Learning Predictive Ambiguity Sets for Decision-Focused Distributionally Robust Optimization

# Learning Predictive Ambiguity Sets for Decision-Focused Distributionally Robust Optimization
Source: [https://arxiv.org/html/2607.09820](https://arxiv.org/html/2607.09820)
###### Abstract

Predict\-then\-optimize systems usually compress uncertainty into a point forecast and then solve a downstream optimization problem as if the forecast were reliable\. Distributionally robust optimization \(DRO\) offers protection against misspecification, but the ambiguity set is often centered at historical samples and uses a fixed radius\. We propose*learned predictive ambiguity sets*\(LPAS\): a deep contextual model outputs a finite nominal scenario distribution, a state\-dependent Wasserstein radius, and optionally an anisotropic ground metric\. These outputs define a contextual ambiguity set that feeds a DRO decision layer\. The radius is trained by a combination of conditional quantile calibration, size regularization, and downstream decision loss, so that robustness is adaptive rather than globally fixed\. We derive the finite dual form used by the decision layer, present a staged training algorithm, and evaluate the method on distributionally robust portfolio optimization with 20 S&P 500 constituents from 2018–2026\. The proposed method substantially improves over equal\-weight, predict\-then\-optimize, and historical Wasserstein DRO baselines, achieving 26\.28% annualized return, Sharpe ratio 1\.30, final wealth 1\.61, and lower tail loss than a deep fixed\-radius DRO baseline while using a smaller average radius\. The results show that learned ambiguity radii can recover most of the performance of strong fixed\-radius DRO while reducing unnecessary conservatism and improving regime adaptivity\.

## 1Introduction

Many machine\-learning decision systems are built around the pipeline

zt⟶ξ^t\+1⟶xt,z\_\{t\}\\longrightarrow\\widehat\{\\xi\}\_\{t\+1\}\\longrightarrow x\_\{t\},\(1\)whereztz\_\{t\}is a context,ξ^t\+1\\widehat\{\\xi\}\_\{t\+1\}is a forecast of an uncertain future quantity, andxtx\_\{t\}is a downstream decision\. In portfolio optimization,ξt\+1\\xi\_\{t\+1\}represents future returns; in inventory control, it represents demand; in network optimization, it may represent edge costs\. This architecture is simple, but it over\-trusts the predictive model\. Errors that are small in prediction space can be large in decision space, especially when the optimizer amplifies mistakes along active constraints or high\-sensitivity directions\.

DRO replaces a single predictive distribution by an ambiguity set and optimizes against the worst plausible distribution\. A common formulation is

minx∈𝒳​supQ∈𝒫𝔼Q​\[ℓ​\(x,ξ\)\],\\min\_\{x\\in\\mathcal\{X\}\}\\sup\_\{Q\\in\\mathcal\{P\}\}\\mathbb\{E\}\_\{Q\}\[\\ell\(x,\\xi\)\],\(2\)whereℓ\\ellis a downstream loss and𝒫\\mathcal\{P\}is an ambiguity set\. Wasserstein DRO is attractive because it provides a geometry\-aware way to perturb an empirical or nominal distribution and often admits tractable convex reformulations\(Mohajerin Esfahani and Kuhn[2018](https://arxiv.org/html/2607.09820#bib.bib17); Gao and Kleywegt[2023](https://arxiv.org/html/2607.09820#bib.bib11); Blanchet and Murthy[2019](https://arxiv.org/html/2607.09820#bib.bib5); Kuhn et al\.[2019](https://arxiv.org/html/2607.09820#bib.bib15)\)\. However, the ambiguity set is commonly hand\-designed: the center is a historical empirical distribution and the radius is a fixed scalar tuned by validation or statistical concentration\. This can be mismatched in contextual environments\. A radius that is safe in volatile periods can be too conservative in stable periods, while a radius tuned for average validation loss can fail under regime shift\.

This paper asks whether the ambiguity set itself can be predicted\. Given contextztz\_\{t\}, a deep model outputs a finite nominal distribution

P^θ\(⋅∣zt\)=∑i=1Npθ,i\(zt\)δξ^θ,i​\(zt\),\\widehat\{P\}\_\{\\theta\}\(\\cdot\\mid z\_\{t\}\)=\\sum\_\{i=1\}^\{N\}p\_\{\\theta,i\}\(z\_\{t\}\)\\delta\_\{\\widehat\{\\xi\}\_\{\\theta,i\}\(z\_\{t\}\)\},\(3\)plus a nonnegative radiusρϕ​\(zt\)\\rho\_\{\\phi\}\(z\_\{t\}\)\. These define the contextual Wasserstein ambiguity set

𝒫θ,ϕ\(zt\)=\{Q:𝖶c\(Q,P^θ\(⋅∣zt\)\)≤ρϕ\(zt\)\}\.\\mathcal\{P\}\_\{\\theta,\\phi\}\(z\_\{t\}\)=\\left\\\{Q:\\mathsf\{W\}\_\{c\}\\bigl\(Q,\\widehat\{P\}\_\{\\theta\}\(\\cdot\\mid z\_\{t\}\)\\bigr\)\\leq\\rho\_\{\\phi\}\(z\_\{t\}\)\\right\\\}\.\(4\)The decision is then computed by a DRO layer\. The key modeling principle is that uncertainty should be both statistically calibrated and decision relevant: the radius should be large when the forecast is unreliable or the decision is sensitive to errors, and small when robustness mainly induces conservatism\.

