Distributionally Robust Listwise Preference Optimization

# Distributionally Robust Listwise Preference Optimization
Source: [https://arxiv.org/html/2607.01715](https://arxiv.org/html/2607.01715)
Xudong Wu The University of Hong Kong Hong Kong SAR &Jian Qian The University of Hong Kong Hong Kong SAR &Pangpang Liu Yale University New Haven, CT, USA &Vaneet Aggarwal Purdue University West Lafayette, IN, USA &Jiayu Chen The University of Hong Kong Hong Kong SAR

###### Abstract

Existing robust preference optimization for language\-model alignment mainly studies pairwise supervision and places robustness at the dataset, prompt, or preference\-pair level\. We instead study listwise preference optimization under ranking\-label uncertainty: given a prompt and a candidate list, the observed ranking over that list may be ambiguous due to annotator inconsistency, near\-ties, lossy rankwise feedback, or reward\-model noise\. We propose a pointwise total\-variation robust Plackett–Luce objective that directly robustifies the ranking label conditional on the candidate list\. The robust loss admits an exact decomposition into the nominal PL loss plus a worst\-case PL correction, and the worst\-case ranking is obtained by sorting current implicit scores in ascending order, reducing the inner maximization fromK\!K\!enumeration toO​\(K​log⁡K\)O\(K\\log K\)\. This tractable structure yields strong offline and online optimization guarantees\. In the offline fixed\-list setting, the robust objective is convex and projected stochastic subgradient reaches globalϵ\\epsilon\-suboptimality withO​\(ϵ−2\)O\(\\epsilon^\{\-2\}\)sample complexity\. In the online policy\-induced setting, where candidate lists are generated by the current policy, we establish weak convexity andO~​\(ϵ−2\)\\widetilde\{O\}\(\\epsilon^\{\-2\}\)Moreau\-envelope stationarity\. Experiments in offline LLM alignment show that the proposed robust correction largely preserves performance under clean labels and improves robustness under noise\. In online alignment, it makes reward\-model\-ranked candidate expansion more reliable and improves both reward\-model and external GPT\-4 judge metrics\.

## 1Introduction

Learning from human preferences has become a central mechanism for aligning large language models, with reinforcement\-learning\-from\-human\-feedback \(RLHF\) and direct preference optimization \(DPO\) serving as standard recipes for instruction\-tuned models\(Christianoet al\.,[2017](https://arxiv.org/html/2607.01715#bib.bib18); Ouyanget al\.,[2022](https://arxiv.org/html/2607.01715#bib.bib19); Rafailovet al\.,[2023](https://arxiv.org/html/2607.01715#bib.bib9)\)\. Most existing analyses and algorithms formulate preference feedback as*pairwise*supervision under a Bradley–Terry \(BT\) model\(Bradley and Terry,[1952](https://arxiv.org/html/2607.01715#bib.bib16); Rafailovet al\.,[2023](https://arxiv.org/html/2607.01715#bib.bib9)\), where each training example identifies a chosen response and a rejected response\. However, many modern preference datasets are naturally*listwise*: for a single prompt, multiple candidate responses are available, and the supervision may contain a full or partial ranking\. This has motivated Plackett–Luce \(PL\) listwise preference objectives\(Plackett,[1975](https://arxiv.org/html/2607.01715#bib.bib17); Xiaet al\.,[2008](https://arxiv.org/html/2607.01715#bib.bib13); Liuet al\.,[2025](https://arxiv.org/html/2607.01715#bib.bib10); Songet al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib14)\), which exploit the relative ordering of multiple candidates rather than reducing the feedback to isolated pairwise comparisons\.

While these works establish listwise preference optimization, they do not address robustness to uncertainty in the observed ranking label itself\. To the best of our knowledge, this is the first work to study robust listwise preference optimization for LLM alignment under conditional ranking\-label ambiguity\.

Robust preference optimization has developed largely along a different axis\. Recent robust DPO\-style methods introduce distributional uncertainty over the empirical preference dataset, the prompt distribution, or the preference\-pair distribution\(Wuet al\.,[2025](https://arxiv.org/html/2607.01715#bib.bib8); Mandalet al\.,[2025](https://arxiv.org/html/2607.01715#bib.bib5); Xuet al\.,[2026](https://arxiv.org/html/2607.01715#bib.bib6)\)\. These approaches address an important question: how should alignment behave when the distribution from which prompts or preference pairs are sampled is perturbed? They do not directly address a different and common source of uncertainty in listwise supervision: even after conditioning on the same prompt and the same candidate list, the observed ranking label itself may be unreliable\.

This paper studies this*conditional ranking\-label uncertainty*\. Given a promptxx, a realized candidate listY=\(y1,…,yK\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\), and an observed rankingσ⋆\\sigma^\{\\star\}, we allow the ranking\-label distribution to vary within a pointwise total\-variation ambiguity set around the empirical ranking\. This models local ambiguity caused by annotator inconsistency, near\-ties between candidates, tied or lossy rankwise feedback, and reward\-model\-induced ranking noise in online alignment\. The key distinction from prior robust preference\-optimization work is that we do not robustify which prompts, pairs, or candidate lists are sampled; instead, conditional on a realized candidate list, we robustify the ranking label over that list\.

This candidate\-list\-conditioned formulation has two advantages\. First, it targets the supervision noise directly: for a fixed prompt–candidate\-list instance, annotator disagreement, reward\-model errors, near\-ties, and lossy rank annotations all manifest as uncertainty in the observed ordering among the same alternatives\.

Second, the formulation preserves the full listwise ranking signal\. The PL model is a strict listwise generalization of the pairwise BT/DPO objective: whenK=2K=2, it recovers the standard pairwise preference loss, while forK\>2K\>2, it retains the sequence\-level ordering among multiple candidates\. Prior listwise ranking and preference\-optimization methods have shown that using the relative ordering among multiple candidates can exploit richer supervision than reducing feedback to isolated pairwise comparisons\(Xiaet al\.,[2008](https://arxiv.org/html/2607.01715#bib.bib13); Songet al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib14); Liuet al\.,[2025](https://arxiv.org/html/2607.01715#bib.bib10)\)\. Our robustification keeps this listwise structure intact: it perturbs the ranking over the same candidate set rather than decomposing the supervision into independent pairwise label flips\.

At first glance, this formulation appears computationally expensive because the adversary may choose amongK\!K\!possible rankings\. Our main structural observation is that the PL loss makes this inner maximization exactly tractable: the worst\-case ranking is obtained by sorting the current implicit scores in ascending order\. Consequently, the robust listwise loss reduces to a convex combination of the nominal PL loss and a single adversarial PL loss, computable inO​\(K​log⁡K\)O\(K\\log K\)time\. ForK≥3K\\geq 3, this correction is intrinsically listwise and does not reduce to a collection of independent pairwise BT corrections\.

