)\(w\_\{i\},v\_\{i,j\}\)represents a structural bias\-variance tradeoff analogous to temporal\-difference return estimation in reinforcement learning\. As summarized in Table[1](https://arxiv.org/html/2608.10562#S5.T1), impressions and clicks serve distinct causal roles:

- •Impression Axis \(Bias Control\):An impression is the external stimulus presented to the user\. Weighting towards the*last impression*\(wn=1w\_\{n\}=1\) isolates the causally proximal context that directly triggered conversion, minimizing context attribution bias \(analogous to Monte Carlo return estimation\)\.
- •Click Axis \(Variance Control\):A click sequence within a fixed impression represents a stochastic exploration process\. Selecting the*first click*\(vi,1=1v\_\{i,1\}=1\) captures the user’s immediate intent signature before subsequent taps inject behavioral noise, minimizing intent estimation variance \(analogous toTD​\(0\)\\text\{TD\}\(0\)temporal\-difference learning\)\.

Table 1:Attribution design space and its qualitative rationale: bias, variance, real\-time feasibility, and RL credit\-assignment analogs\.
##### Production Feasibility Constraints\.

MARCO’s credit\-assignment policy balances streaming feasibility on the click axis with context proximity on the impression axis\. The consistency conditions of Section[5\.2](https://arxiv.org/html/2608.10562#S5.SS2)mandate that credited intent be deterministic and instantly resolvable without event buffering, ruling out last\-click and time\-decay rules in favor of first\-click attribution\. This choice carries minimal empirical risk: across the vast majority of impressions, initial and subsequent clicks share the same intent category, ensuring high intent alignment regardless of the click\-axis rule\. On the impression axis, we select the last impression to minimize context attribution bias, as it represents the causally proximal stimulus\. The resulting*last\-impression, first\-click*policy is both statistically robust and instantly resolvable at serving time\.

### 5\.2Cross\-Pipeline Consistency Conditions

Because CTR and CVR training and calibration operate as independent distributed services, all four pipelines must resolve to the exact same credited click intentκ​\(i\)∈\{1,…,K\}\\kappa\(i\)\\in\\\{1,\\dots,K\\\}for every impressioni∈Ii\\in I\. Training consistency requiresκCTRtrain​\(i\)=κCVRtrain​\(i\)\\kappa\_\{\\mathrm\{CTR\}\}^\{\\mathrm\{train\}\}\(i\)=\\kappa\_\{\\mathrm\{CVR\}\}^\{\\mathrm\{train\}\}\(i\), calibration consistency requiresκCTRcali​\(i\)=κCVRcali​\(i\)\\kappa\_\{\\mathrm\{CTR\}\}^\{\\mathrm\{cali\}\}\(i\)=\\kappa\_\{\\mathrm\{CVR\}\}^\{\\mathrm\{cali\}\}\(i\), and the end\-to\-end invariant requires all four to agree asκCTRtrain​\(i\)=κCVRtrain​\(i\)=κCTRcali​\(i\)=κCVRcali​\(i\)\\kappa\_\{\\mathrm\{CTR\}\}^\{\\mathrm\{train\}\}\(i\)=\\kappa\_\{\\mathrm\{CVR\}\}^\{\\mathrm\{train\}\}\(i\)=\\kappa\_\{\\mathrm\{CTR\}\}^\{\\mathrm\{cali\}\}\(i\)=\\kappa\_\{\\mathrm\{CVR\}\}^\{\\mathrm\{cali\}\}\(i\)\. The first two align the training and calibration stages internally, but they do not link training to calibration\. The shared single\-source dedup cache of Section[5\.3](https://arxiv.org/html/2608.10562#S5.SS3)supplies that link and enforces the invariant by construction\.

### 5\.3Production System Engineering

To enforce global consistency across distributed streaming environments \(Figure[1](https://arxiv.org/html/2608.10562#S1.F1)\), MARCO deploys two core infrastructure services: a*Click Metadata Table*recording event\-level payloads and a low\-latency, durable key\-value*Deduplication Cache*\.

##### Concurrent Click Attribution\.

When interaction events arrive concurrently across distributed edge servers, the engine executes an atomic*first\-write\-wins*check on the Deduplication Cache\. The earliest click event within the last attributed impression is registered as the credited intentκ​\(i\)\\kappa\(i\), while subsequent clicks are marked as deduplicated\. This guarantees key uniqueness and write\-atomicity under heavy write concurrency\.

##### Schema\-Agnostic Conversion Attribution\.

Offsite conversions arrive via asynchronous pipelines with heterogeneous user identifier schemas\. When a conversion event triggers a lookup, the attribution system queries the Deduplication Cache using whichever identifier is present\. Because all identifier paths resolve to the unified underlying cache entry established during click processing, all downstream consumers \(CTR/CVR trainers and calibrators\) observe the identical intent label, fulfilling consistency conditions of Section[5\.2](https://arxiv.org/html/2608.10562#S5.SS2)\.

##### Composition\-Level Fallback Guardrail\.

To protect auction delivery against model dilution or upstream prediction outages in concurrent A/B testing environments, MARCO incorporates a zero\-latency fallback operator:

Φ=ΦMARCO\+𝕀​\(ΦMARCO=0\)⋅Φstd\\Phi=\\Phi\_\{\\mathrm\{MARCO\}\}\+\\mathbb\{I\}\(\\Phi\_\{\\mathrm\{MARCO\}\}=0\)\\cdot\\Phi\_\{\\mathrm\{std\}\}\(7\)If experimental dilution causes intent\-specific predictions to vanish \(e\.g\.,ΦMARCO=0\\Phi\_\{\\mathrm\{MARCO\}\}=0\), the system reverts to the standard scalar scoreΦstd=λ​μ\\Phi\_\{\\mathrm\{std\}\}=\\lambda\\mu, guaranteeing no\-worse\-than\-baseline robustness\.

##### Cold\-Start Protocol\.

MARCO addresses two operational cold\-start scenarios:

1. 1\.Unseen Click Types:New UI interaction features default to the low\-intent group, a low\-risk strategy since high\-intent actions represent established, high\-conversion navigation paths\.
2. 2\.Unwarmed Calibration Services:To resolve the circular dependency where the real\-time calibrator requires prediction traffic before MARCO goes live, we deploy a two\-phase dummy composition:Φphase1=Φstd\+0⋅ΦMARCO\\Phi\_\{\\mathrm\{phase1\}\}=\\Phi\_\{\\mathrm\{std\}\}\+0\\cdot\\Phi\_\{\\mathrm\{MARCO\}\}\. In Phase 1, scores remain identical to production baseline while intent\-specific heads generate shadow predictions to warm up the calibration service\. Once calibration variance converges \(typically 2–3 days\), Phase 2 activates the full MARCO composition \(Eq\.[7](https://arxiv.org/html/2608.10562#S5.E7)\) with zero delivery perturbation\.

## 6Experiments

We evaluate MARCO across both offline log benchmarks and large\-scale online A/B experiments on Meta’s production advertising platforms\. All experiments instantiate the framework at binary intent granularity \(K=2K=2\), capturing the primary contrast based on conversion likelihood\. Online experiments span multi\-week evaluation windows covering hundreds of millions of users with user\-level randomization\. All reported online gains are statistically significant atp<0\.05p<0\.05under95%95\\%confidence intervals\. Automated fallback \(§[5\.3](https://arxiv.org/html/2608.10562#S5.SS3)\) triggers on<0\.1%<0\.1\\%of traffic, ensuring near\-complete treatment exposure\. Our empirical evaluation addresses four primaryResearchQuestions:

- •𝐑𝐐𝟏\\mathbf\{RQ1\}Per\-Intent Calibration:Does the observability barrier of Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)hold in practice? Do progressively stronger single\-output methods such as auxiliary tasks, historical features, and MoE gating remain unable to reduce per\-intent miscalibration, while a model that exposes per\-intent outputs, such as MARCO, eliminates it?
- •𝐑𝐐𝟐\\mathbf\{RQ2\}Composed Post\-Impression NE:Does MARCO improve end\-to\-end post\-impression conversion prediction, and do empirical results align with theoretical headroom \(Δℱ\\Delta\_\{\\mathcal\{F\}\}\) and routing efficiency \(η^\\hat\{\\eta\}\)?
- •𝐑𝐐𝟑\\mathbf\{RQ3\}Pre\-Impression Funnel Recall:Does restoring per\-intent calibration improve candidate retrieval fidelity in early ranking stages?
- •𝐑𝐐𝟒\\mathbf\{RQ4\}Online Business Impact:What is the topline business impact in live auctions, and how does score separation alter candidate ranking?

### 6\.1Offline Evaluation

#### 6\.1\.1Post\-click: per\-intent calibration\(𝐑𝐐𝟏\)\\mathbf\{\(RQ1\)\}

We compare MARCO against four progressively stronger paradigms for utilizing intent signals short of structural decomposition:

1. 1\.Baseline \(Pooled CVR\):An ESMM\-style vertical funnel model\[[1](https://arxiv.org/html/2608.10562#bib.bib1),[2](https://arxiv.org/html/2608.10562#bib.bib2)\]fitting a single conversion head over all clicks\.
2. 2\.\+ Auxiliary Intent Tasks:Appends auxiliary task heads to predict per\-intent distribution, testing whether auxiliary representation learning resolves pooled head miscalibration\.
3. 3\.\+ Historical Intent Feature:Feeds historical intent distributions as impression\-time input features—the strongest intent signal observable prior to interaction \(Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)\)\.
4. 4\.\+ MoE on Historical Intent:Deploys a Mixture\-of\-Experts \(MoE\) architecture gated on historical intent features, testing whether architectural gating eliminates subgroup bias\.