#### Contributions\.

This work makes four contributions\. First, it introduces learned predictive ambiguity sets, a contextual bridge between probabilistic deep forecasting and Wasserstein DRO\. Second, it develops a radius\-learning objective that combines prediction loss, quantile\-style calibration, radius\-size regularization, and realized decision loss\. Third, it gives a tractable DRO layer through the Wasserstein dual and an implementable staged training algorithm\. Fourth, it provides a portfolio optimization study showing that adaptive ambiguity radii can strongly outperform nonrobust and historical\-DRO baselines, and can match much of the performance of a deep fixed\-radius DRO model with a smaller learned radius\.

## 2Related Work

#### Robust optimization and DRO\.

Classical robust optimization protects decisions against deterministic uncertainty sets and provides tractable reformulations for many conic and linear models\(Ben\-Tal, El Ghaoui, and Nemirovski[2009](https://arxiv.org/html/2607.09820#bib.bib3); Bertsimas, Brown, and Caramanis[2011](https://arxiv.org/html/2607.09820#bib.bib4)\)\. DRO extends this idea from uncertain parameters to uncertain probability laws\. Moment\-based ambiguity sets provide early data\-driven DRO formulations\(Delage and Ye[2010](https://arxiv.org/html/2607.09820#bib.bib8); Goh and Sim[2010](https://arxiv.org/html/2607.09820#bib.bib13)\), while modern Wasserstein ambiguity sets use optimal transport geometry to compare empirical and perturbed distributions\(Mohajerin Esfahani and Kuhn[2018](https://arxiv.org/html/2607.09820#bib.bib17); Gao and Kleywegt[2023](https://arxiv.org/html/2607.09820#bib.bib11); Blanchet and Murthy[2019](https://arxiv.org/html/2607.09820#bib.bib5)\)\. Wasserstein DRO has also been connected to statistical regularization and adversarial robustness in machine learning\(Shafieezadeh\-Abadeh, Mohajerin Esfahani, and Kuhn[2015](https://arxiv.org/html/2607.09820#bib.bib20); Sinha, Namkoong, and Duchi[2018](https://arxiv.org/html/2607.09820#bib.bib21); Duchi and Namkoong[2021](https://arxiv.org/html/2607.09820#bib.bib9); Gao, Chen, and Kleywegt[2022](https://arxiv.org/html/2607.09820#bib.bib12)\)\. LPAS keeps the tractable worst\-case\-expectation machinery of Wasserstein DRO, but makes the center and radius contextual and learnable\.

#### Decision\-focused learning\.

Smart predict\-then\-optimize trains prediction models by downstream decision quality rather than standard predictive error\(Elmachtoub and Grigas[2022](https://arxiv.org/html/2607.09820#bib.bib10)\)\. Related decision\-focused approaches differentiate through combinatorial or continuous optimization layers so that the predictor is optimized for the final task\(Wilder, Dilkina, and Tambe[2019](https://arxiv.org/html/2607.09820#bib.bib25); Amos and Kolter[2017](https://arxiv.org/html/2607.09820#bib.bib2); Agrawal et al\.[2019](https://arxiv.org/html/2607.09820#bib.bib1)\)\. LPAS follows this principle, but the learned object is not merely a point estimate or deterministic cost vector\. It is a predictive distribution together with a state\-dependent ambiguity radius, so the downstream layer can decide both where to optimize and how much robustness is needed\.

#### Learning uncertainty sets and calibration\.

Recent work learns robust uncertainty sets from data and differentiates through robust optimization problems\(Wang et al\.[2023](https://arxiv.org/html/2607.09820#bib.bib24)\)\. Predict\-then\-calibrate methods construct robust contextual feasible sets after fitting a predictor\(Sun, Liu, and Li[2023](https://arxiv.org/html/2607.09820#bib.bib22)\), while end\-to\-end conditional robust optimization directly trains robust contextual decisions with a coverage\-sensitive objective\(Chenreddy and Delage[2024](https://arxiv.org/html/2607.09820#bib.bib7)\)\. LPAS differs by learning a finite predictive distribution and a Wasserstein radius jointly\. The radius\-learning loss is also related to quantile regression and conformal calibration, which provide tools for adaptive predictive uncertainty\(Koenker and Bassett[1978](https://arxiv.org/html/2607.09820#bib.bib14); Romano, Patterson, and Candes[2019](https://arxiv.org/html/2607.09820#bib.bib19)\)\.