We analyze the resulting objective in both offline and online settings naturally induced by listwise alignment\. In the offline fixed\-list setting with log\-linear scores, the robust objective is convex and reaches globalϵ\\epsilon\-suboptimality withO​\(ϵ−2\)O\(\\epsilon^\{\-2\}\)sample complexity\. In the online policy\-induced setting, where candidate lists are sampled from the current policy, the objective is no longer globally convex; we establish weak convexity and anO~​\(ϵ−2\)\\widetilde\{O\}\(\\epsilon^\{\-2\}\)Moreau\-envelope stationarity guarantee using an explicit ascending\-sort Clarke\-subgradient oracle\. Empirically, the proposed correction behaves as a conservative ranking\-label regularizer: it preserves clean\-label performance, improves stability under structured ranking\-label corruption, and helps larger listwise candidate sets become more reliable in online reward\-model\-driven alignment\.

### 1\.1Contributions

Our contributions are as follows\.

- •A robust listwise preference\-optimization formulation\.To the best of our knowledge, we are the first to study robust listwise preference optimization for LLM alignment under conditional ranking\-label uncertainty\. Unlike prior robust DPO methods that perturb the data, prompt, or pair distribution, our ambiguity set targets the ranking label conditional on a realized candidate list\. The formulation recovers the standard pairwise BT/DPO setting whenK=2K=2, while providing a genuinely listwise robustness model whenK\>2K\>2\.
- •Exact and tractable robust PL loss\.We show that the robust PL loss admits an exact decomposition into the nominal PL loss and a worst\-case PL loss\. Although the inner maximization is overK\!K\!rankings, the PL structure implies that the worst\-case ranking is simply the ascending\-score order, giving anO​\(K​log⁡K\)O\(K\\log K\)evaluation algorithm\.
- •Strong offline and online optimization theory\.In the offline fixed\-list log\-linear setting, the robust objective is convex and admits anO​\(ϵ−2\)O\(\\epsilon^\{\-2\}\)stochastic subgradient guarantee\. In the online policy\-induced setting, we prove weak convexity and anO~​\(ϵ−2\)\\widetilde\{O\}\(\\epsilon^\{\-2\}\)Moreau\-envelope stationarity bound using an explicit Clarke\-subgradient oracle\. These guarantees improve or match closely related robust preference\-optimization rates under comparable log\-linear oracle models, as summarized in Table[1](https://arxiv.org/html/2607.01715#S1.T1)\.
- •Empirical validation under ranking\-label uncertainty\.Offline experiments show that robust PL is most useful when listwise labels are structurally corrupted, especially under severe top\-rank noise\. Online LLM alignment experiments show that robustness helps larger candidate lists become more reliable when rankings are generated by a reward model\.

### 1\.2Relation to Prior Robust Preference Optimization

Prior robust preference\-optimization methods mainly study pairwise BT/DPO objectives and place robustness on the data, prompt, pair, or oracle distribution\. In contrast, we study PL listwise preference optimization and place uncertainty on the ranking label conditional on a realized candidate list\. This distinction is central: forK≥3K\\geq 3, our robust correction is a genuinely listwise PL max\-gap over permutations, with an exact ascending\-score solution rather than a reduction to independent pairwise BT corrections\.

[Table˜1](https://arxiv.org/html/2607.01715#S1.T1)summarizes how our setting differs from representative robust preference\-optimization theories in terms of preference model, robustness source, robust loss structure, and sample/oracle complexity\. Additional discussion is provided in Appendix[A](https://arxiv.org/html/2607.01715#A1)\.

Table 1:Comparison with representative robust preference\-optimization theory\.*Note\.*The Main guarantee row reports the total sample/oracle complexity under the corresponding oracle model\. Detailed comparisons withMandalet al\.\([2025](https://arxiv.org/html/2607.01715#bib.bib5)\)andLiet al\.\([2026](https://arxiv.org/html/2607.01715#bib.bib7)\)are provided in[Remarks˜2](https://arxiv.org/html/2607.01715#Thmremark2)and[4](https://arxiv.org/html/2607.01715#Thmremark4)\.

## 2Background: From Pairwise DPO to Listwise PL\-DPO

Given a promptxxand two responsesy\+,y−y^\{\+\},y^\{\-\}, the Bradley–Terry \(BT\) model assumesℙ​\(y\+≻y−∣x\)=σ​\(r⋆​\(x,y\+\)−r⋆​\(x,y−\)\)\\mathbb\{P\}\(y^\{\+\}\\succ y^\{\-\}\\mid x\)=\\sigma\\\!\\big\(r^\{\\star\}\(x,y^\{\+\}\)\-r^\{\\star\}\(x,y^\{\-\}\)\\big\), whereσ\\sigmais the logistic sigmoid andr⋆r^\{\\star\}is the latent ground\-truth reward\(Bradley and Terry,[1952](https://arxiv.org/html/2607.01715#bib.bib16)\)\. Under KL\-regularized RLHF the optimal policy admits the Gibbs formπ⋆​\(y∣x\)∝πref​\(y∣x\)​exp⁡\(r⋆​\(x,y\)/β\)\\pi^\{\\star\}\(y\\mid x\)\\propto\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\\exp\(r^\{\\star\}\(x,y\)/\\beta\), sor⋆​\(x,y\)=β​log⁡\[π⋆​\(y∣x\)/πref​\(y∣x\)\]\+β​log⁡Z​\(x\)r^\{\\star\}\(x,y\)=\\beta\\log\[\\pi^\{\\star\}\(y\\mid x\)/\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\]\+\\beta\\log Z\(x\)\. The partitionβ​log⁡Z​\(x\)\\beta\\log Z\(x\)cancels in the BT difference, motivating the implicit DPO scoregθ​\(x,y\):=β​log⁡\[πθ​\(y∣x\)/πref​\(y∣x\)\]g\_\{\\theta\}\(x,y\):=\\beta\\log\[\\pi\_\{\\theta\}\(y\\mid x\)/\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\]and the pairwise DPO lossℓDPO​\(θ;x,y\+,y−\)=−log⁡σ​\(gθ​\(x,y\+\)−gθ​\(x,y−\)\)\\ell\_\{\\mathrm\{DPO\}\}\(\\theta;x,y^\{\+\},y^\{\-\}\)=\-\\log\\sigma\(g\_\{\\theta\}\(x,y^\{\+\}\)\-g\_\{\\theta\}\(x,y^\{\-\}\)\)\(Rafailovet al\.,[2023](https://arxiv.org/html/2607.01715#bib.bib9)\)\.