Table 2:Per\-intent calibration\-error reduction relative to the pooled baseline \(%\), higher is better\. Single\-output architectures are bounded by Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1), whereas MARCO eliminates over99%99\\%of per\-intent error\.Table[2](https://arxiv.org/html/2608.10562#S6.T2)presents the evaluation ladder\. Extra capacity through the auxiliary task leaves per\-intent calibration unchanged \(0%0\\%\)\. Historical intent features and MoE gating yield marginal improvements \(≤8%\\leq 8\\%\)\. Because these baselines emit a single outputμ\\mufor a mixed click population, Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)imposes a strict observational upper bound: a single prediction cannot simultaneously match distinct conditional targetsμk\\mu\_\{k\}\. MARCO removes essentially all of the per\-intent calibration error, more than an order of magnitude above the nearest alternative, so the effect is unambiguous\.

This evaluation ladder focuses on single\-output architectures to empirically validate the observability limit in Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)\. Because these models emit a single prediction prior to interaction, their calibration ceiling remains fixed regardless of added tasks, historical features, or MoE gating\. While any architecture with per\-intent output heads can bypass this barrier, MARCO provides the minimal, zero\-latency design needed at industrial scale\.

#### 6\.1\.2Composed Post\-impression NE\(𝐑𝐐𝟐\)\\mathbf\{\(RQ2\)\}

The post\-impression evaluation assesses the end\-to\-end conversion predictionΦ\\Phiacross the global impression space, composing CTR and CVR scores via Eq\.[1](https://arxiv.org/html/2608.10562#S1.E1)for the baseline and Eq\.[2](https://arxiv.org/html/2608.10562#S2.E2)for MARCO\. We evaluate offline performance on production data\[[17](https://arxiv.org/html/2608.10562#bib.bib17)\]across a 7\-day evaluation window\. Table[3](https://arxiv.org/html/2608.10562#S6.T3)reports post\-impression Normalized Entropy gains across a comprehensive evaluation matrix, systematically varying CTR backbone capacity \(Base, Large 5×\\times, and an Oracle router evaluating intent routing under ground\-truthP​\(k∣click\)P\(k\\mid\\text\{click\}\)while using predicted click rates\), composition architecture \(MARCO vs CTR\-split\-only\), and calibration adjustment strategies \(None, CVR\-only, CTR\-only, and joint CTR & CVR\)\.

Table 3:Composed\-model post\-impression NE gain across evaluation settings \(%\), higher is better, with 95% confidence intervals reported against standard composition\.Related RQComposition LogicCalibration AdjustmentNE Gain \(%\) \[95% CI\]Ablation Analysis \(Base Model\)2\.1MARCONone\+0\.054\+0\.054\[\+0\.043,\+0\.067\]\[\+0\.043,\+0\.067\]2\.1, 2\.2MARCOCTR & CVR\+0\.223\+0\.223\[\+0\.210,\+0\.235\]\[\+0\.210,\+0\.235\]2\.2\.1MARCOCVR only\+0\.181\+0\.181\[\+0\.168,\+0\.194\]\[\+0\.168,\+0\.194\]2\.2\.2MARCOCTR only\+0\.097\+0\.097\[\+0\.088,\+0\.110\]\[\+0\.088,\+0\.110\]2\.3CTR\-split\-onlyNone\+0\.002\+0\.002\[−0\.020,\+0\.007\]\[\-0\.020,\+0\.007\]Capacity Scaling & Theoretical Bound2\.4MARCO \(Large CTR,5×5\\times\)None\+0\.092\+0\.092\[\+0\.067,\+0\.112\]\[\+0\.067,\+0\.112\]2\.4MARCO \(Oracle Router\)None\+2\.268\+2\.268\[\+2\.235,\+2\.302\]\[\+2\.235,\+2\.302\]Table[3](https://arxiv.org/html/2608.10562#S6.T3)reveals four key findings directly validating our theoretical framework:

1. 1\.𝐑𝐐𝟐​\.1\\mathbf\{RQ2\.1\}Empirical Weak Dominance:MARCO strictly outperforms the pooled baseline in post\-impression NE across all model capacities and calibration settings in Table[3](https://arxiv.org/html/2608.10562#S6.T3)\(e\.g\.,\+0\.054%\+0\.054\\%pre\-calibration under Base CTR\)\. This universal, strictly positive lift empirically validates Weak Dominance \(Δℱ≥0\\Delta\_\{\\mathcal\{F\}\}\\geq 0, Theorem[1](https://arxiv.org/html/2608.10562#Thmtheorem1)\(i\)\), confirming that structural intent decomposition expands the hypothesis class without inflating population risk\.
2. 2\.𝐑𝐐𝟐​\.2\\mathbf\{RQ2\.2\}Calibration Service Synergy:Pre\-calibration gains jump by over4×4\\timeswhen passing through the online calibration service \(\+0\.054%\+0\.054\\%to\+0\.223%\+0\.223\\%for Base CTR under joint CTR & CVR calibration\)\. Isolating individual calibration components reveals that CVR\-only calibration accounts for the majority of this lift \(\+0\.181%\+0\.181\\%\), whereas CTR\-only calibration contributes\+0\.097%\+0\.097\\%\. Unmixing per\-intent outputs allows CVR calibrators to adjust each intent head against its homogeneous sub\-population, resolving group\-conditional errors that aggregate calibrators are structurally blind to \(Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)\)\.
3. 3\.𝐑𝐐𝟐​\.3\\mathbf\{RQ2\.3\}Estimation Coupling Bias \(εcouple\\varepsilon\_\{\\mathrm\{couple\}\}\):The CTR\-split\-only ablation yields a statistically negligible pre\-calibration gain \(\+0\.002%\+0\.002\\%\[−0\.020%,\+0\.007%\]\[\-0\.020\\%,\+0\.007\\%\]\)\. Because∑kλk=λ\\sum\_\{k\}\\lambda\_\{k\}=\\lambda\(Lemma[9](https://arxiv.org/html/2608.10562#Thmlemma9)\), splitting the CTR head without CVR\-side decomposition provides zero CVR headroom while introducing estimation coupling biasεcouple\\varepsilon\_\{\\mathrm\{couple\}\}\. This confirms that CVR\-side structural intent decomposition is strictly required to capture post\-impression gains\.
4. 4\.𝐑𝐐𝟐​\.4\\mathbf\{RQ2\.4\}Routing Efficiency Expansion \(η^\\hat\{\\eta\}\):Expanding CTR model capacity from1×1\\times\(Base\) to5×5\\times\(Large\) increases the pre\-calibration composed NE gain from\+0\.054%\+0\.054\\%to\+0\.092%\+0\.092\\%in Table[3](https://arxiv.org/html/2608.10562#S6.T3), demonstrating that scaling intent prediction capacity directly unlocks higher post\-impression accuracy\. When intent routing is driven by an Oracle router \(π^=π\\widehat\{\\pi\}=\\pi\), the NE gain reaches\+2\.268%\+2\.268\\%\[\+2\.235%,\+2\.302%\]\[\+2\.235\\%,\+2\.302\\%\], establishing the empirical upper bound for realizable theoretical headroomΔℱ\\Delta\_\{\\mathcal\{F\}\}\. Benchmarking pre\-calibration gains against Oracle headroom \(Δℱ=\+2\.268%\\Delta\_\{\\mathcal\{F\}\}=\+2\.268\\%\) confirms that scaling CTR capacity increases realized routing efficiencyη^\\hat\{\\eta\}from2\.38%2\.38\\%\(Base\) to4\.06%4\.06\\%\(Large\)\. This capacity progression directly validates Proposition[7](https://arxiv.org/html/2608.10562#Thmproposition7): expanding CTR capacity enlarges the realizable routing classΠ\(c\)\\Pi^\{\(c\)\}, systematically reducing the heterogeneity\-weighted routing costεroute\\varepsilon\_\{\\mathrm\{route\}\}and raising serving efficiencyη^\\hat\{\\eta\}toward100%100\\%\.

#### 6\.1\.3Pre\-Impression Funnel Recall\(𝐑𝐐𝟑\)\\mathbf\{\(RQ3\)\}

The pre\-impression stage governs early\-stage candidate retrieval, deciding which ad candidates survive to full auction scoring\. We evaluate candidate retrieval fidelity on offline production logs across a 3\-day evaluation window by truncating candidate sets at an early\-funnel scale of𝒪​\(102\)\\mathcal\{O\}\(10^\{2\}\)\. Retrieval quality is measured using value\-weighted recall \(the ratio of value delivered by the truncated candidate list to the ideal unconstrained list\)\. MARCO achieves a\+0\.38%\+0\.38\\%lift in value\-weighted candidate retrieval recall over the pooled baseline\. Restoring per\-intent score calibration at inference improves rank\-order fidelity early in the funnel, ensuring higher\-quality candidates advance to final auction scoring\.

### 6\.2Online Business Impact\(𝐑𝐐𝟒\)\\mathbf\{\(RQ4\)\}

MARCO has been incrementally deployed across multiple ad surfaces, conversion types, and traffic segments on Meta’s social media platforms\.

#### 6\.2\.1Cumulative Production Impact

MARCO was deployed through several successive production launches\. Each launch was conducted as an independent experiment on a disjoint traffic segment, measured against its own concurrent user\-randomized holdback\. Every launch yielded positive gains, collectively covering a substantial fraction of Meta’s ad traffic\. Evaluated globally across all combined launch segments, MARCO delivered a\+0\.98%\+0\.98\\%cumulative lift in topline metrics \(95%95\\%CI:\[\+0\.86%,\+1\.12%\]\[\+0\.86\\%,\+1\.12\\%\]\)\. The confidence interval is a bootstrap percentile interval computed across launches, reflecting performance stability across distinct rollout segments\.