#### Portfolio optimization\.

The experimental task is rooted in mean\-variance portfolio selection\(Markowitz[1952](https://arxiv.org/html/2607.09820#bib.bib16)\)and risk\-sensitive portfolio design\. Wasserstein DRO has been studied for robust mean\-variance portfolios and related financial decision problems\(Blanchet, Chen, and Zhou[2022](https://arxiv.org/html/2607.09820#bib.bib6)\)\. Our portfolio layer uses the predictive scenario model to estimate returns, a Wasserstein ambiguity penalty to temper aggressive forecasts, and standard turnover and covariance regularization\. Tail metrics such as CVaR are reported because average returns alone can hide downside risk\(Rockafellar and Uryasev[2000](https://arxiv.org/html/2607.09820#bib.bib18)\)\.

## 3Problem Setup

Let\(z,ξ\)∼P⋆\(z,\\xi\)\\sim P^\{\\star\}, wherez∈𝒵z\\in\\mathcal\{Z\}is the observed context andξ∈Ξ⊆ℝd\\xi\\in\\Xi\\subseteq\\mathbb\{R\}^\{d\}is an uncertain quantity realized after the decision\. A decision maker selectsx∈𝒳⊆ℝmx\\in\\mathcal\{X\}\\subseteq\\mathbb\{R\}^\{m\}and incurs lossℓ​\(x,ξ\)\\ell\(x,\\xi\)\. The goal is to learn a contextual decision rulez↦x​\(z\)z\\mapsto x\(z\)with low out\-of\-sample loss and stable tail behavior\.

The proposed architecture has four components: a nominal scenario modelP^θ\(⋅∣z\)\\widehat\{P\}\_\{\\theta\}\(\\cdot\\mid z\), a radius networkρϕ​\(z\)\\rho\_\{\\phi\}\(z\), an optional ground metriccψ​\(ξ,ξ′;z\)c\_\{\\psi\}\(\\xi,\\xi^\{\\prime\};z\), and a DRO decision layer\. The nominal model returns scenariosξ^θ,i​\(z\)\\widehat\{\\xi\}\_\{\\theta,i\}\(z\)and probabilitiespθ,i​\(z\)p\_\{\\theta,i\}\(z\)with∑ipθ,i​\(z\)=1\\sum\_\{i\}p\_\{\\theta,i\}\(z\)=1\. The radius network is parameterized as

ρϕ​\(z\)=ρmin\+softplus⁡\(gϕ​\(z\)\),\\rho\_\{\\phi\}\(z\)=\\rho\_\{\\min\}\+\\operatorname\{softplus\}\(g\_\{\\phi\}\(z\)\),\(5\)which ensures nonnegativity\. A bounded alternative is

ρϕ​\(z\)=ρmin\+\(ρmax−ρmin\)​σ​\(gϕ​\(z\)\)\.\\rho\_\{\\phi\}\(z\)=\\rho\_\{\\min\}\+\(\\rho\_\{\\max\}\-\\rho\_\{\\min\}\)\\sigma\(g\_\{\\phi\}\(z\)\)\.\(6\)The optional metric may be fixed, such asc​\(ξ,ξ′\)=‖ξ−ξ′‖2c\(\\xi,\\xi^\{\\prime\}\)=\\\|\\xi\-\\xi^\{\\prime\}\\\|\_\{2\}, or contextual and anisotropic, such as

cψ​\(ξ,ξ′;z\)=‖Aψ​\(z\)​\(ξ−ξ′\)‖2\.c\_\{\\psi\}\(\\xi,\\xi^\{\\prime\};z\)=\\\|A\_\{\\psi\}\(z\)\(\\xi\-\\xi^\{\\prime\}\)\\\|\_\{2\}\.\(7\)The experiments in this paper use a fixed Euclidean transport geometry; the learned\-metric extension is left as a modular component\.

## 4Learned Predictive Ambiguity Sets

### 4\.1Discrete Nominal Distributions

For each contextztz\_\{t\}, the model outputs

P^θ,t=∑i=1Npθ,i​\(zt\)​δξ^θ,i​\(zt\)\.\\widehat\{P\}\_\{\\theta,t\}=\\sum\_\{i=1\}^\{N\}p\_\{\\theta,i\}\(z\_\{t\}\)\\delta\_\{\\widehat\{\\xi\}\_\{\\theta,i\}\(z\_\{t\}\)\}\.\(8\)The predictive mean is

μ^θ,t=𝔼P^θ,t​\[ξ\]=∑i=1Npθ,i​\(zt\)​ξ^θ,i​\(zt\)\.\\widehat\{\\mu\}\_\{\\theta,t\}=\\mathbb\{E\}\_\{\\widehat\{P\}\_\{\\theta,t\}\}\[\\xi\]=\\sum\_\{i=1\}^\{N\}p\_\{\\theta,i\}\(z\_\{t\}\)\\widehat\{\\xi\}\_\{\\theta,i\}\(z\_\{t\}\)\.\(9\)A point\-prediction baseline uses onlyμ^θ,t\\widehat\{\\mu\}\_\{\\theta,t\}, while LPAS uses the full finite distribution as the center of a Wasserstein ball\.