#### Listwise generalization via Plackett–Luce\.

For each promptxx, suppose we observe a realized candidate list𝒴=\{y1,…,yK\}\\mathcal\{Y\}=\\\{y\_\{1\},\\dots,y\_\{K\}\\\}and a deterministic empirical rankingσ⋆∈SK\\sigma^\{\\star\}\\in S\_\{K\}, whereσi⋆\\sigma^\{\\star\}\_\{i\}is the index of the response placed at rankii\. The listwise analogue of BT is the Plackett–Luce model\(Plackett,[1975](https://arxiv.org/html/2607.01715#bib.bib17)\):

ℙ​\(σ⋆∣x,𝒴\)=∏i=1Kexp⁡\(r⋆​\(x,yσi⋆\)\)∑j=iKexp⁡\(r⋆​\(x,yσj⋆\)\)\.\\mathbb\{P\}\(\\sigma^\{\\star\}\\mid x,\\mathcal\{Y\}\)=\\prod\_\{i=1\}^\{K\}\\frac\{\\exp\(r^\{\\star\}\(x,y\_\{\\sigma^\{\\star\}\_\{i\}\}\)\)\}\{\\sum\_\{j=i\}^\{K\}\\exp\(r^\{\\star\}\(x,y\_\{\\sigma^\{\\star\}\_\{j\}\}\)\)\}\.\(1\)Substituting the same DPO reparameterization, the partitionβ​log⁡Z​\(x\)\\beta\\log Z\(x\)cancels at every stage of the product, yielding the listwise PL\-DPO loss

ℓPL​\(θ;x,y1:K,σ\)=−∑i=1Kgθ​\(x,yσi\)\+∑i=1Klog⁡\(∑j=iKexp⁡\(gθ​\(x,yσj\)\)\)\.\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)=\-\\sum\_\{i=1\}^\{K\}g\_\{\\theta\}\(x,y\_\{\\sigma\_\{i\}\}\)\+\\sum\_\{i=1\}^\{K\}\\log\\\!\\Big\(\\sum\_\{j=i\}^\{K\}\\exp\(g\_\{\\theta\}\(x,y\_\{\\sigma\_\{j\}\}\)\)\\Big\)\.\(2\)AtK=2K=2andσ=\(1,2\)\\sigma=\(1,2\), \([2](https://arxiv.org/html/2607.01715#S2.E2)\) reduces to−log⁡σ​\(gθ​\(x,y1\)−gθ​\(x,y2\)\)\-\\log\\sigma\(g\_\{\\theta\}\(x,y\_\{1\}\)\-g\_\{\\theta\}\(x,y\_\{2\}\)\), recovering pairwise DPO exactly\. ForK≥3K\\geq 3,ℓPL\\ell\_\{\\mathrm\{PL\}\}aggregates information fromK−1K\-1stagewise PL choices and is strictly more informative than any single pairwise comparison drawn from the same list\. A formal curvature view of this stagewise information aggregation is provided in[Proposition˜H\.2](https://arxiv.org/html/2607.01715#Thmproposition2a)\.

The*nominal*listwise objective on a fixed offline dataset𝒟=\{\(x,y1:K,σ⋆\)\}\\mathcal\{D\}=\\\{\(x,y\_\{1:K\},\\sigma^\{\\star\}\)\\\}isJnom​\(θ\):=𝔼\(x,y1:K,σ⋆\)∼𝒟​\[ℓPL​\(θ;x,y1:K,σ⋆\)\]J\_\{\\mathrm\{nom\}\}\(\\theta\):=\\mathbb\{E\}\_\{\(x,y\_\{1:K\},\\sigma^\{\\star\}\)\\sim\\mathcal\{D\}\}\[\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\)\]\. We add ranking\-label robustness next\.

## 3Pointwise TV\-Robust Listwise Objective

We treat each empirical ranking labelσ⋆\\sigma^\{\\star\}as a Dirac point mass onSKS\_\{K\}and let an adversary perturb this label distribution within a TV ball of radiusρ∈\[0,1\]\\rho\\in\[0,1\]\.

###### Definition 1\(Pointwise TV ambiguity\)\.

For sample\(x,y1:K,σ⋆\)\(x,y\_\{1:K\},\\sigma^\{\\star\}\),

𝒰TV​\(δσ⋆,ρ\):=\{P∈Δ​\(SK\):TV​\(P,δσ⋆\)≤ρ\},TV​\(P,Q\)=12​‖P−Q‖1\.\\mathcal\{U\}\_\{\\mathrm\{TV\}\}\(\\delta\_\{\\sigma^\{\\star\}\},\\rho\):=\\big\\\{P\\in\\Delta\(S\_\{K\}\):\\mathrm\{TV\}\(P,\\delta\_\{\\sigma^\{\\star\}\}\)\\leq\\rho\\big\\\},\\quad\\mathrm\{TV\}\(P,Q\)=\\tfrac\{1\}\{2\}\\\|P\-Q\\\|\_\{1\}\.\(3\)

###### Definition 2\(Robust listwise objective\)\.

The pointwise robust loss and offline robust objective are

ℓrob​\(θ;x,y1:K,σ⋆\)\\displaystyle\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\):=maxP∈𝒰TV​\(δσ⋆,ρ\)⁡𝔼σ∼P​\[ℓPL​\(θ;x,y1:K,σ\)\],\\displaystyle:=\\max\_\{P\\in\\mathcal\{U\}\_\{\\mathrm\{TV\}\}\(\\delta\_\{\\sigma^\{\\star\}\},\\rho\)\}\\mathbb\{E\}\_\{\\sigma\\sim P\}\[\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)\],Jrob​\(θ\)\\displaystyle J\_\{\\mathrm\{rob\}\}\(\\theta\):=𝔼𝒟​\[ℓrob​\(θ;x,y1:K,σ⋆\)\]\.\\displaystyle:=\\mathbb\{E\}\_\{\\mathcal\{D\}\}\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\)\]\.\(4\)

###### Lemma 1\(Exact pointwise TV decomposition\)\.

For every sample\(x,y1:K,σ⋆\)\(x,y\_\{1:K\},\\sigma^\{\\star\}\), parameterθ\\theta, andρ∈\[0,1\]\\rho\\in\[0,1\],

ℓrob​\(θ;x,y1:K,σ⋆\)=\(1−ρ\)​ℓPL​\(θ;x,y1:K,σ⋆\)\+ρ​maxσ∈SK⁡ℓPL​\(θ;x,y1:K,σ\)\.\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\)=\(1\-\\rho\)\\,\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\)\+\\rho\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)\.\(5\)Equivalently,ℓrob=ℓPL​\(σ⋆\)\+ρ​\(maxσ⁡ℓPL​\(σ\)−ℓPL​\(σ⋆\)\)\\ell\_\{\\mathrm\{rob\}\}=\\ell\_\{\\mathrm\{PL\}\}\(\\sigma^\{\\star\}\)\+\\rho\\big\(\\max\_\{\\sigma\}\\ell\_\{\\mathrm\{PL\}\}\(\\sigma\)\-\\ell\_\{\\mathrm\{PL\}\}\(\\sigma^\{\\star\}\)\\big\)\.