#### 6\.2\.2Detailed Holdback Analysis

To evaluate MARCO’s production performance under live auction dynamics, we conducted a multi\-week, user\-randomized holdback experiment on a primary advertising surface, comparing the proposed intent\-decomposed composition against the standard single\-click baseline\. MARCO achieved a\+2\.80%\+2\.80\\%lift in conversions per click with a95%95\\%confidence interval of\[\+2\.50%,\+3\.10%\]\[\+2\.50\\%,\+3\.10\\%\], confirming substantial end\-to\-end performance gains in live production traffic\.

The online conversion rate improvement \(\+2\.80%\+2\.80\\%\) significantly exceeds the offline impression\-space NE gain \(\+0\.223%\+0\.223\\%\)\. This divergence is expected and attributable to two fundamental structural mechanisms:

1. 1\.Probability Space Compression:Offline NE evaluates the global impression space, where the served scoreΦ=λ​μ\\Phi=\\lambda\\muscales on\-click conversion ratesμ\\muby the baseline click\-through rateλ\\lambda\. Under squared\-loss score headroom analysis \(Proposition[6](https://arxiv.org/html/2608.10562#Thmproposition6)\), impression\-level risk gaps scale proportionally toλ2\\lambda^\{2\}\. Because click\-through rates are small \(λ≪1\\lambda\\ll 1\), offline impression\-space metrics undergo geometric compression relative to raw on\-click conversion gains\.
2. 2\.Auction Reallocation Dynamics:Modern real\-time ad auctions operate as competitive winner\-take\-all mechanisms\. By eliminating group\-conditional miscalibration, MARCO achieves superior score separation across intent strata \(Section[6\.2\.3](https://arxiv.org/html/2608.10562#S6.SS2.SSS3)\)\. This enhanced score fidelity allows high\-intent candidate ads to systematically win auctions while suppressing low\-converting traffic, dynamically reallocating delivery budget toward higher\-converting opportunities and amplifying calibrated scoring gains into non\-linear online business lifts\.

#### 6\.2\.3Prediction Distribution Analysis

![Refer to caption](https://arxiv.org/html/2608.10562v1/figs/ecvr-distribution.png)Figure 3:Cumulative Distribution Function \(CDF\) of predicted conversion rates \(eCVR\) at the pre\-impression stage, illustrating score separation under MARCO\.Figure[3](https://arxiv.org/html/2608.10562#S6.F3)illustrates the mechanism driving online auction gains\. For impressions prone to low\-intent clicks, MARCO depresses expected conversion scores, causing the treatment CDF to rise faster in the low\-eCVR regime \(<0\.04<0\.04\)\. Conversely, for impressions prone to high\-intent CTA taps, MARCO amplifies scores, shifting the treatment CDF below the baseline in the high\-eCVR regime \(\>0\.08\>0\.08\)\. This score separation provides the auction mechanism with high\-fidelity conversion signals, systematically suppressing low\-converting candidates and winning auctions for high\-intent traffic\.

## 7Discussion and Conclusion

In this section, we address privacy and ethical considerations, demonstrate the generalizability of intent decomposition to adjacent domains, and outline future technical directions\.

##### Ethical Considerations and Advertiser Alignment\.

MARCO’s design adheres strictly to platform privacy standards and ethical AI practices\. First, regardingdata privacy, MARCO relies exclusively on physical UI interactions already logged in standard interaction pipelines, requiring no new sensitive user profiling or off\-platform tracking\. Second, to ensurestatistical fairness across advertisers, restoring subgroup calibration corrects systematic under\-prediction on high\-intent traffic and over\-prediction on low\-intent traffic, eliminating unfair auction penalties for advertisers optimizing for high\-converting direct actions\. Finally, MARCO remainsobjective\-agnostic, dynamically routing candidate value based on advertiser\-defined campaign goals rather than hard\-coding a preference for direct sales over brand engagement\.

##### Broad Generalizability and Diagnostic Framework\.

The principle of structural decomposition extends beyond click\-intent prediction\. Whenever a single supervised target aggregates events with structurally heterogeneous downstream success rates, single\-output models inherit subgroup bias masked by aggregate calibration\. Representative applications include app install campaigns \(pooling organic\-attributed and ad\-driven installs with distinct 7\-day retention rates\), e\-commerce funnels \(merging quick\-add interactions from product grids with considered\-adds from detailed review pages\), and video recommendation platforms \(combining lightweight preview plays with intentional full\-screen watch sessions\)\.

Industrial practitioners can evaluate whether a target domain warrants intent decomposition prior to model engineering\. By evaluating the impression\-weighted Kullback\-Leibler \(KL\) divergence spread across candidate interaction strata \(following Lemma[3](https://arxiv.org/html/2608.10562#Thmlemma3)and Lemma[11](https://arxiv.org/html/2608.10562#Thmlemma11)\), one can quantify the exact theoretical headroomΔℱ\\Delta\_\{\\mathcal\{F\}\}opened by decomposition directly from historical interaction logs\.

##### Limitations and Future Directions\.

While MARCO achieves substantial production impact at binary intent granularity \(K=2K=2\), theoretical headroomΔℱ\\Delta\_\{\\mathcal\{F\}\}increases monotonically with finer intent partitions\. We outline three key vectors for future research:

1. 1\.Taxonomy Scaling \(Depth & Breadth\):ExpandingKKcan proceed along two dimensions:*depth*\(partitioning CTA clicks into sub\-navigation types\) and*breadth*\(incorporating post\-impression interaction signals such as dwell time, video watch percentage, and organic engagement\)\. Because Weak Dominance holds for arbitrary partitions \(Theorem[1](https://arxiv.org/html/2608.10562#Thmtheorem1)\(i\)\), finer taxonomies never degrade population risk\.
2. 2\.Routing Efficiency Leverage \(η^\\hat\{\\eta\}\):Realized online gain is governed by routing efficiencyη^=1−εroute/Δℱ\\hat\{\\eta\}=1\-\\varepsilon\_\{\\mathrm\{route\}\}/\\Delta\_\{\\mathcal\{F\}\}\. Because routing error concentrates where per\-intent conversion rates diverge \(Lemma[11](https://arxiv.org/html/2608.10562#Thmlemma11)\), scaling the capacity of the*CTR intent predictor*—rather than the CVR model—is the highest\-leverage vector to close the remaining headroom gap\.
3. 3\.Sparse\-Stratum Calibration Infrastructure:AsKKincreases, interaction volume is divided across finer intent buckets\. Rare interaction categories risk calibration variance due to sample sparsity\. Developing dynamic calibrators capable of robust real\-time convergence under long\-tailed intent distributions remains an important system engineering challenge\.

Conventional ads ranking architectures conflate distinct user behaviors into a single click label, imposing a structural observability barrier that no model capacity can resolve\. MARCO demonstrates that decoupling predictions by intent via zero\-cost behavioral supervision restores subgroup calibration, resolves auction score distortions, and delivers substantial topline growth at industrial scale\.

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## Appendix ADerivation of the Headroom and Weak Dominance

This appendix gives the full statements and proofs behind the theory section\. We prove the observability barrier, the Bayes equivalence, the excess\-risk identity, weak dominance in any function class, the exact squared\-loss headroom, the genericity of strict improvement, the score bridge, and the signed decomposition of the realized gain\. The squared\-loss results carry a closed form\. The normalized\-entropy results keep only the loss\-free parts, for the reason given in the mixture\-of\-experts lemma \(Lemma[6](https://arxiv.org/html/2608.10562#Thmlemma6)\)\.

### A\.1Setup and notation

We condition on a clickC=1C=1unless the impression level is named\. LetXXbe impression\-time features with impression distributionρ\\rho\. Letk∈\{1,…,K\}k\\in\\\{1,\\dots,K\\\}index the click intent overKKdisjoint categories, observed at training from the logged click type\. LetY∈\{0,1\}Y\\in\\\{0,1\\\}be the conversion\. Define

μk\\displaystyle\\mu\_\{k\}=𝔼\[Y∣X=x,K=k\],πk=P\(K=k∣X=x,C=1\),\\displaystyle=\\mathbb\{E\}\[Y\\mid X\{=\}x,K\{=\}k\],\\quad\\pi\_\{k\}=P\(K\{=\}k\\mid X\{=\}x,C\{=\}1\),\(8\)μ\\displaystyle\\mu=𝔼​\[Y∣X=x,C=1\]\.\\displaystyle=\\mathbb\{E\}\[Y\\mid X\{=\}x,C\{=\}1\]\.The law of total probability gives the exact identityμ=∑k=1Kπk​μk\\mu=\\sum\_\{k=1\}^\{K\}\\pi\_\{k\}\\mu\_\{k\}\. This identity assumes no homogeneity of clicks\. Letλ=P​\(C=1∣X=x\)\\lambda=P\(C\{=\}1\\mid X\{=\}x\)be the click\-through rate andλk=λ​πk\\lambda\_\{k\}=\\lambda\\pi\_\{k\}the per\-intent click rate, so∑kλk=λ\\sum\_\{k\}\\lambda\_\{k\}=\\lambdaandπk=λk/λ\\pi\_\{k\}=\\lambda\_\{k\}/\\lambda\.

Letℱ\\mathcal\{F\}be the hypothesis class of a single pooled conversion head\. The intent\-decomposed class is

𝒢=\{g=∑k=1Kπk​fk:fk∈ℱ\}\.\\mathcal\{G\}=\\Big\\\{\\,g=\\sum\_\{k=1\}^\{K\}\\pi\_\{k\}f\_\{k\}:\\ f\_\{k\}\\in\\mathcal\{F\}\\,\\Big\\\}\.\(9\)MARCO serves the impression\-level scoreΦMARCO=∑kλk​μk\\Phi\_\{\\mathrm\{MARCO\}\}=\\sum\_\{k\}\\lambda\_\{k\}\\mu\_\{k\}\. The standard model servesΦstd=λ​μ\\Phi\_\{\\mathrm\{std\}\}=\\lambda\\mu\.