### 4\.2Wasserstein Ambiguity Set

For a distributionQ∈𝒫​\(Ξ\)Q\\in\\mathcal\{P\}\(\\Xi\)and nominal distributionP^θ,t\\widehat\{P\}\_\{\\theta,t\}, define

𝖶cψ​\(Q,P^θ,t;zt\)=infγ∈Π​\(Q,P^θ,t\)∫Ξ×Ξcψ​\(ξ,ξ^;zt\)​dγ​\(ξ,ξ^\),\\mathsf\{W\}\_\{c\_\{\\psi\}\}\(Q,\\widehat\{P\}\_\{\\theta,t\};z\_\{t\}\)=\\inf\_\{\\gamma\\in\\Pi\(Q,\\widehat\{P\}\_\{\\theta,t\}\)\}\\int\_\{\\Xi\\times\\Xi\}c\_\{\\psi\}\(\\xi,\\widehat\{\\xi\};z\_\{t\}\)\\mathrm\{d\}\\gamma\(\\xi,\\widehat\{\\xi\}\),\(10\)whereΠ​\(Q,P^θ,t\)\\Pi\(Q,\\widehat\{P\}\_\{\\theta,t\}\)is the set of couplings with marginalsQQandP^θ,t\\widehat\{P\}\_\{\\theta,t\}\. The learned ambiguity set is

𝒫θ,ϕ,ψ,t=\{Q∈𝒫​\(Ξ\):𝖶cψ​\(Q,P^θ,t;zt\)≤ρϕ​\(zt\)\}\.\\mathcal\{P\}\_\{\\theta,\\phi,\\psi,t\}=\\\{Q\\in\\mathcal\{P\}\(\\Xi\):\\mathsf\{W\}\_\{c\_\{\\psi\}\}\(Q,\\widehat\{P\}\_\{\\theta,t\};z\_\{t\}\)\\leq\\rho\_\{\\phi\}\(z\_\{t\}\)\\\}\.\(11\)IfQ=∑j=1Mqj​δξjQ=\\sum\_\{j=1\}^\{M\}q\_\{j\}\\delta\_\{\\xi\_\{j\}\}is also discrete, then the transport discrepancy is the linear program

minπ≥0​∑j=1M∑i=1Ncψ​\(ξj,ξ^θ,i;zt\)​πj​i\\min\_\{\\pi\\geq 0\}\\sum\_\{j=1\}^\{M\}\\sum\_\{i=1\}^\{N\}c\_\{\\psi\}\(\\xi\_\{j\},\\widehat\{\\xi\}\_\{\\theta,i\};z\_\{t\}\)\\pi\_\{ji\}\(12\)subject to∑jπj​i=pθ,i​\(zt\)\\sum\_\{j\}\\pi\_\{ji\}=p\_\{\\theta,i\}\(z\_\{t\}\)and∑iπj​i=qj\\sum\_\{i\}\\pi\_\{ji\}=q\_\{j\}\. Thus the radius controls the amount of adversarial mass transportation away from the predicted scenarios\.

### 4\.3DRO Decision Layer

Given the ambiguity set, the decision is

xt⋆=arg​minx∈𝒳⁡\{supQ∈𝒫θ,ϕ,ψ,t𝔼Q​\[ℓ​\(x,ξ\)\]\+r​\(x\)\},x\_\{t\}^\{\\star\}=\\operatorname\*\{arg\\,min\}\_\{x\\in\\mathcal\{X\}\}\\left\\\{\\sup\_\{Q\\in\\mathcal\{P\}\_\{\\theta,\\phi,\\psi,t\}\}\\mathbb\{E\}\_\{Q\}\[\\ell\(x,\\xi\)\]\+r\(x\)\\right\\\},\(13\)wherer​\(x\)r\(x\)is a deterministic regularizer such as a risk penalty, transaction\-cost term, or strong\-convexity penalty\.

The standard Wasserstein dual gives

supQ:𝖶cψ​\(Q,P^θ,t\)≤ρt𝔼Q​\[ℓ​\(x,ξ\)\]\\displaystyle\\sup\_\{Q:\\mathsf\{W\}\_\{c\_\{\\psi\}\}\(Q,\\widehat\{P\}\_\{\\theta,t\}\)\\leq\\rho\_\{t\}\}\\mathbb\{E\}\_\{Q\}\[\\ell\(x,\\xi\)\]\(14\)=infη≥0\{η​ρt\+∑i=1Npθ,i​\(zt\)​si​\(x,η\)\},\\displaystyle\\quad=\\inf\_\{\\eta\\geq 0\}\\left\\\{\\eta\\rho\_\{t\}\+\\sum\_\{i=1\}^\{N\}p\_\{\\theta,i\}\(z\_\{t\}\)\\,s\_\{i\}\(x,\\eta\)\\right\\\},whereρt=ρϕ​\(zt\)\\rho\_\{t\}=\\rho\_\{\\phi\}\(z\_\{t\}\)and

si​\(x,η\)=supξ∈Ξ\[ℓ​\(x,ξ\)−η​cψ​\(ξ,ξ^θ,i;zt\)\]\.s\_\{i\}\(x,\\eta\)=\\sup\_\{\\xi\\in\\Xi\}\\bigl\[\\ell\(x,\\xi\)\-\\eta c\_\{\\psi\}\(\\xi,\\widehat\{\\xi\}\_\{\\theta,i\};z\_\{t\}\)\\bigr\]\.\(15\)Under affine loss and norm\-based costs, the inner supremum has a finite\-dimensional conic representation or a closed form\. This is the computational bridge from learned ambiguity sets to differentiable robust decisions\.