The full proof is in Appendix[C](https://arxiv.org/html/2607.01715#A3)\.

## 4Tractable Worst\-Case Ranking

By[Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1), evaluating the pointwise robust loss requires solving

maxσ∈SK⁡ℓPL​\(θ;x,y1:K,σ\),\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\),which is naively a maximization overK\!K\!possible rankings\. The key structural observation is that, under the PL loss, the adversarial ranking is not arbitrary: it is obtained by placing low\-score candidates before high\-score candidates\. Thus the inner maximization reduces to a single sorting operation\.

###### Theorem 1\(Worst\-case ranking by ascending scores\)\.

Fix\(θ,x,y1:K\)\(\\theta,x,y\_\{1:K\}\)and letgi:=gθ​\(x,yi\)g\_\{i\}:=g\_\{\\theta\}\(x,y\_\{i\}\)\. Ifσwc∈SK\\sigma\_\{\\rm wc\}\\in S\_\{K\}sorts the scores in nondecreasing order,gσwc​\(1\)≤⋯≤gσwc​\(K\)g\_\{\\sigma\_\{\\rm wc\}\(1\)\}\\leq\\cdots\\leq g\_\{\\sigma\_\{\\rm wc\}\(K\)\}, then

σwc∈arg⁡maxσ∈SK⁡ℓPL​\(θ;x,y1:K,σ\)\.\\sigma\_\{\\rm wc\}\\in\\arg\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)\.Hence the inner maximization in the robust PL loss is solved by sorting and costsO​\(K​log⁡K\)O\(K\\log K\)\. With ties, any deterministic tie\-breaking rule within tied groups is valid\.

The full proof is in Appendix[D](https://arxiv.org/html/2607.01715#A4)\.

Algorithm 1Exact robust PL loss1:

x,Y=\(y1,…,yK\),σ⋆,θ,ρx,Y=\(y\_\{1\},\\ldots,y\_\{K\}\),\\sigma^\{\\star\},\\theta,\\rho
2:

gi←gθ​\(x,yi\)g\_\{i\}\\leftarrow g\_\{\\theta\}\(x,y\_\{i\}\)for

i∈\[K\]i\\in\[K\]
3:

σwc←argsorti∈\[K\]⁡\(gi\)\\sigma\_\{\\rm wc\}\\leftarrow\\operatorname\{argsort\}\_\{i\\in\[K\]\}\(g\_\{i\}\)in nondecreasing order

4:

ℓnom←ℓPL​\(θ;x,Y,σ⋆\)\\ell\_\{\\rm nom\}\\leftarrow\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\),

ℓwc←ℓPL​\(θ;x,Y,σwc\)\\ell\_\{\\rm wc\}\\leftarrow\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\_\{\\rm wc\}\)
5:return

\(1−ρ\)​ℓnom\+ρ​ℓwc\(1\-\\rho\)\\ell\_\{\\rm nom\}\+\\rho\\ell\_\{\\rm wc\}

Algorithm 2Offline Robust PL\-DPO

1:Dataset

𝒟=\{\(xi,Yi,σi⋆\)\}i=1n\\mathcal\{D\}=\\\{\(x\_\{i\},Y\_\{i\},\\sigma\_\{i\}^\{\\star\}\)\\\}\_\{i=1\}^\{n\},

θ0∈Θ\\theta\_\{0\}\\in\\Theta, stepsize

η\\eta, batch size

BsB\_\{s\}, radius

ρ\\rho\.

2:for

t=0,…,T−1t=0,\\ldots,T\-1do

3:Sample mini\-batch

ℬt⊂𝒟\\mathcal\{B\}\_\{t\}\\subset\\mathcal\{D\}\.

4:

ℒ^RPL​\(θt\)←Bs−1​∑\(xi,Yi,σi⋆\)∈ℬtℓrob​\(θt;xi,Yi,σi⋆\)\\widehat\{\\mathcal\{L\}\}\_\{\\rm RPL\}\(\\theta\_\{t\}\)\\leftarrow B\_\{s\}^\{\-1\}\\\!\\\!\\sum\_\{\(x\_\{i\},Y\_\{i\},\\sigma\_\{i\}^\{\\star\}\)\\in\\mathcal\{B\}\_\{t\}\}\\ell\_\{\\rm rob\}\(\\theta\_\{t\};x\_\{i\},Y\_\{i\},\\sigma\_\{i\}^\{\\star\}\), with

ℓrob\\ell\_\{\\rm rob\}from[Algorithm˜1](https://arxiv.org/html/2607.01715#alg1)\.

5:Choose

g^t∈∂θℒ^RPL​\(θt\)\\widehat\{g\}\_\{t\}\\in\\partial\_\{\\theta\}\\widehat\{\\mathcal\{L\}\}\_\{\\rm RPL\}\(\\theta\_\{t\}\)\.

6:

θt\+1←ΠΘ​\(θt−η​g^t\)\\theta\_\{t\+1\}\\leftarrow\\Pi\_\{\\Theta\}\(\\theta\_\{t\}\-\\eta\\widehat\{g\}\_\{t\}\)\.

7:endfor

8:return

θ¯T=T−1​∑t=0T−1θt\\bar\{\\theta\}\_\{T\}=T^\{\-1\}\\sum\_\{t=0\}^\{T\-1\}\\theta\_\{t\}\.

Algorithm 3Online Robust PL\-SAIL

1:Initial parameter

θ0∈Θ\\theta\_\{0\}\\in\\Theta, stepsize

η\\eta, batch size

BsB\_\{s\}, list size

KK, radius

ρ\\rho\.