### A\.2Bayes equivalence

###### Lemma 1\(Bayes equivalence\)

At the true conditionalsΦMARCO=Φstd\\Phi\_\{\\mathrm\{MARCO\}\}=\\Phi\_\{\\mathrm\{std\}\}\.

###### Proof 1

Useλk=λ​πk\\lambda\_\{k\}=\\lambda\\pi\_\{k\}and∑kπk=1\\sum\_\{k\}\\pi\_\{k\}=1\. ThenΦMARCO=∑kλ​πk​μk=λ​∑kπk​μk=λ​μ=Φstd\\Phi\_\{\\mathrm\{MARCO\}\}=\\sum\_\{k\}\\lambda\\pi\_\{k\}\\mu\_\{k\}=\\lambda\\sum\_\{k\}\\pi\_\{k\}\\mu\_\{k\}=\\lambda\\mu=\\Phi\_\{\\mathrm\{std\}\}\.

At the population level the two scores coincide\. The Bayes\-optimal impression score is unchanged by the decomposition\. Any deployed gain is therefore a finite\-capacity estimation and calibration effect\. The click\-rate decomposition adds no independent factor\. It supplies the routingπk=λk/λ\\pi\_\{k\}=\\lambda\_\{k\}/\\lambdaand the totalλ=∑kλk\\lambda=\\sum\_\{k\}\\lambda\_\{k\}\. The served conversion factor is∑kπk​μk\\sum\_\{k\}\\pi\_\{k\}\\mu\_\{k\}, an element of𝒢\\mathcal\{G\}whose routing is produced by the click\-rate head\. The analysis reduces to the single conversion head of \([9](https://arxiv.org/html/2608.10562#A1.E9)\)\.

### A\.3Excess risk

We first record the squared\-loss identity, then the cross\-entropy identity\.

Squared loss uses the riskℛ​\(f\)=𝔼​\[\(Y−f\)2\]\\mathcal\{R\}\(f\)=\\mathbb\{E\}\[\(Y\-f\)^\{2\}\]\. Write⟨⋅,⋅⟩\\langle\\cdot,\\cdot\\rangleand∥⋅∥\\lVert\\cdot\\rVertfor theL2​\(ρ\)L^\{2\}\(\\rho\)inner product and norm\. he identity below is the standardL2​\(ρ\)L^\{2\}\(\\rho\)projection and bias\-variance decomposition\[[18](https://arxiv.org/html/2608.10562#bib.bib18)\]\.

###### Lemma 2\(Squared\-loss excess risk\)

For anyff,ℛ​\(f\)=ℛ​\(μ\)\+∥f−μ∥2\\mathcal\{R\}\(f\)=\\mathcal\{R\}\(\\mu\)\+\\lVert f\-\\mu\\rVert^\{2\}\. The infimum ofℛ\\mathcal\{R\}over a closed set is attained at theL2​\(ρ\)L^\{2\}\(\\rho\)projection ofμ\\muonto that set\.

###### Proof 2

Expand

ℛ​\(f\)=\\displaystyle\\mathcal\{R\}\(f\)=\{\}𝔼​\[\(Y−μ\)2\]\\displaystyle\\mathbb\{E\}\[\(Y\-\\mu\)^\{2\}\]\(10\)\+2​𝔼​\[\(Y−μ\)​\(μ−f\)\]\\displaystyle\+2\\,\\mathbb\{E\}\[\(Y\-\\mu\)\(\\mu\-f\)\]\+𝔼​\[\(μ−f\)2\]\.\\displaystyle\+\\mathbb\{E\}\[\(\\mu\-f\)^\{2\}\]\.The cross term vanishes because𝔼​\[Y−μ∣X\]=0\\mathbb\{E\}\[Y\-\\mu\\mid X\]=0\. The first term isℛ​\(μ\)\\mathcal\{R\}\(\\mu\)\. The last term is∥f−μ∥2\\lVert f\-\\mu\\rVert^\{2\}\. Minimizing∥f−μ∥2\\lVert f\-\\mu\\rVert^\{2\}over a closed set is projection\.

Cross\-entropy uses the riskℛ​\(f\)=𝔼​\[−Y​log⁡f−\(1−Y\)​log⁡\(1−f\)\]\\mathcal\{R\}\(f\)=\\mathbb\{E\}\[\-Y\\log f\-\(1\-Y\)\\log\(1\-f\)\]forffvalued in\(0,1\)\(0,1\)\. Normalized entropy isNE​\(f\)=ℛ​\(f\)/H​\(y¯\)\\mathrm\{NE\}\(f\)=\\mathcal\{R\}\(f\)/H\(\\bar\{y\}\)withy¯=𝔼​\[Y\]\\bar\{y\}=\\mathbb\{E\}\[Y\]andHHthe binary entropy\.H​\(y¯\)H\(\\bar\{y\}\)is a fixed positive constant, so NE and cross\-entropy share minimizers and orderings\. Every statement below applies to both\.

###### Lemma 3\(Cross\-entropy excess\-risk identity\)

For any\(0,1\)\(0,1\)\-valuedff,

ℛ​\(f\)=ℛ​\(μ\)\+𝔼X​\[KL​\(Bern​\(μ\)∥Bern​\(f\)\)\]\.\\mathcal\{R\}\(f\)=\\mathcal\{R\}\(\\mu\)\+\\mathbb\{E\}\_\{X\}\\big\[\\mathrm\{KL\}\(\\mathrm\{Bern\}\(\\mu\)\\,\\\|\\,\\mathrm\{Bern\}\(f\)\)\\big\]\.\(11\)Henceℛ​\(f\)≥ℛ​\(μ\)\\mathcal\{R\}\(f\)\\geq\\mathcal\{R\}\(\\mu\)with equality ifff=μf=\\mualmost everywhere\.

###### Proof 3

Cross\-entropy is the Bregman divergence generated byψ​\(p\)=p​log⁡p\+\(1−p\)​log⁡\(1−p\)\\psi\(p\)=p\\log p\+\(1\-p\)\\log\(1\-p\)\[[19](https://arxiv.org/html/2608.10562#bib.bib19),[20](https://arxiv.org/html/2608.10562#bib.bib20)\]\. WriteDψ​\(y,f\)=ψ​\(y\)−ψ​\(f\)−ψ′​\(f\)​\(y−f\)D\_\{\\psi\}\(y,f\)=\\psi\(y\)\-\\psi\(f\)\-\\psi^\{\\prime\}\(f\)\(y\-f\)\. Then𝔼​\[Dψ​\(Y,f\)∣X\]=𝔼​\[ψ​\(Y\)∣X\]−ψ​\(f\)−ψ′​\(f\)​\(μ−f\)\\mathbb\{E\}\[D\_\{\\psi\}\(Y,f\)\\mid X\]=\\mathbb\{E\}\[\\psi\(Y\)\\mid X\]\-\\psi\(f\)\-\\psi^\{\\prime\}\(f\)\(\\mu\-f\)because𝔼​\[Y∣X\]=μ\\mathbb\{E\}\[Y\\mid X\]=\\mu\. Subtracting the same expression atf=μf=\\muleavesKL​\(Bern​\(μ\)∥Bern​\(f\)\)\\mathrm\{KL\}\(\\mathrm\{Bern\}\(\\mu\)\\,\\\|\\,\\mathrm\{Bern\}\(f\)\)\. Take the expectation overXX\. Nonnegativity and the equality case follow from strict convexity ofψ\\psi\.

The excess risk under cross\-entropy is an averaged Kullback\-Leibler divergence between the true per\-impression conversion rate and the prediction\. Under the classical calibration–refinement decomposition of a proper scoring rule\[[21](https://arxiv.org/html/2608.10562#bib.bib21)\], this gap splits into a calibration term, which measures how far predictions sit from the conditional conversion rate they are grouped with, and a refinement term, which measures the residual heterogeneity within those groups\. A per\-intent calibration step drives the calibration term to zero within each intent, while decomposition attacks the refinement term by splitting the pooled group into homogeneous intent slices\. The log\-loss gap and per\-intent calibration are thus the same object measured on each intent slice\.

### A\.4Observability barrier

###### Proof 4\(Proof of Proposition[1](https://arxiv.org/html/2608.10562#Thmproposition1)\)

Fixxxand condition on a click\. Being calibrated on intent categorykkmeans the emitted value equals that category’s conditional mean,μ=μk\\mu=\\mu\_\{k\}\. Simultaneous calibration on allKKcategories thus requiresμ=μk\\mu=\\mu\_\{k\}for everykk, which is impossible whenever theμk\\mu\_\{k\}are not all equal, since a single scalar equals at most one element of a set of distinct reals; the same fact gives the at\-most\-one claim\. Under a strictly proper scoring rule the risk\-minimizing single output is the pooled conditional mean𝔼​\[Y∣X=x,C=1\]=∑kπk​μk\\mathbb\{E\}\[Y\\mid X\{=\}x,C\{=\}1\]=\\sum\_\{k\}\\pi\_\{k\}\\mu\_\{k\}\(Lemmas[2](https://arxiv.org/html/2608.10562#Thmlemma2)and[3](https://arxiv.org/html/2608.10562#Thmlemma3)\), which is aggregate\-calibrated yet, as a convex combination of theμk\\mu\_\{k\}, coincides with at most one of them\.

### A\.5Inclusion

###### Lemma 4\(Inclusion\)

ℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}\.

###### Proof 5

Givenf∈ℱf\\in\\mathcal\{F\}, setfk≡ff\_\{k\}\\equiv ffor allkkin \([9](https://arxiv.org/html/2608.10562#A1.E9)\)\. Then∑kπk​f=f\\sum\_\{k\}\\pi\_\{k\}f=fbecause∑kπk=1\\sum\_\{k\}\\pi\_\{k\}=1\. This is set\-theoretic and uses neither the loss nor any structure onℱ\\mathcal\{F\}\.