### 4\.4Portfolio Special Case

The portfolio task provides a useful closed\-form instance of the decision layer\. Letr∈ℝdr\\in\\mathbb\{R\}^\{d\}be next\-period returns and letw∈Δdw\\in\\Delta\_\{d\}be long\-only portfolio weights,

Δd=\{w∈ℝ\+d:𝟏⊤​w=1\}\.\\Delta\_\{d\}=\\\{w\\in\\mathbb\{R\}\_\{\+\}^\{d\}:\\mathbf\{1\}^\{\\top\}w=1\\\}\.\(16\)A transaction\-cost and risk\-regularized loss is

ℓ​\(w,r\)=−r⊤​w\+λrisk​w⊤​Σ^t​w\+λtc​‖w−wt−1‖1\.\\ell\(w,r\)=\-r^\{\\top\}w\+\\lambda\_\{\\mathrm\{risk\}\}w^\{\\top\}\\widehat\{\\Sigma\}\_\{t\}w\+\\lambda\_\{\\mathrm\{tc\}\}\\\|w\-w\_\{t\-1\}\\\|\_\{1\}\.\(17\)The corresponding DRO decision is

wt⋆=arg​minw∈Δd\\displaystyle w\_\{t\}^\{\\star\}=\\operatorname\*\{arg\\,min\}\_\{w\\in\\Delta\_\{d\}\}supQ:𝖶c​\(Q,P^θ,t\)≤ρϕ​\(zt\)𝔼Q​\[−r⊤​w\]\\displaystyle\\sup\_\{Q:\\mathsf\{W\}\_\{c\}\(Q,\\widehat\{P\}\_\{\\theta,t\}\)\\leq\\rho\_\{\\phi\}\(z\_\{t\}\)\}\\mathbb\{E\}\_\{Q\}\[\-r^\{\\top\}w\]\(18\)\+λrisk​w⊤​Σ^t​w\+λtc​‖w−wt−1‖1\.\\displaystyle\+\\lambda\_\{\\mathrm\{risk\}\}w^\{\\top\}\\widehat\{\\Sigma\}\_\{t\}w\+\\lambda\_\{\\mathrm\{tc\}\}\\\|w\-w\_\{t\-1\}\\\|\_\{1\}\.Ifc​\(r,r^\)=‖r−r^‖2c\(r,\\widehat\{r\}\)=\\\|r\-\\widehat\{r\}\\\|\_\{2\}and only the linear return term is robustified, then duality gives

supQ:𝖶1​\(Q,P^θ,t\)≤ρt𝔼Q​\[−r⊤​w\]=−μ^θ,t⊤​w\+ρt​‖w‖2\.\\sup\_\{Q:\\mathsf\{W\}\_\{1\}\(Q,\\widehat\{P\}\_\{\\theta,t\}\)\\leq\\rho\_\{t\}\}\\mathbb\{E\}\_\{Q\}\[\-r^\{\\top\}w\]=\-\\widehat\{\\mu\}\_\{\\theta,t\}^\{\\top\}w\+\\rho\_\{t\}\\\|w\\\|\_\{2\}\.\(19\)Thus the robust portfolio can be written as

maxw∈Δd\\displaystyle\\max\_\{w\\in\\Delta\_\{d\}\}μ^θ,t⊤​w−ρϕ​\(zt\)​‖w‖2−λrisk​w⊤​Σ^t​w\\displaystyle\\widehat\{\\mu\}\_\{\\theta,t\}^\{\\top\}w\-\\rho\_\{\\phi\}\(z\_\{t\}\)\\\|w\\\|\_\{2\}\-\\lambda\_\{\\mathrm\{risk\}\}w^\{\\top\}\\widehat\{\\Sigma\}\_\{t\}w\(20\)−λtc​‖w−wt−1‖1\.\\displaystyle\-\\lambda\_\{\\mathrm\{tc\}\}\\\|w\-w\_\{t\-1\}\\\|\_\{1\}\.The radius has a direct interpretation: largerρϕ​\(zt\)\\rho\_\{\\phi\}\(z\_\{t\}\)makes the optimizer less aggressive when the predictive distribution is unreliable\.

### 4\.5Radius Calibration

The radius should not be merely large; it should be calibrated\. Let

et=‖Sθ,t−1/2​\(ξt\+1−μ^θ,t\)‖2,e\_\{t\}=\\left\\\|S\_\{\\theta,t\}^\{\-1/2\}\(\\xi\_\{t\+1\}\-\\widehat\{\\mu\}\_\{\\theta,t\}\)\\right\\\|\_\{2\},\(21\)whereSθ,tS\_\{\\theta,t\}is a predicted or empirical regularized scale matrix\. A statistically interpretable radius approximates a conditional high quantile,

ρϕ​\(zt\)≈Qτ​\(et∣zt\),\\rho\_\{\\phi\}\(z\_\{t\}\)\\approx Q\_\{\\tau\}\(e\_\{t\}\\mid z\_\{t\}\),\(22\)with quantile levelτ\\tau\. We use the pinball loss

ℓτ​\(e,ρ\)=\(τ−𝟏​\{e≤ρ\}\)​\(e−ρ\)\.\\ell\_\{\\tau\}\(e,\\rho\)=\(\\tau\-\\mathbf\{1\}\\\{e\\leq\\rho\\\}\)\(e\-\\rho\)\.\(23\)This gives the learned radius a forecast\-calibration interpretation before it is fine\-tuned for downstream decision quality\.