2:for

t=0,…,T−1t=0,\\ldots,T\-1do

3:Sample

xi∼𝒟xx\_\{i\}\\sim\\mathcal\{D\}\_\{x\}, generate

Yi∼πθt⊗K\(⋅\|xi\)Y\_\{i\}\\sim\\pi\_\{\\theta\_\{t\}\}^\{\\otimes K\}\(\\cdot\|x\_\{i\}\), and obtain

σi⋆∼p⋆\(⋅\|xi,Yi\)\\sigma\_\{i\}^\{\\star\}\\sim p^\{\\star\}\(\\cdot\|x\_\{i\},Y\_\{i\}\), for

i∈\[Bs\]i\\in\[B\_\{s\}\]\.

4:Evaluate

ℓrob​\(θt;xi,Yi,σi⋆\)\\ell\_\{\\rm rob\}\(\\theta\_\{t\};x\_\{i\},Y\_\{i\},\\sigma\_\{i\}^\{\\star\}\)by[Algorithm˜1](https://arxiv.org/html/2607.01715#alg1)\.

5:Form

G^t←Bs−1​∑i=1BsG​\(θt;Zi\)\\widehat\{G\}\_\{t\}\\leftarrow B\_\{s\}^\{\-1\}\\sum\_\{i=1\}^\{B\_\{s\}\}G\(\\theta\_\{t\};Z\_\{i\}\), where

Zi=\(xi,Yi,σi⋆\)Z\_\{i\}=\(x\_\{i\},Y\_\{i\},\\sigma\_\{i\}^\{\\star\}\)and

GGis defined in \([10](https://arxiv.org/html/2607.01715#S5.E10)\)\.

6:

θt\+1←ΠΘ​\(θt−η​G^t\)\\theta\_\{t\+1\}\\leftarrow\\Pi\_\{\\Theta\}\(\\theta\_\{t\}\-\\eta\\widehat\{G\}\_\{t\}\)\.

7:endfor

8:return

θR\\theta\_\{R\},

R∼Uniform​\{0,…,T−1\}R\\sim\{\\rm Uniform\}\\\{0,\\ldots,T\-1\\\}\.

## 5Optimization Theory

We analyze two settings: an*offline*fixed\-list setting in which\(y1:K,σ⋆\)\(y\_\{1:K\},\\sigma^\{\\star\}\)is independent ofθ\\theta, and an*online*policy\-induced setting in whichYYis sampled from the current policyπθ⊗K\\pi\_\{\\theta\}^\{\\otimes K\}\.

The offline and online optimization procedures are summarized in[Algorithms˜2](https://arxiv.org/html/2607.01715#alg2)and[3](https://arxiv.org/html/2607.01715#alg3)\.

### 5\.1Offline Fixed\-List Setting

###### Assumption 1\(Log\-linear policy class\)\.

The response space𝒴\\mathcal\{Y\}is finite\. Letψ:𝒳×𝒴→ℝdp\\psi:\\mathcal\{X\}\\times\\mathcal\{Y\}\\to\\mathbb\{R\}^\{d\_\{p\}\}satisfysupx,y‖ψ​\(x,y\)‖2≤Bψ\\sup\_\{x,y\}\\\|\\psi\(x,y\)\\\|\_\{2\}\\leq B\_\{\\psi\}, whereBψ=1B\_\{\\psi\}=1can be obtained by rescaling\. ForB\>0B\>0, letΘ:=\{θ∈ℝdp:‖θ‖2≤B\}\\Theta:=\\\{\\theta\\in\\mathbb\{R\}^\{d\_\{p\}\}:\\\|\\theta\\\|\_\{2\}\\leq B\\\}, and consider

Π=\{πθ:πθ​\(y\|x\)=exp⁡\(θ⊤​ψ​\(x,y\)\)∑y′∈𝒴exp⁡\(θ⊤​ψ​\(x,y′\)\),θ∈Θ\}\.\\Pi=\\left\\\{\\pi\_\{\\theta\}:\\pi\_\{\\theta\}\(y\|x\)=\\frac\{\\exp\(\\theta^\{\\top\}\\psi\(x,y\)\)\}\{\\sum\_\{y^\{\\prime\}\\in\\mathcal\{Y\}\}\\exp\(\\theta^\{\\top\}\\psi\(x,y^\{\\prime\}\)\)\},\\ \\theta\\in\\Theta\\right\\\}\.Assumeπref=πθref\\pi\_\{\\rm ref\}=\\pi\_\{\\theta\_\{\\rm ref\}\}for some fixedθref∈Θ\\theta\_\{\\rm ref\}\\in\\Theta, and setD:=supθ∈Θ‖θ−θref‖2<∞D:=\\sup\_\{\\theta\\in\\Theta\}\\\|\\theta\-\\theta\_\{\\rm ref\}\\\|\_\{2\}<\\infty\.

###### Proposition 1\(Convexity of the offline robust objective\)\.

Suppose[Assumption˜1](https://arxiv.org/html/2607.01715#Thmassumption1)holds,θ↦ℓPL​\(θ;x,y1:K,σ\)\\theta\\mapsto\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)is convex for every\(x,y1:K,σ\)\(x,y\_\{1:K\},\\sigma\)\. Hence by[Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1),θ↦ℓrob​\(θ;x,y1:K,σ⋆\)\\theta\\mapsto\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\)is convex onΘ\\Theta\. The offline robust objectiveJrobJ\_\{\\mathrm\{rob\}\}is convex\.

The proof is in Appendix[E\.1](https://arxiv.org/html/2607.01715#A5.SS1)\. We use the projected stochastic subgradient method

θt\+1=ΠΘ​\(θt−η​g^t\),t=0,…,T−1,\\theta\_\{t\+1\}=\\Pi\_\{\\Theta\}\\big\(\\theta\_\{t\}\-\\eta\\,\\widehat\{g\}\_\{t\}\\big\),\\qquad t=0,\\dots,T\-1,\(6\)whereg^t\\widehat\{g\}\_\{t\}is a mini\-batch unbiased subgradient estimator with conditional variance bounded byσg2/Bs\\sigma\_\{g\}^\{2\}/B\_\{s\}\(mini\-batch sizeBsB\_\{s\}\)\.