Inclusion gives the headroom used throughout\. Define

Δℱ=inff∈ℱℛ​\(f\)−infg∈𝒢ℛ​\(g\),\\Delta\_\{\\mathcal\{F\}\}=\\inf\_\{f\\in\\mathcal\{F\}\}\\mathcal\{R\}\(f\)\-\\inf\_\{g\\in\\mathcal\{G\}\}\\mathcal\{R\}\(g\),\(12\)the risk gap between the best\-in\-class pooled and decomposed models\. Weak dominance \(Δℱ≥0\\Delta\_\{\\mathcal\{F\}\}\\geq 0\) is the special case of the responsibility\-gated dominance theorem \(Theorem[1](https://arxiv.org/html/2608.10562#Thmtheorem1)\(i\)\) at the supervised gateπk=P\(K=k∣X=x,C=1\)\\pi\_\{k\}=P\(K\{=\}k\\mid X\{=\}x,C\{=\}1\): sinceℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}, the infimum over the superset can only decrease\. It holds for squared loss and for cross\-entropy, needs no linear structure and no capacity assumption, so the decomposed objective is never worse than the pooled objective in any function class\. This is why the improvement is structural rather than a capacity effect\.

### A\.6Exact squared\-loss headroom

Takeℱ\\mathcal\{F\}to be a finite\-dimensional linear subspace ofL2​\(ρ\)L^\{2\}\(\\rho\)with basis\{e1,…,eD\}\\\{e\_\{1\},\\dots,e\_\{D\}\\\}\. This is the finite\-capacity model for a single scalar head\.

###### Lemma 5\(𝒢\\mathcal\{G\}is a subspace\)

𝒢=span​\{πk​ej:1≤k≤K,1≤j≤D\}\\mathcal\{G\}=\\mathrm\{span\}\\\{\\pi\_\{k\}e\_\{j\}:\\ 1\\leq k\\leq K,\\ 1\\leq j\\leq D\\\}, a subspace of dimension at mostK​DKD, andℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}\. Moreover𝒢=ℱ\\mathcal\{G\}=\\mathcal\{F\}iffπk​ej∈ℱ\\pi\_\{k\}e\_\{j\}\\in\\mathcal\{F\}for allkkandjj\. Equivalently𝒢⊋ℱ\\mathcal\{G\}\\supsetneq\\mathcal\{F\}iff there existkkandjjwithπk​ej∉ℱ\\pi\_\{k\}e\_\{j\}\\notin\\mathcal\{F\}\.

###### Proof 6

Writefk=∑jak​j​ejf\_\{k\}=\\sum\_\{j\}a\_\{kj\}e\_\{j\}\. Theng=∑k,jak​j​\(πk​ej\)g=\\sum\_\{k,j\}a\_\{kj\}\(\\pi\_\{k\}e\_\{j\}\), so𝒢=span​\{πk​ej\}\\mathcal\{G\}=\\mathrm\{span\}\\\{\\pi\_\{k\}e\_\{j\}\\\}\. This is a subspace\. The inclusion lemma givesℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}\. If everyπk​ej∈ℱ\\pi\_\{k\}e\_\{j\}\\in\\mathcal\{F\}then𝒢⊆ℱ\\mathcal\{G\}\\subseteq\\mathcal\{F\}, so𝒢=ℱ\\mathcal\{G\}=\\mathcal\{F\}\. Conversely𝒢=ℱ\\mathcal\{G\}=\\mathcal\{F\}forces each generatorπk​ej∈ℱ\\pi\_\{k\}e\_\{j\}\\in\\mathcal\{F\}\.

###### Proposition 2\(Non\-constant routing is necessary but not sufficient\)

Constantπk\\pi\_\{k\}givesπk​ej∈ℱ\\pi\_\{k\}e\_\{j\}\\in\\mathcal\{F\}, so non\-constant routing is necessary for𝒢⊋ℱ\\mathcal\{G\}\\supsetneq\\mathcal\{F\}\. It is not sufficient\. Letρ\\rhohave atomsx1,x2,x3x\_\{1\},x\_\{2\},x\_\{3\}and letℱ=span​\{𝟏x1,𝟏x2\}\\mathcal\{F\}=\\mathrm\{span\}\\\{\\mathbf\{1\}\_\{x\_\{1\}\},\\mathbf\{1\}\_\{x\_\{2\}\}\\\}\. Letπ1\\pi\_\{1\}take distinct values onx1x\_\{1\}andx2x\_\{2\}and letπ2=1−π1\\pi\_\{2\}=1\-\\pi\_\{1\}\. Multiplication byπk\\pi\_\{k\}is diagonal, soπk​ej∈ℱ\\pi\_\{k\}e\_\{j\}\\in\\mathcal\{F\}and𝒢=ℱ\\mathcal\{G\}=\\mathcal\{F\}\. The correct condition is the one in the subspace lemma\.

###### Proposition 3\(Exact headroom\)

LetPℱ,P𝒢P\_\{\\mathcal\{F\}\},P\_\{\\mathcal\{G\}\}be theL2​\(ρ\)L^\{2\}\(\\rho\)projections ontoℱ,𝒢\\mathcal\{F\},\\mathcal\{G\}\. Then

Δℱ\\displaystyle\\Delta\_\{\\mathcal\{F\}\}=inff∈ℱℛ​\(f\)−infg∈𝒢ℛ​\(g\)\\displaystyle=\\inf\_\{f\\in\\mathcal\{F\}\}\\mathcal\{R\}\(f\)\-\\inf\_\{g\\in\\mathcal\{G\}\}\\mathcal\{R\}\(g\)\(13\)=∥μ−Pℱ​μ∥2−∥μ−P𝒢​μ∥2\\displaystyle=\\lVert\\mu\-P\_\{\\mathcal\{F\}\}\\mu\\rVert^\{2\}\-\\lVert\\mu\-P\_\{\\mathcal\{G\}\}\\mu\\rVert^\{2\}=∥P𝒢​μ−Pℱ​μ∥2≥0,\\displaystyle=\\lVert P\_\{\\mathcal\{G\}\}\\mu\-P\_\{\\mathcal\{F\}\}\\mu\\rVert^\{2\}\\ \\geq 0,withΔℱ=0\\Delta\_\{\\mathcal\{F\}\}=0iffP𝒢​μ=Pℱ​μP\_\{\\mathcal\{G\}\}\\mu=P\_\{\\mathcal\{F\}\}\\mu\.

###### Proof 7

By the squared\-loss excess\-risk lemma the two infima areℛ​\(μ\)\+∥μ−Pℱ​μ∥2\\mathcal\{R\}\(\\mu\)\+\\lVert\\mu\-P\_\{\\mathcal\{F\}\}\\mu\\rVert^\{2\}andℛ​\(μ\)\+∥μ−P𝒢​μ∥2\\mathcal\{R\}\(\\mu\)\+\\lVert\\mu\-P\_\{\\mathcal\{G\}\}\\mu\\rVert^\{2\}\. Sinceℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}we havePℱ=Pℱ​P𝒢P\_\{\\mathcal\{F\}\}=P\_\{\\mathcal\{F\}\}P\_\{\\mathcal\{G\}\}\. The Pythagorean identity on nested subspaces \(the classical projection theorem\[[18](https://arxiv.org/html/2608.10562#bib.bib18)\]\) gives∥μ−Pℱ​μ∥2=∥μ−P𝒢​μ∥2\+∥P𝒢​μ−Pℱ​μ∥2\\lVert\\mu\-P\_\{\\mathcal\{F\}\}\\mu\\rVert^\{2\}=\\lVert\\mu\-P\_\{\\mathcal\{G\}\}\\mu\\rVert^\{2\}\+\\lVert P\_\{\\mathcal\{G\}\}\\mu\-P\_\{\\mathcal\{F\}\}\\mu\\rVert^\{2\}\.

No assumption thatμk∈ℱ\\mu\_\{k\}\\in\\mathcal\{F\}orμ∈ℱ\\mu\\in\\mathcal\{F\}is used\. The headroom is the energy ofμ\\mualong the intent\-modulated directions in𝒢\\mathcal\{G\}that are orthogonal toℱ\\mathcal\{F\}\.

###### Lemma 6\(No closed form under normalized entropy\)

Under cross\-entropy the classℱ\\mathcal\{F\}cannot be a linear subspace\. A finite\-dimensional subspace contains functions valued outside\(0,1\)\(0,1\)on a set of positive probability, where cross\-entropy is infinite\. Modeling in logit space withf=σ​\(z\)f=\\sigma\(z\)makes the decomposed predictor∑kπk​σ​\(zk\)\\sum\_\{k\}\\pi\_\{k\}\\sigma\(z\_\{k\}\)a mixture of experts, becauseσ​\(∑kπk​zk\)≠∑kπk​σ​\(zk\)\\sigma\(\\sum\_\{k\}\\pi\_\{k\}z\_\{k\}\)\\neq\\sum\_\{k\}\\pi\_\{k\}\\sigma\(z\_\{k\}\)\. So𝒢\\mathcal\{G\}is not a subspace: there are no orthogonal projections and no Pythagorean identity, hence no closed\-form projection expression for the cross\-entropy headroomΔℱ\\Delta\_\{\\mathcal\{F\}\}analogous to the squared\-loss form\. The cross\-entropy risk itself remains well defined and finite; only its best\-in\-class gap lacks a closed form\. Inclusion and weak dominance \(Δℱ≥0\\Delta\_\{\\mathcal\{F\}\}\\geq 0\) still hold because they are set\-theoretic\. We therefore measureΔℱ\\Delta\_\{\\mathcal\{F\}\}empirically as the held\-out NE gap between the best pooled and the best decomposed model of the same architecture and capacity\.