## 5Training Objective and Algorithm

The complete objective is

minθ,ϕ,ψℒ​\(θ,ϕ,ψ\)\\displaystyle\\min\_\{\\theta,\\phi,\\psi\}\\quad\\mathcal\{L\}\(\\theta,\\phi,\\psi\)=ℒdec\+λpred​ℒpred\+λcal​ℒcal\\displaystyle=\\mathcal\{L\}\_\{\\mathrm\{dec\}\}\+\\lambda\_\{\\mathrm\{pred\}\}\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\+\\lambda\_\{\\mathrm\{cal\}\}\\mathcal\{L\}\_\{\\mathrm\{cal\}\}\(24\)\+λsize​ℒsize\+λstab​ℒstab\.\\displaystyle\\quad\+\\lambda\_\{\\mathrm\{size\}\}\\mathcal\{L\}\_\{\\mathrm\{size\}\}\+\\lambda\_\{\\mathrm\{stab\}\}\\mathcal\{L\}\_\{\\mathrm\{stab\}\}\.The decision loss is

ℒdec=1T​∑t=1Tℓeval​\(xt⋆,ξt\+1\)\.\\mathcal\{L\}\_\{\\mathrm\{dec\}\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\ell\_\{\\mathrm\{eval\}\}\(x\_\{t\}^\{\\star\},\\xi\_\{t\+1\}\)\.\(25\)The prediction loss trains the nominal scenario distribution\. Depending on the scenario model, it can be a negative log\-likelihood, energy score, or scenario reconstruction loss

ℒpred=1T​∑t=1Tmini⁡‖ξt\+1−ξ^θ,i​\(zt\)‖22\.\\mathcal\{L\}\_\{\\mathrm\{pred\}\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\min\_\{i\}\\\|\\xi\_\{t\+1\}\-\\widehat\{\\xi\}\_\{\\theta,i\}\(z\_\{t\}\)\\\|\_\{2\}^\{2\}\.\(26\)The calibration, size, and stability losses are

ℒcal=1T​∑t=1Tℓτ​\(et,ρϕ​\(zt\)\),ℒsize=1T​∑t=1Tρϕ​\(zt\),\\mathcal\{L\}\_\{\\mathrm\{cal\}\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\ell\_\{\\tau\}\(e\_\{t\},\\rho\_\{\\phi\}\(z\_\{t\}\)\),\\quad\\mathcal\{L\}\_\{\\mathrm\{size\}\}=\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\rho\_\{\\phi\}\(z\_\{t\}\),\(27\)ℒstab=1T−1​∑t=2T\|ρϕ​\(zt\)−ρϕ​\(zt−1\)\|2\.\\mathcal\{L\}\_\{\\mathrm\{stab\}\}=\\frac\{1\}\{T\-1\}\\sum\_\{t=2\}^\{T\}\|\\rho\_\{\\phi\}\(z\_\{t\}\)\-\\rho\_\{\\phi\}\(z\_\{t\-1\}\)\|^\{2\}\.\(28\)The size term prevents radius inflation, while the calibration term prevents radius collapse\. Algorithm[1](https://arxiv.org/html/2607.09820#alg1)summarizes the training procedure\.

Algorithm 1Training Learned Predictive Ambiguity Sets0:Context\-outcome pairs

\{\(zt,ξt\+1\)\}t=1T\\\{\(z\_\{t\},\\xi\_\{t\+1\}\)\\\}\_\{t=1\}^\{T\}, scenario count

NN, quantile level

τ\\tau
1:Pretrain

P^θ\(⋅∣z\)\\widehat\{P\}\_\{\\theta\}\(\\cdot\\mid z\)by

ℒpred\\mathcal\{L\}\_\{\\mathrm\{pred\}\}
2:Compute normalized forecast errors

ete\_\{t\}on a calibration split

3:Train

ρϕ​\(z\)\\rho\_\{\\phi\}\(z\)by the pinball loss

ℒcal\\mathcal\{L\}\_\{\\mathrm\{cal\}\}with size regularization

4:foreach training epochdo

5:foreach minibatch of contextsdo

6:Output scenarios

ξ^θ,i​\(zt\)\\widehat\{\\xi\}\_\{\\theta,i\}\(z\_\{t\}\), probabilities

pθ,i​\(zt\)p\_\{\\theta,i\}\(z\_\{t\}\), and radius

ρϕ​\(zt\)\\rho\_\{\\phi\}\(z\_\{t\}\)
7:Construct

𝒫θ,ϕ,ψ,t\\mathcal\{P\}\_\{\\theta,\\phi,\\psi,t\}by Eq\. \([11](https://arxiv.org/html/2607.09820#S4.E11)\)