###### Theorem 2\(Offline suboptimality of projected stochastic subgradient\)\.

Suppose[Assumption˜1](https://arxiv.org/html/2607.01715#Thmassumption1)holds, a mini\-batch oracle with second moment bounded as𝔼​\[‖g^t‖22∣θt\]≤4​K2​Bψ2\+σg2/Bs\\mathbb\{E\}\[\\\|\\widehat\{g\}\_\{t\}\\\|\_\{2\}^\{2\}\\mid\\theta\_\{t\}\]\\leq 4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}, withη=2​B/T​\(4​K2​Bψ2\+σg2/Bs\)\\eta=2B/\\sqrt\{T\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}\)\}the averaged iterateθ¯T:=1T​∑t=0T−1θt\\bar\{\\theta\}\_\{T\}:=\\tfrac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\theta\_\{t\}satisfies

𝔼​\[Jrob​\(θ¯T\)\]−minθ∈Θ⁡Jrob​\(θ\)≤2​B​4​K2​Bψ2\+σg2/BsT\.\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\bar\{\\theta\}\_\{T\}\)\]\-\\min\_\{\\theta\\in\\Theta\}J\_\{\\mathrm\{rob\}\}\(\\theta\)\\leq\\frac\{2B\\sqrt\{4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}\}\}\{\\sqrt\{T\}\}\.\(7\)ConsequentlyT=O​\(\(4​K2​Bψ2\+σg2/Bs\)/ε2\)T=O\(\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}\)/\\varepsilon^\{2\}\)iterations suffice forε\\varepsilon\-suboptimality\. With fixedBs=Θ​\(1\)B\_\{s\}=\\Theta\(1\), the total sample complexity isBs​T=O​\(ε−2\)B\_\{s\}T=O\(\\varepsilon^\{\-2\}\)\.

The full proof is in Appendix[E\.3](https://arxiv.org/html/2607.01715#A5.SS3)\.

### 5\.2Online Policy\-Induced Setting

We now study the on\-policy setting where, for each promptx∼𝒟xx\\sim\\mathcal\{D\}\_\{x\}, the candidate listY=\(y1,…,yK\)∈𝒴KY=\(y\_\{1\},\\dots,y\_\{K\}\)\\in\\mathcal\{Y\}^\{K\}is sampled iid asY∼πθ⊗K\(⋅∣x\)Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\\mid x\)\. The ranking oracle returnsσ⋆∼p⋆\(⋅∣x,Y\)\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\), conditionally independent ofθ\\thetagiven\(x,Y\)\(x,Y\)\.*Notation:*in the offline log\-linear setting \([Section˜5\.1](https://arxiv.org/html/2607.01715#S5.SS1)\) we wrote scores asgθ​\(x,y\)=θ⊤​ϕ​\(x,y\)g\_\{\\theta\}\(x,y\)=\\theta^\{\\top\}\\phi\(x,y\)\. In the online setting we instead parameterize the policy asπθ​\(y∣x\)∝exp⁡\(θ⊤​ψ​\(x,y\)\)\\pi\_\{\\theta\}\(y\\mid x\)\\propto\\exp\(\\theta^\{\\top\}\\psi\(x,y\)\)and letsθ​\(x,y\):=log⁡\[πθ​\(y∣x\)/πref​\(y∣x\)\]s\_\{\\theta\}\(x,y\):=\\log\[\\pi\_\{\\theta\}\(y\\mid x\)/\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\]be the induced log\-ratio score\. Both reduce to affine functions ofθ\\theta\. The online objectives are

Jnomon​\(θ\):=𝔼x,Y∼πθ⊗K,σ⋆∼p⋆​\[ℓPL​\(θ;x,Y,σ⋆\)\],Jrobon​\(θ\):=𝔼x,Y∼πθ⊗K,σ⋆∼p⋆​\[ℓrob​\(θ;x,Y,σ⋆\)\]\.J\_\{\\mathrm\{nom\}\}^\{\\mathrm\{on\}\}\(\\theta\):=\\mathbb\{E\}\_\{x,Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\},\\sigma^\{\\star\}\\sim p^\{\\star\}\}\[\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\],\\quad J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\):=\\mathbb\{E\}\_\{x,Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\},\\sigma^\{\\star\}\\sim p^\{\\star\}\}\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\]\.\(8\)By[Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1),Jrobon=Jnomon\+ρ​A​\(θ\)J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}=J\_\{\\mathrm\{nom\}\}^\{\\mathrm\{on\}\}\+\\rho A\(\\theta\)withA​\(θ\)=𝔼​\[maxσ⁡ℓPL​\(θ;x,Y,σ\)−ℓPL​\(θ;x,Y,σ⋆\)\]A\(\\theta\)=\\mathbb\{E\}\\big\[\\max\_\{\\sigma\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\-\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\big\]\. This connects the SAIL\-style bilevel formulation of online alignment\(Dinget al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib15)\)with our pointwise\-TV listwise robustness; see[Appendix˜G](https://arxiv.org/html/2607.01715#A7)for the bilevel\-to\-single\-level reduction we use\.

Becauseπθ⊗K\\pi\_\{\\theta\}^\{\\otimes K\}depends onθ\\theta,JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}is*not*globally convex in general\. Our analysis therefore departs from the offline case: we work in the weakly convex framework\(Davis and Drusvyatskiy,[2019](https://arxiv.org/html/2607.01715#bib.bib20); Drusvyatskiy and Lewis,[2018](https://arxiv.org/html/2607.01715#bib.bib21)\), with the Clarke subdifferential∂C\\partial\_\{C\}\(Clarke,[1990](https://arxiv.org/html/2607.01715#bib.bib22)\)\.

###### Assumption 2\(Online policy\-induced sampling and well\-posedness\)\.

For eachθ∈Θ\\theta\\in\\Theta, drawx∼𝒟xx\\sim\\mathcal\{D\}\_\{x\}, sampleY=\(y1,…,yK\)∼πθ⊗K\(⋅\|x\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\)\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\|x\), and drawσ⋆∼p⋆\(⋅\|x,Y\)\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\|x,Y\), where𝒟x\\mathcal\{D\}\_\{x\}is independent ofθ\\thetaandσ⋆\\sigma^\{\\star\}is conditionally independent ofθ\\thetagiven\(x,Y\)\(x,Y\)\. Define

F​\(θ\):=Jrobon​\(θ\)\+IΘ​\(θ\),IΘ​\(θ\)=\{0,θ∈Θ,\+∞,θ∉Θ,Finf:=infθ∈ℝdpF​\(θ\)\>−∞\.F\(\\theta\):=J\_\{\\rm rob\}^\{\\rm on\}\(\\theta\)\+I\_\{\\Theta\}\(\\theta\),\\qquad I\_\{\\Theta\}\(\\theta\)=\\begin\{cases\}0,&\\theta\\in\\Theta,\\\\ \+\\infty,&\\theta\\notin\\Theta,\\end\{cases\}\\qquad F\_\{\\inf\}:=\\inf\_\{\\theta\\in\\mathbb\{R\}^\{d\_\{p\}\}\}F\(\\theta\)\>\-\\infty\.AssumeFFis proper and lower semicontinuous and bounded below\.