### A\.7Genericity of strict improvement

###### Lemma 7\(Zero set of a real\-analytic function\)

Ifh:U→ℝh:U\\to\\mathbb\{R\}is real\-analytic on a connected openU⊆ℝmU\\subseteq\\mathbb\{R\}^\{m\}andh≢0h\\not\\equiv 0, then\{h=0\}\\\{h=0\\\}has Lebesgue measure zero\[[22](https://arxiv.org/html/2608.10562#bib.bib22)\]\.

###### Proposition 4\(Genericity, squared loss\)

LetΘ⊆ℝm\\Theta\\subseteq\\mathbb\{R\}^\{m\}be open and connected\. Fixρ\\rhoand the routingπ\\pi, henceℱ,𝒢,Pℱ,P𝒢\\mathcal\{F\},\\mathcal\{G\},P\_\{\\mathcal\{F\}\},P\_\{\\mathcal\{G\}\}\. Assume

- \(A1\)θ↦μθ\\theta\\mapsto\\mu\_\{\\theta\}is real\-analytic intoL2​\(ρ\)L^\{2\}\(\\rho\), with a localL2​\(ρ\)L^\{2\}\(\\rho\)envelope dominatingθ↦μθ\\theta\\mapsto\\mu\_\{\\theta\}, so Hilbert\-valued analyticity holds\.
- \(A2\)there isθ0\\theta\_\{0\}withP𝒢​μθ0≠Pℱ​μθ0P\_\{\\mathcal\{G\}\}\\mu\_\{\\theta\_\{0\}\}\\neq P\_\{\\mathcal\{F\}\}\\mu\_\{\\theta\_\{0\}\}\.

Then\{θ:Δℱ​\(θ\)=0\}\\\{\\theta:\\Delta\_\{\\mathcal\{F\}\}\(\\theta\)=0\\\}has Lebesgue measure zero\. Under any prior absolutely continuous with respect to Lebesgue measure,Δℱ\>0\\Delta\_\{\\mathcal\{F\}\}\>0almost surely\.

###### Proof 8

Letr​\(θ\)=∥P𝒢​μθ−Pℱ​μθ∥2r\(\\theta\)=\\lVert P\_\{\\mathcal\{G\}\}\\mu\_\{\\theta\}\-P\_\{\\mathcal\{F\}\}\\mu\_\{\\theta\}\\rVert^\{2\}\. Sinceπ\\piandρ\\rhoare fixed,PℱP\_\{\\mathcal\{F\}\}andP𝒢P\_\{\\mathcal\{G\}\}are fixed bounded linear operators\. By \(A1\)θ↦\(P𝒢−Pℱ\)​μθ\\theta\\mapsto\(P\_\{\\mathcal\{G\}\}\-P\_\{\\mathcal\{F\}\}\)\\mu\_\{\\theta\}is real\-analytic intoL2​\(ρ\)L^\{2\}\(\\rho\)\. Composing with the continuous bilinear inner product makesrrreal\-analytic\. By \(A2\),r​\(θ0\)\>0r\(\\theta\_\{0\}\)\>0, sor≢0r\\not\\equiv 0\. The analytic zero\-set lemma makes\{r=0\}\\\{r=0\\\}null\. The exact\-headroom proposition givesΔℱ=r\\Delta\_\{\\mathcal\{F\}\}=r\.

###### Corollary 1\(Affine instance\)

Letℱ\\mathcal\{F\}be affine,K=2K=2, letμH,μL\\mu\_\{H\},\\mu\_\{L\}be affine, and letπH\\pi\_\{H\}be affine\. Fromμ=μL\+πH​\(μH−μL\)\\mu=\\mu\_\{L\}\+\\pi\_\{H\}\(\\mu\_\{H\}\-\\mu\_\{L\}\), a product of two affine functions is affine iff a factor is constant\. Soμ∈ℱ\\mu\\in\\mathcal\{F\}iffπH\\pi\_\{H\}is constant orμH−μL\\mu\_\{H\}\-\\mu\_\{L\}is constant\. That is a finite union of proper affine subspaces of parameter space, hence null\. Anyθ\\thetaoutside it witnesses \(A2\)\. No analytic machinery is needed here\.

###### Proposition 5\(Genericity, cross\-entropy\)

Parametrize the conditionals byθ\\thetain a connected openΘ⊆ℝm\\Theta\\subseteq\\mathbb\{R\}^\{m\}\. Supposeθ↦inff∈ℱℛθ​\(f\)\\theta\\mapsto\\inf\_\{f\\in\\mathcal\{F\}\}\\mathcal\{R\}\_\{\\theta\}\(f\)andθ↦infg∈𝒢ℛθ​\(g\)\\theta\\mapsto\\inf\_\{g\\in\\mathcal\{G\}\}\\mathcal\{R\}\_\{\\theta\}\(g\)are real\-analytic, which holds for a smooth link, a strictly convex loss, and a unique interior minimizer with non\-singular Hessian by the implicit function theorem\. SupposeΔℱ​\(θ0\)\>0\\Delta\_\{\\mathcal\{F\}\}\(\\theta\_\{0\}\)\>0for someθ0\\theta\_\{0\}\. Then\{θ:Δℱ​\(θ\)=0\}\\\{\\theta:\\Delta\_\{\\mathcal\{F\}\}\(\\theta\)=0\\\}has Lebesgue measure zero\.

###### Proof 9

Δℱ​\(θ\)=infℱℛθ−inf𝒢ℛθ\\Delta\_\{\\mathcal\{F\}\}\(\\theta\)=\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\_\{\\theta\}\-\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}\_\{\\theta\}is real\-analytic by hypothesis and non\-zero atθ0\\theta\_\{0\}\. The analytic zero\-set lemma makes its zero set null\. The passage from a null set to probability zero is absolute continuity\. The content is the analyticity of the two infima, which for cross\-entropy is a genuine implicit\-function argument rather than a projection identity\.

###### Lemma 8\(Genericity is not a deployment guarantee\)

Genericity rules out the knife\-edge in which the pooled class already fitsμ\\muas well as the decomposed class does\. The deployed system is a single unknown truthθ∗\\theta^\{\\ast\}, which can lie on the null set\. Structured configurations such as coarse intent buckets, symmetries, or sparsity are exactly where it might\. The operative claim for deployment is the measured headroom, not the prior\.

### A\.8Responsibility\-gated dominance

###### Theorem 1\(Responsibility\-gated dominance\)

Letℱ\\mathcal\{F\}be any single\-expert hypothesis class\. Let the gateπ=\(π1,…,πK\)\\pi=\(\\pi\_\{1\},\\dots,\\pi\_\{K\}\)withπk=P\(K=k∣X=x,C=1\)\\pi\_\{k\}=P\(K\{=\}k\\mid X\{=\}x,C\{=\}1\)be a supervised, input\-dependent responsibility over an intentKKobserved at training, with∑kπk=1\\sum\_\{k\}\\pi\_\{k\}=1, and define𝒢=\{∑kπk​fk:fk∈ℱ\}\\mathcal\{G\}=\\\{\\sum\_\{k\}\\pi\_\{k\}f\_\{k\}:f\_\{k\}\\in\\mathcal\{F\}\\\}\. Then for any riskℛ\\mathcal\{R\}:

- \(i\)no worse:inf𝒢ℛ≤infℱℛ\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}\\leq\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\.
- \(ii\)generically better: if the truth is parametrized byθ\\thetain a connected open set withθ↦infℱℛθ,inf𝒢ℛθ\\theta\\mapsto\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\_\{\\theta\},\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}\_\{\\theta\}real\-analytic andinf𝒢ℛ<infℱℛ\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}<\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}at oneθ0\\theta\_\{0\}, then\{θ:inf𝒢ℛ=infℱℛ\}\\\{\\theta:\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}=\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\\\}is Lebesgue\-null\.
- \(iii\)not BMA: the gate is a fixed per\-example responsibility, not a posterior over models, so the minimizerg⋆=∑kπk​fk⋆g^\{\\star\}=\\sum\_\{k\}\\pi\_\{k\}f\_\{k\}^\{\\star\}does not collapse to a single expert asn→∞n\\to\\inftywheneverπ\\piis non\-degenerate and theμk\\mu\_\{k\}differ\.

###### Proof 10\(Proof sketch\)

\(i\) Settingfk≡ff\_\{k\}\\equiv fgives∑kπk​f=f\\sum\_\{k\}\\pi\_\{k\}f=fsince∑kπk=1\\sum\_\{k\}\\pi\_\{k\}=1, soℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}, and the infimum over a superset can only decrease\. \(ii\) Letr​\(θ\)=infℱℛθ−inf𝒢ℛθ≥0r\(\\theta\)=\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\_\{\\theta\}\-\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}\_\{\\theta\}\\geq 0, real\-analytic by hypothesis and positive atθ0\\theta\_\{0\}, sor≢0r\\not\\equiv 0, and by Mityagin’s theorem its zero set is Lebesgue\-null\. Under squared lossr​\(θ\)=∥\(P𝒢−Pℱ\)​μθ∥2r\(\\theta\)=\\lVert\(P\_\{\\mathcal\{G\}\}\-P\_\{\\mathcal\{F\}\}\)\\mu\_\{\\theta\}\\rVert^\{2\}is analytic directly\. \(iii\) Bayesian model averaging weightswk=P​\(Mk∣Dn\)w\_\{k\}=P\(M\_\{k\}\\mid D\_\{n\}\)concentrate on one model by posterior consistency, so∑kwk​fk→fk⋆\\sum\_\{k\}w\_\{k\}f\_\{k\}\\to f\_\{k^\{\\star\}\}, which is selection, not combination\. Hereπk\\pi\_\{k\}is the data\-generating responsibility, constant innnand varying inxx, sog⋆g^\{\\star\}stays a genuine input\-dependent mixture\.