8:Solve the DRO layer Eq\. \([13](https://arxiv.org/html/2607.09820#S4.E13)\)

9:Evaluate realized decision loss and update parameters using Eq\. \([24](https://arxiv.org/html/2607.09820#S5.E24)\)

10:endfor

11:endfor

12:returnTrained scenario generator, radius network, and DRO decision rule

## 6Experiments

### 6\.1Portfolio Optimization Setup

We evaluate LPAS on daily portfolio optimization using 20 S&P 500 constituents over January 2018–June 2026\. The experiment uses daily returns from TMUS, PPG, LYB, ALB, GPC, STE, WRB, VRTX, BLK, VEEV, INCY, PAYX, TXT, CSCO, MOS, OXY, HON, MAS, AMGN, and IEX\. After feature construction, the sample contains 2054 usable feature rows\. The chronological split has 1129 training observations, 410 validation observations, and 515 test observations\. The portfolio is long\-only and fully invested\.

The predictive model is a Transformer scenario generator\(Vaswani et al\.[2017](https://arxiv.org/html/2607.09820#bib.bib23)\)with lookback length 63, two Transformer layers, four attention heads, hidden dimension 48, andN=7N=7scenarios\. Lagged returns, rolling volatility, and market\-state features are used as contexts\. The selected candidate uses ridge coefficient 1\.0, risk penalty 8\.0, transaction\-cost objective weight 0\.0015, realized transaction cost 0\.001, quantile levelτ=0\.9\\tau=0\.9, and radius floorρmin=10−4\\rho\_\{\\min\}=10^\{\-4\}\.

The baselines are equal weight \(EW\), predict\-then\-optimize \(P2O\) using the predictive mean, historical Wasserstein DRO \(Hist\-WDRO\) centered at historical returns, and deep predictive fixed\-radius DRO \(Fixed\-DRO\) centered at the same Transformer scenario distribution but using a scalar validation\-selected radius\. LPAS\-W uses the same scenario generator as Fixed\-DRO but replaces the global radius with a learned context\-dependent radius\.

### 6\.2Hyperparameter Selection

Hyperparameters are selected on the validation window by a decision\-aware score that combines average loss, tail loss, calibration, and radius size\. Table[1](https://arxiv.org/html/2607.09820#S6.T1)shows the candidate\-level validation results\. The selected model does not minimize prediction MSE alone; it minimizes the downstream decision\-aware validation score\.

Table 1:Transformer hyperparameter selection\. The selected model minimizes the decision\-aware validation score rather than prediction MSE alone\.
### 6\.3Main Results

Table[2](https://arxiv.org/html/2607.09820#S6.T2)reports the main out\-of\-sample portfolio metrics\. LPAS strongly improves over equal weight, predict\-then\-optimize, and historical Wasserstein DRO\. Compared with deep fixed\-radius DRO, LPAS gives slightly lower annualized return and Sharpe ratio, but uses a substantially smaller average radius and obtains slightly better worst\-month and tail\-loss metrics\. This suggests that the learned radius reduces conservatism while retaining most of the performance of the best fixed\-radius robust model\.

Table 2:Main out\-of\-sample portfolio results\. EW: equal weight; P2O: predict\-then\-optimize; Hist\-WDRO: historical Wasserstein DRO; Fixed\-DRO: deep predictive fixed\-radius DRO; LPAS\-W: learned predictive Wasserstein ambiguity set\. Higher annualized return, Sharpe ratio, and wealth are better; lower maximum drawdown, turnover, worst\-month loss magnitude, CVaR95 loss, and radius are better\.Figures[1](https://arxiv.org/html/2607.09820#S6.F1)and[2](https://arxiv.org/html/2607.09820#S6.F2)show cumulative wealth and drawdown separately\. Predict\-then\-optimize suffers from aggressive forecast amplification, while robust methods produce more stable wealth paths\. The learned\-radius method tracks the fixed\-radius robust portfolio closely while avoiding a globally large ambiguity radius\.

![Refer to caption](https://arxiv.org/html/2607.09820v1/x1.png)Figure 1:Out\-of\-sample cumulative wealth\. Robust methods stabilize wealth accumulation relative to predict\-then\-optimize\.![Refer to caption](https://arxiv.org/html/2607.09820v1/x2.png)Figure 2:Out\-of\-sample drawdown\. LPAS\-W remains close to the strong fixed\-radius DRO baseline while avoiding a globally large ambiguity radius\.
### 6\.4Ablation Results

Table[3](https://arxiv.org/html/2607.09820#S6.T3)isolates the role of the learned radius components\. Removing calibration collapses empirical coverage to zero and leads to negative annualized return, large drawdown, and high turnover\. Removing decision\-focused fine\-tuning has only a small effect in this implementation, suggesting that prediction pretraining and radius calibration already explain most of the gain\. Removing size regularization has little effect at the selected hyperparameter scale, but it remains important as a safeguard against radius inflation\.