We do not place the existence of a Clarke\-subdifferential stochastic oracle as an assumption; instead we*construct*one from ascending\-sort below\.

###### Proposition 2\(Online weak convexity\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold\.JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}isκ\\kappa\-weakly convex onΘ\\Thetawith

κ=K2​Bψ2​\(8\+log⁡K\+2​D​Bψ\)\.\\kappa=K^\{2\}B\_\{\\psi\}^\{2\}\\big\(8\+\\log K\+2DB\_\{\\psi\}\\big\)\.\(9\)

The bound \([9](https://arxiv.org/html/2607.01715#S5.E9)\) is polynomial inKKand the model\-boundedness constants; the full argument is deferred to Appendix[F](https://arxiv.org/html/2607.01715#A6)\.

#### Stochastic oracle from ascending\-sort\.

Letσsel​\(θ;x,Y\)\\sigma^\{\\mathrm\{sel\}\}\(\\theta;x,Y\)be the deterministic ascending\-score maximizer ofℓPL​\(θ;x,Y,σ\)\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\), with fixed tie\-breaking, as given by[Theorem˜1](https://arxiv.org/html/2607.01715#Thmtheorem1)\. ForZ=\(x,Y,σ⋆\)Z=\(x,Y,\\sigma^\{\\star\}\), define

G​\(θ;Z\):=\(1−ρ\)​∇ℓPL​\(θ;x,Y,σ⋆\)\+ρ​∇ℓPL​\(θ;x,Y,σsel\)\+ℓrob​\(θ;x,Y,σ⋆\)​Sθ​\(x,Y\)\.G\(\\theta;Z\):=\(1\-\\rho\)\\nabla\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\nabla\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\mathrm\{sel\}\}\)\+\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)S\_\{\\theta\}\(x,Y\)\.\(10\)The last term is the score\-function correction for policy\-induced sampling\.[Lemma˜F\.9](https://arxiv.org/html/2607.01715#Thmlemma9)verifies thatGGis a valid stochastic Clarke\-subgradient oracle with bounded second moment\.

###### Theorem 3\(Online robust convergence\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold\. Letκ\\kappabe as in \([9](https://arxiv.org/html/2607.01715#S5.E9)\) andGtot2G\_\{\\mathrm\{tot\}\}^\{2\}as in \([F\.5](https://arxiv.org/html/2607.01715#A6.E5)\)\. ForF=Jrobon\+IΘF=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\+I\_\{\\Theta\}andλ^∈\(0,1/κ\)\\hat\{\\lambda\}\\in\(0,1/\\kappa\), defineFλ^​\(θ\)=minu∈Θ⁡\{Jrobon​\(u\)\+12​λ^​‖u−θ‖2\}F\_\{\\hat\{\\lambda\}\}\(\\theta\)=\\min\_\{u\\in\\Theta\}\\\{J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(u\)\+\\tfrac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\\\}\. Fixλ^∈\(0,1/κ\)\\hat\{\\lambda\}\\in\(0,1/\\kappa\)and setΔ0:=Fλ^​\(θ0\)−Finf\\Delta\_\{0\}:=F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{0\}\)\-F\_\{\\inf\}\. Run[Algorithm˜3](https://arxiv.org/html/2607.01715#alg3)with the explicit ascending\-sort oracle \([10](https://arxiv.org/html/2607.01715#S5.E10)\) and constant stepsizeη=2​λ^​Δ0/\(Gtot2​T\)\\eta=\\sqrt\{2\\hat\{\\lambda\}\\Delta\_\{0\}/\(G\_\{\\mathrm\{tot\}\}^\{2\}T\)\}\. Then

𝔼​\[‖∇Fλ^​\(θR\)‖2\]≤21−κ​λ^​2​Δ0​Gtot2λ^​T\.\\mathbb\{E\}\\\!\\big\[\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{R\}\)\\\|^\{2\}\\big\]\\leq\\frac\{2\}\{1\-\\kappa\\hat\{\\lambda\}\}\\sqrt\{\\frac\{2\\,\\Delta\_\{0\}\\,G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\hat\{\\lambda\}\\,T\}\}\.\(11\)

###### Corollary 1\(Sample / oracle complexity\)\.

𝔼​‖∇Fλ^​\(θR\)‖2≤ε\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{R\}\)\\\|^\{2\}\\leq\\varepsilonholds forT≥8​Δ0​Gtot2/\[λ^​\(1−κ​λ^\)2​ε2\]T\\geq 8\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}/\[\\hat\{\\lambda\}\(1\-\\kappa\\hat\{\\lambda\}\)^\{2\}\\varepsilon^\{2\}\]\. Atλ^=1/\(2​κ\)\\hat\{\\lambda\}=1/\(2\\kappa\),

T=O​\(κ​Δ0​Gtot2ε2\)=O~​\(K6​Bψ4​\(log⁡K\+D​Bψ\)3ε2\)\.T=O\\\!\\left\(\\frac\{\\kappa\\,\\Delta\_\{0\}\\,G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\\;=\\;\\tilde\{O\}\\\!\\left\(\\frac\{K^\{6\}B\_\{\\psi\}^\{4\}\(\\log K\+DB\_\{\\psi\}\)^\{3\}\}\{\\varepsilon^\{2\}\}\\right\)\.\(12\)

The proofs of[Theorems˜3](https://arxiv.org/html/2607.01715#Thmtheorem3)and[1](https://arxiv.org/html/2607.01715#Thmcorollary1)are in Appendix[F](https://arxiv.org/html/2607.01715#A6)\.

## 6Experiments

#### Evaluation questions\.

Our experiments test the behavior predicted by the proposed ambiguity model rather than claiming that robustness monotonically improves all metrics\. We ask whether the robust correction: \(i\) preserves performance when rankings are reliable; \(ii\) reduces the failure modes of PL learning when listwise labels are corrupted; and \(iii\) makes larger candidate lists more reliable in online alignment, where rankings are generated by a reward model\.

We evaluate the proposed pointwise\-TV robust PL objective in two settings matching our theory: offline fixed\-list ranking with corrupted labels, and online policy\-induced alignment with reward\-model\-generated rankings\. Full experimental details are deferred to Appendix[J](https://arxiv.org/html/2607.01715#A10)\.

#### Setup\.

We use UltraFeedback\(Cuiet al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib23)\), where each prompt has four candidate responses, yielding a natural listwise preference problem withK=4K=4\. We compare three objectives:*Nominal BT*, the standard pairwise DPO/BT baseline using chosen–rejected comparisons;*Nominal PL*, the non\-robust listwise Plackett–Luce objective; and*Robust PL*, our pointwise\-TV robust version of the PL objective\. In the online setting, candidate responses are generated by the current policy and ranked by a frozen reward model; the corresponding evaluation metrics are defined in the online result table\.