Part \(iii\) records the standard mixture\-of\-experts versus Bayesian model averaging distinction\[[23](https://arxiv.org/html/2608.10562#bib.bib23),[24](https://arxiv.org/html/2608.10562#bib.bib24)\]\. A fitted input\-dependent gate combines rather than selects, whereas BMA weights concentrate on a single model, so BMA does not enlarge the class\. We include it for completeness and do not claim it as novel, since combination rather than selection is the defining property of a mixture of experts\. What is distinctive to MARCO is not combination versus selection but that the gate is a supervised observed responsibility read from the logged click type, which lets each expert be exposed and separately calibrated, a property that sits outside both latent\-gate mixtures of experts and Bayesian model averaging\. Parts \(i\) and \(ii\) are the weak\-dominance and genericity results, here stated for a general supervised responsibility gate\.

### A\.9Realized gain is a signed decomposition

Δℱ\\Delta\_\{\\mathcal\{F\}\}is a population best\-in\-class quantity\. The deployed model uses estimated routingπ^\\widehat\{\\pi\}, finite\-sample heads, and a stratified training objective\. We express the realized gain as an exact telescoping identity through reference predictors: the best\-in\-class pooled modelPℱ​μP\_\{\\mathcal\{F\}\}\\muwith riskinfℱℛ\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}, the best\-in\-class decomposed modelP𝒢​μP\_\{\\mathcal\{G\}\}\\muwith riskinf𝒢ℛ\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}, the stratified predictorgstrat=∑kπk​Pℱ​μk∈𝒢g\_\{\\mathrm\{strat\}\}=\\sum\_\{k\}\\pi\_\{k\}P\_\{\\mathcal\{F\}\}\\mu\_\{k\}\\in\\mathcal\{G\}that fits each head on its own intent slice and routes by the trueπ\\pi, and the deployed modelg^\\widehat\{g\}with estimated heads and routingπ^\\widehat\{\\pi\}\. Writingf^\\widehat\{f\}for the fitted pooled model,

ℛ​\(f^\)−ℛ​\(g^\)=\\displaystyle\\mathcal\{R\}\(\\widehat\{f\}\)\-\\mathcal\{R\}\(\\widehat\{g\}\)=\{\}Δℱ\+\(ℛ​\(f^\)−infℱℛ\)⏟estℱ≥0\\displaystyle\\Delta\_\{\\mathcal\{F\}\}\+\\underbrace\{\\big\(\\mathcal\{R\}\(\\widehat\{f\}\)\-\\textstyle\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\\big\)\}\_\{\\mathrm\{est\}\_\{\\mathcal\{F\}\}\\geq 0\}\(14\)−\(ℛ​\(gstrat\)−inf𝒢ℛ\)⏟εcouple≥0−\(ℛ​\(g^\)−ℛ​\(gstrat\)\)⏟εroute\.\\displaystyle\-\\underbrace\{\\big\(\\mathcal\{R\}\(g\_\{\\mathrm\{strat\}\}\)\-\\textstyle\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}\\big\)\}\_\{\\varepsilon\_\{\\mathrm\{couple\}\}\\geq 0\}\-\\underbrace\{\\big\(\\mathcal\{R\}\(\\widehat\{g\}\)\-\\mathcal\{R\}\(g\_\{\\mathrm\{strat\}\}\)\\big\)\}\_\{\\varepsilon\_\{\\mathrm\{route\}\}\}\.The identity is exact by cancellation, for any choice of reference predictors\. Three terms carry a fixed sign by construction:Δℱ≥0\\Delta\_\{\\mathcal\{F\}\}\\geq 0is weak dominance,estℱ≥0\\mathrm\{est\}\_\{\\mathcal\{F\}\}\\geq 0is the pooled estimation error, andεcouple≥0\\varepsilon\_\{\\mathrm\{couple\}\}\\geq 0becausegstrat∈𝒢g\_\{\\mathrm\{strat\}\}\\in\\mathcal\{G\}cannot beat the joint minimizerP𝒢​μP\_\{\\mathcal\{G\}\}\\mu, whose heads solve a coupled least\-squares problem rather than the per\-slice projections\. It is the price of training heads separately rather than jointly \(Assumption[1](https://arxiv.org/html/2608.10562#Thmassumption1)\)\. The remaining termεroute=ℛ​\(g^\)−ℛ​\(gstrat\)\\varepsilon\_\{\\mathrm\{route\}\}=\\mathcal\{R\}\(\\widehat\{g\}\)\-\\mathcal\{R\}\(g\_\{\\mathrm\{strat\}\}\)collects the finite\-sample head error and the estimated\-routing error; its routing component is controlled by the heterogeneity\-weighted cost of Lemma[11](https://arxiv.org/html/2608.10562#Thmlemma11),εroute≤𝔼​\[∥π^−π∥2​s2\]\\varepsilon\_\{\\mathrm\{route\}\}\\leq\\mathbb\{E\}\[\\lVert\\widehat\{\\pi\}\-\\pi\\rVert^\{2\}s^\{2\}\], and is nonnegative in expectation when the heads and routing are consistent, vanishing as routing becomes exact or intents become homogeneous\. The realized gain is thusΔℱ\\Delta\_\{\\mathcal\{F\}\}inflated by the pooled model’s own estimation error and deflated by the coupling and routing penalties, so it can exceedΔℱ\\Delta\_\{\\mathcal\{F\}\}or turn negative; we report the measured headroom empirically\. In the deployed system the decomposed heads share the backbone and add fewer than10−7%10^\{\-7\}\\%of parameters, so bothestℱ\\mathrm\{est\}\_\{\\mathcal\{F\}\}and the estimation part ofεroute\\varepsilon\_\{\\mathrm\{route\}\}are small rather than theKK\-fold blow\-up ofKKindependent models, which is why the realized gains are positive rather than variance\-swamped\.

###### Assumption 1\(Objective\)

Weak dominance and the headroom refer to minimizing the composite riskℛ​\(g\)\\mathcal\{R\}\(g\)over𝒢\\mathcal\{G\}\. Training per\-intent heads separately yields a suboptimal member of𝒢\\mathcal\{G\}that is not guaranteed to beat the pooled optimum\. Carryεcouple\\varepsilon\_\{\\mathrm\{couple\}\}explicitly or reframe training as joint minimization of∑kπk​fk\\sum\_\{k\}\\pi\_\{k\}f\_\{k\}\.

### A\.10Bridging to the served score

###### Lemma 9\(Score bridge\)

The conversion decomposition supplies the per\-intent targetsμk\\mu\_\{k\}that enlarge the class and create the headroom, and the click\-rate decomposition supplies the routingπk=λk/λ\\pi\_\{k\}=\\lambda\_\{k\}/\\lambda\. Neither head alone forms the composition∑kλk​μk\\sum\_\{k\}\\lambda\_\{k\}\\mu\_\{k\}, so both are required by construction\. There is no separate click\-rate approximation headroom, because∑kλk=λ\\sum\_\{k\}\\lambda\_\{k\}=\\lambda\.

###### Proposition 6\(Score headroom, squared loss\)

Work at the impression level with impression distributionρimp\\rho\_\{\\mathrm\{imp\}\}\. Suppose both models share the same CTR estimateλ^=λ\\widehat\{\\lambda\}=\\lambda, so only the conversion factor differs\. Under squared loss on the score,

infℱμℛscore−inf𝒢μℛscore\\displaystyle\\inf\_\{\\mathcal\{F\}\_\{\\mu\}\}\\mathcal\{R\}\_\{\\mathrm\{score\}\}\-\\inf\_\{\\mathcal\{G\}\_\{\\mu\}\}\\mathcal\{R\}\_\{\\mathrm\{score\}\}\(15\)=∥P𝒢μ​μ−Pℱμ​μ∥λ2​ρimp2\.\\displaystyle\\quad=\\lVert P\_\{\\mathcal\{G\}\_\{\\mu\}\}\\mu\-P\_\{\\mathcal\{F\}\_\{\\mu\}\}\\mu\\rVert^\{2\}\_\{\\lambda^\{2\}\\rho\_\{\\mathrm\{imp\}\}\}\.The score\-level headroom equals the conversion\-head headroom of the exact\-headroom proposition, computed in the click\-magnitude\-weighted measureλ2​ρimp\\lambda^\{2\}\\rho\_\{\\mathrm\{imp\}\}, with routingπk=λk/λ\\pi\_\{k\}=\\lambda\_\{k\}/\\lambdasupplied by the CTR head\.

###### Proof 11

Withλ^=λ\\widehat\{\\lambda\}=\\lambdathe score predictor isλ​μ^\\lambda\\widehat\{\\mu\}for a conversion predictorμ^\\widehat\{\\mu\}, and the score target isλ​μ\\lambda\\mu\. The squared score\-risk excess is𝔼ρimp​\[\(λ​μ^−λ​μ\)2\]=𝔼ρimp​\[λ2​\(μ^−μ\)2\]=∥μ^−μ∥λ2​ρimp2\\mathbb\{E\}\_\{\\rho\_\{\\mathrm\{imp\}\}\}\[\(\\lambda\\widehat\{\\mu\}\-\\lambda\\mu\)^\{2\}\]=\\mathbb\{E\}\_\{\\rho\_\{\\mathrm\{imp\}\}\}\[\\lambda^\{2\}\(\\widehat\{\\mu\}\-\\mu\)^\{2\}\]=\\lVert\\widehat\{\\mu\}\-\\mu\\rVert^\{2\}\_\{\\lambda^\{2\}\\rho\_\{\\mathrm\{imp\}\}\}\. Minimize overℱμ\\mathcal\{F\}\_\{\\mu\}and over𝒢μ\\mathcal\{G\}\_\{\\mu\}and subtract\. Apply the exact\-headroom proposition inL2​\(λ2​ρimp\)L^\{2\}\(\\lambda^\{2\}\\rho\_\{\\mathrm\{imp\}\}\)\.