Table 3:Ablation study for the learned radius\. Calibration is essential: without it, the learned radius collapses and robust decisions degrade\.
### 6\.5Adaptive Radius and Calibration

Figure[3](https://arxiv.org/html/2607.09820#S6.F3)visualizes the learned radius against market volatility\. The learned radius rises in high\-uncertainty regimes and falls in calmer regimes, which is the desired behavior of a contextual ambiguity set\. The empirical coverage of LPAS\-W is lower than the nominal quantile target because decision\-aware validation and size regularization trade exact coverage for performance and a smaller robust radius\. If strict coverage is required, the radius can be post\-adjusted by split conformal calibration\.

![Refer to caption](https://arxiv.org/html/2607.09820v1/x3.png)Figure 3:Learned LPAS\-W radius across market\-volatility regimes\. The radius increases in uncertain periods and decreases in calmer periods\.
### 6\.6Regime Analysis

Table[4](https://arxiv.org/html/2607.09820#S6.T4)reports a compact regime comparison for the main methods\. Each regime contains 129 test observations\. LPAS\-W is particularly competitive in drawdown and high learned\-radius regimes\. In drawdown regimes it obtains annualized return 0\.5167 and Sharpe ratio 2\.0312, slightly above deep fixed\-radius DRO\. In high learned\-radius periods it also improves over the fixed\-radius robust baseline in annualized return and Sharpe ratio\. These results support the central hypothesis: adaptive radii are most valuable when uncertainty is state dependent and decision sensitive\.

Table 4:Regime results on the test set\. Hist denotes historical WDRO, Fixed denotes fixed\-radius DRO, and LPAS denotes LPAS\-W\. Bold indicates the best value within each regime\.Figures[4](https://arxiv.org/html/2607.09820#S6.F4),[5](https://arxiv.org/html/2607.09820#S6.F5), and[6](https://arxiv.org/html/2607.09820#S6.F6)give additional diagnostics\. Calibration by volatility bin checks whether the learned radius responds to market state rather than acting as a constant penalty\. The coverage\-radius plot shows the conservatism required by each method to obtain empirical coverage, and the turnover plot verifies that the performance gain is not simply produced by uncontrolled trading intensity\.

![Refer to caption](https://arxiv.org/html/2607.09820v1/x4.png)Figure 4:Empirical coverage by market\-volatility quartile\. The diagnostic compares LPAS\-W and ablations against historical and fixed\-radius DRO baselines, with the dashed line marking the target coverage level\.![Refer to caption](https://arxiv.org/html/2607.09820v1/x5.png)Figure 5:Coverage versus radius conservatism\. LPAS\-W attains high empirical coverage with a smaller mean ambiguity radius than fixed\-radius robust optimization, illustrating the benefit of state\-dependent radius learning\.![Refer to caption](https://arxiv.org/html/2607.09820v1/x6.png)Figure 6:Average turnover by method\. The comparison checks whether LPAS\-W’s performance is driven by uncontrolled trading intensity\.
### 6\.7Decision Sensitivity Diagnostic

Figure[7](https://arxiv.org/html/2607.09820#S6.F7)plots decision regret against prediction error\. The diagnostic illustrates why prediction error alone is not enough: decision loss depends on whether forecast error lies in a decision\-sensitive direction\. This supports the decision\-focused design of the learned ambiguity radius\.

![Refer to caption](https://arxiv.org/html/2607.09820v1/x7.png)Figure 7:Decision regret versus prediction error\. Prediction error alone does not determine downstream loss because optimization amplifies errors differently across directions\.

## 7Discussion and Limitations

The experiments show a clear hierarchy\. Nonrobust predict\-then\-optimize performs poorly because it turns noisy return forecasts into aggressive portfolios\. Historical Wasserstein DRO is stable but underuses contextual information\. Deep fixed\-radius DRO is a strong baseline and achieves the highest overall Sharpe ratio in the current portfolio experiment\. LPAS\-W does not dominate this fixed\-radius baseline on every average metric, but it obtains competitive returns with a much smaller average radius, slightly better worst\-month and CVaR95 loss, and stronger adaptivity in high\-radius and drawdown regimes\.

The current implementation has three limitations\. First, the experiment is a single rolling split over one 20\-asset S&P 500 universe; broader universes, multiple seeds, and additional rolling folds should be included for stronger empirical claims\. Second, the learned anisotropic metric is part of the framework but not evaluated in the present experiment\. Third, empirical coverage is below the nominal quantile level after decision\-aware tuning, so strict risk\-control applications should add conformal post\-calibration\.

## 8Conclusion

We proposed learned predictive ambiguity sets for decision\-focused DRO\. The method learns a finite nominal scenario distribution and a context\-dependent Wasserstein radius, then solves a robust decision layer whose ambiguity set adapts to the current state\. The portfolio experiment shows that this approach substantially outperforms nonrobust and historical robust baselines, and remains competitive with a strong deep fixed\-radius DRO model while using a smaller radius\. The results suggest that the right object to learn in robust decision\-making is not only a prediction, but also how much the optimizer should distrust that prediction\.

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