Sinceλ\\lambdais small, the score\-level headroom is scaled down byλ2\\lambda^\{2\}relative to the on\-click conversion headroom\. This is why impression\-space gains look modest even when the on\-click miscalibration is large\. The CTR head enters only through the routing and the shared magnitudeλ\\lambda\. There is no separate CTR approximation headroom because∑kλk=λ\\sum\_\{k\}\\lambda\_\{k\}=\\lambda\.

###### Corollary 2\(Score bridge, cross\-entropy\)

Under cross\-entropy, withλ^=λ\\widehat\{\\lambda\}=\\lambda, the served MARCO score isλ⋅\(∑kπk​μk\)\\lambda\\cdot\(\\sum\_\{k\}\\pi\_\{k\}\\mu\_\{k\}\), an element ofλ⋅𝒢\\lambda\\cdot\\mathcal\{G\}, and the standard score is an element ofλ⋅ℱ\\lambda\\cdot\\mathcal\{F\}\. Sinceℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\}givesλ​ℱ⊆λ​𝒢\\lambda\\mathcal\{F\}\\subseteq\\lambda\\mathcal\{G\}, weak dominance carries to the score andΔΦ≥0\\Delta\_\{\\Phi\}\\geq 0\. There is no closed form\. Report the impression\-space NE gap directly\.

###### Proof 12

λ​ℱ⊆λ​𝒢\\lambda\\mathcal\{F\}\\subseteq\\lambda\\mathcal\{G\}by the inclusion lemma scaled pointwise byλ\\lambda\. Take the infimum over the superset\. The absence of a closed form is the mixture\-of\-experts lemma \(Lemma[6](https://arxiv.org/html/2608.10562#Thmlemma6)\)\.

### A\.11Scope of the claims

###### Lemma 10\(Vanishing at infinite capacity\)

For nested classes whose union is dense,infℱℛ→ℛ​\(μ\)\\inf\_\{\\mathcal\{F\}\}\\mathcal\{R\}\\to\\mathcal\{R\}\(\\mu\)\. Sinceℱ⊆𝒢\\mathcal\{F\}\\subseteq\\mathcal\{G\},inf𝒢ℛ\\inf\_\{\\mathcal\{G\}\}\\mathcal\{R\}is squeezed toℛ​\(μ\)\\mathcal\{R\}\(\\mu\)as well, soΔℱ→0\\Delta\_\{\\mathcal\{F\}\}\\to 0\. The headroom is a strictly finite\-capacity phenomenon\. This is consistent with the Bayes equivalence of the pooled and decomposed scores at the population level\.

###### Proof 13

Letℱ1⊆ℱ2⊆⋯\\mathcal\{F\}\_\{1\}\\subseteq\\mathcal\{F\}\_\{2\}\\subseteq\\cdotsbe the nested classes with⋃nℱn¯=L2​\(ρ\)\\overline\{\\bigcup\_\{n\}\\mathcal\{F\}\_\{n\}\}=L^\{2\}\(\\rho\), and𝒢n⊇ℱn\\mathcal\{G\}\_\{n\}\\supseteq\\mathcal\{F\}\_\{n\}the corresponding decomposed classes\. By the excess\-risk identities \(Lemmas[2](https://arxiv.org/html/2608.10562#Thmlemma2)and[3](https://arxiv.org/html/2608.10562#Thmlemma3)\),infℱnℛ−ℛ​\(μ\)\\inf\_\{\\mathcal\{F\}\_\{n\}\}\\mathcal\{R\}\-\\mathcal\{R\}\(\\mu\)equals the approximation error ofμ\\mubyℱn\\mathcal\{F\}\_\{n\}, which decreases to0by density \(the standard universal\-approximation/denseness property\[[25](https://arxiv.org/html/2608.10562#bib.bib25)\]\)\. Sinceℛ​\(μ\)≤inf𝒢nℛ≤infℱnℛ\\mathcal\{R\}\(\\mu\)\\leq\\inf\_\{\\mathcal\{G\}\_\{n\}\}\\mathcal\{R\}\\leq\\inf\_\{\\mathcal\{F\}\_\{n\}\}\\mathcal\{R\}, the middle term is squeezed toℛ​\(μ\)\\mathcal\{R\}\(\\mu\), soΔℱn=infℱnℛ−inf𝒢nℛ→0\\Delta\_\{\\mathcal\{F\}\_\{n\}\}=\\inf\_\{\\mathcal\{F\}\_\{n\}\}\\mathcal\{R\}\-\\inf\_\{\\mathcal\{G\}\_\{n\}\}\\mathcal\{R\}\\to 0\.

### A\.12Routing realizes the headroom

###### Lemma 11\(Routing cost is heterogeneity\-weighted\)

The routing cost obeysεroute≤𝔼​\[∥π^−π∥2​s2\]\\varepsilon\_\{\\mathrm\{route\}\}\\leq\\mathbb\{E\}\[\\lVert\\widehat\{\\pi\}\-\\pi\\rVert^\{2\}\\,s^\{2\}\]withs2=∑k\(μk−μ\)2s^\{2\}=\\sum\_\{k\}\(\\mu\_\{k\}\-\\mu\)^\{2\}\. It vanishes where the per\-intent rates are homogeneous \(s=0s=0\), regardless of routing error, and concentrates where the intents disagree\.

###### Proof 14

The served conversion factor isg^=∑kπ^k​μk=μ\+δ\\widehat\{g\}=\\sum\_\{k\}\\widehat\{\\pi\}\_\{k\}\\mu\_\{k\}=\\mu\+\\deltawithδ=∑k\(π^k−πk\)​μk\\delta=\\sum\_\{k\}\(\\widehat\{\\pi\}\_\{k\}\-\\pi\_\{k\}\)\\mu\_\{k\}\. Sinceπ^\\widehat\{\\pi\}andπ\\piare distributions,∑k\(π^k−πk\)=0\\sum\_\{k\}\(\\widehat\{\\pi\}\_\{k\}\-\\pi\_\{k\}\)=0, so subtractingμ​∑k\(π^k−πk\)=0\\mu\\sum\_\{k\}\(\\widehat\{\\pi\}\_\{k\}\-\\pi\_\{k\}\)=0givesδ=∑k\(π^k−πk\)​\(μk−μ\)\\delta=\\sum\_\{k\}\(\\widehat\{\\pi\}\_\{k\}\-\\pi\_\{k\}\)\(\\mu\_\{k\}\-\\mu\)\. Cauchy\-Schwarz overkkgives\|δ\|≤∥π^−π∥​s\|\\delta\|\\leq\\lVert\\widehat\{\\pi\}\-\\pi\\rVert\\,swiths2=∑k\(μk−μ\)2s^\{2\}=\\sum\_\{k\}\(\\mu\_\{k\}\-\\mu\)^\{2\}\. Squaring and taking𝔼ρ\\mathbb\{E\}\_\{\\rho\}yields the bound\. Whens=0s=0everyμk=μ\\mu\_\{k\}=\\mu, soδ=0\\delta=0\.

###### Proposition 7\(Routing monotonicity\)

As CTR capacityccgrows, the best\-in\-class routing costεroute⋆​\(c\)\\varepsilon^\{\\star\}\_\{\\mathrm\{route\}\}\(c\)is non\-increasing and the routing efficiencyη⋆​\(c\)=1−εroute⋆​\(c\)/Δℱ\\eta^\{\\star\}\(c\)=1\-\\varepsilon^\{\\star\}\_\{\\mathrm\{route\}\}\(c\)/\\Delta\_\{\\mathcal\{F\}\}is non\-decreasing\.

###### Proof 15

LetΠ\(c\)\\Pi^\{\(c\)\}be the routings realizable at CTR capacitycc, nested soc≤c′c\\leq c^\{\\prime\}impliesΠ\(c\)⊆Π\(c′\)\\Pi^\{\(c\)\}\\subseteq\\Pi^\{\(c^\{\\prime\}\)\}\. The best\-in\-class routing costεroute⋆​\(c\)=infπ′∈Π\(c\)𝔼​\[\(∑k\(πk′−πk\)​μk\)2\]\\varepsilon^\{\\star\}\_\{\\mathrm\{route\}\}\(c\)=\\inf\_\{\\pi^\{\\prime\}\\in\\Pi^\{\(c\)\}\}\\mathbb\{E\}\\big\[\(\\sum\_\{k\}\(\\pi^\{\\prime\}\_\{k\}\-\\pi\_\{k\}\)\\mu\_\{k\}\)^\{2\}\\big\]is the infimum of a fixed nonnegative functional over a growing set, hence non\-increasing incc\. WithΔℱ\>0\\Delta\_\{\\mathcal\{F\}\}\>0fixed,η⋆​\(c\)=1−εroute⋆​\(c\)/Δℱ\\eta^\{\\star\}\(c\)=1\-\\varepsilon^\{\\star\}\_\{\\mathrm\{route\}\}\(c\)/\\Delta\_\{\\mathcal\{F\}\}is non\-decreasing andΔℱ​η⋆​\(c\)\\Delta\_\{\\mathcal\{F\}\}\\,\\eta^\{\\star\}\(c\)is non\-decreasing\. The served cost adds a nonnegative estimation\-excess term over this floor, so the served efficiency inherits the monotonicity only under an estimation\-negligible or a bias\-dominates\-variance condition\. Under cross\-entropy the same infimum argument applies to the exact functionalπ′↦𝔼​\[KL​\(Bern​\(μ\)∥Bern​\(∑kπk′​μk\)\)\]\\pi^\{\\prime\}\\mapsto\\mathbb\{E\}\[\\mathrm\{KL\}\(\\mathrm\{Bern\}\(\\mu\)\\\|\\mathrm\{Bern\}\(\\sum\_\{k\}\\pi^\{\\prime\}\_\{k\}\\mu\_\{k\}\)\)\], so the floor monotonicity holds with no approximation\.