### 6\.1Offline fixed\-list evaluation

The offline setting directly matches our ambiguity model: the prompt and candidate list are fixed, while the ranking label may be corrupted\. Since binary chosen–rejected label flipping has no unique canonical analogue for a full ranking, we introduce two listwise corruptions\.*Near\-tie*corruption swaps the adjacent pair with the closest annotation scores, modeling local ambiguity between nearly indistinguishable responses\.*Top\-rank*corruption promotes a lower\-ranked response to the first position, modeling a more severe error because the first PL stage selects from the full candidate list\. The noise levelϵ\\epsilonis the fraction of corrupted training rankings; evaluation labels are always clean\.

We also include two pairwise robust\-DPO baselines, TV\-DR\-DPO\(Mandalet al\.,[2025](https://arxiv.org/html/2607.01715#bib.bib5)\)and KLDPO\(Xuet al\.,[2026](https://arxiv.org/html/2607.01715#bib.bib6)\), reimplemented in the same pipeline using their loss\-level robust DPO objectives\. Their hyperparameters are selected from held\-out sweeps, with the sweep results reported in Appendix[I\.2](https://arxiv.org/html/2607.01715#A9.SS2)\. We evaluate all methods by Kendall’sτ\\tauon clean held\-out rankings\. Each method assigns scalar scores to the four candidate responses, which induce a predicted ranking; Kendall’sτ\\taumeasures the rank correlation with the clean UltraFeedback reference ranking\.

#### Offline observations\.

[Table˜2](https://arxiv.org/html/2607.01715#S6.T2)shows that Robust PL incurs only a small degradation under clean labels while providing clear gains when the ranking labels are corrupted\. The improvement is most pronounced under severe top\-rank corruption: whenϵ=1\.0\\epsilon=1\.0, Robust PL substantially improves over Nominal PL for both Qwen3\-0\.6B and Qwen3\-8B\. This matches the PL structure: top\-rank errors corrupt the early stagewise choices that dominate the likelihood, whereas near\-tie corruption often preserves much of the global ordering\. Thus, in the offline fixed\-list setting, Robust PL behaves as a conservative ranking\-label regularizer: it largely preserves clean\-label ranking performance while improving robustness to structured listwise label noise\.

Appendix[I](https://arxiv.org/html/2607.01715#A9)provides further support: additional clean\-label metrics show limited performance degradation,ρ\\rho\-sweeps validate the robustness–over\-regularization tradeoff, and Qwen2\.5\-0\.5B/7B results show consistent trends across model families\.

Table 2:Main offline ranking results under synthetic ranking\-label corruption\. We report Kendall’sτ\\tauon clean held\-out UltraFeedback rankings; higher is better\. For each prompt, model scores induce a ranking over the four candidate responses, which is compared with the clean reference ranking\. The corruption levelϵ\\epsilonapplies only to training labels\.

### 6\.2Online policy\-induced alignment

We next evaluate the online setting, where the current policy generates candidate responses and a frozen reward model ranks them to provide the training signal\. This setting naturally introduces ranking\-label uncertainty: increasing the list size fromK=2K=2toK=4K=4provides richer preference information, but also requires the reward model to make finer\-grained comparisons over more candidates\. Thus, largerKKis not automatically beneficial\.

Our method is a robust listwise extension of the SAIL\-style online preference\-optimization pipeline\. We therefore use the setting\-matched binary SAIL baseline, recovered byK=2,ρ=0K=2,\\rho=0\. The non\-robust listwise extension isK=4,ρ=0K=4,\\rho=0, whileρ\>0\\rho\>0isolates the effect of the proposed robust ranking\-label correction\. We do not include PPO\-style online RLHF or offline robust\-DPO baselines as direct comparisons because they optimize different signals or robustify different objects\. For external evaluation, we follow the LLM\-as\-a\-judge protocol\(Zhenget al\.,[2023](https://arxiv.org/html/2607.01715#bib.bib11)\)and use GPT\-4\(OpenAIet al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib12)\)as the judge to compare model outputs against the dataset chosen responses\.

#### Online observations\.

[Table˜3](https://arxiv.org/html/2607.01715#S6.T3)shows that simply increasing the candidate\-list size is not sufficient\. The non\-robust listwise variantK=4,ρ=0K=4,\\rho=0does not consistently improve over the binary baselineK=2,ρ=0K=2,\\rho=0, suggesting that larger lists provide richer preference information but also introduce finer\-grained reward\-model ranking noise\. Robustness mitigates this issue\. Withρ\>0\\rho\>0, theK=4K=4variants become more reliable: for Qwen3\-0\.6B,K=4,ρ=0\.02K=4,\\rho=0\.02gives the best reward\-model preference and GPT\-4 judge scores; for Qwen3\-8B, robustK=4K=4variants achieve the strongest reward\-model performance and ranking\-agreement metrics\. These results support our main interpretation that Robust PL helps convert larger candidate lists from a noisier supervision source into useful listwise preference signal\. The GPT\-4 judge gains further suggest that the improvement transfers beyond the reward model used for training\.

Table 3:Online Qwen3 results on the U10 held\-out evaluation set\.K=2,ρ=0K=2,\\rho=0is the binary SAIL baseline;K=4,ρ=0K=4,\\rho=0is the non\-robust listwise extension;ρ\>0\\rho\>0gives the robust variant\.Δ\\DeltaReward is the average reward\-model gain over the SFT reference, and Rwd% vs SFT is the fraction of prompts where the reward model prefers the trained response to the SFT response\. GPT% vs Chosen and GPT\+Tie% compare the trained response with the dataset chosen response using GPT\-4 as an external judge\. Top\-1, Pairwise, and Kendall’sτ\\taumeasure agreement between the model\-induced ranking and the reward\-model ranking over candidate lists\. Higher is better for all metrics\.

## 7Conclusion

We introduced a listwise\-native notion of preference uncertainty: a pointwise total\-variation ambiguity set on the ranking label over a realized candidate list, combined with the Plackett–Luce listwise loss\. The resulting robust PL objective has three key guarantees\. First, the inner worst\-case ranking problem overK\!K\!permutations is exactly solved by ascending\-score sorting, givingO​\(K​log⁡K\)O\(K\\log K\)evaluation\. Second, in the offline fixed\-list log\-linear setting, the robust objective is convex and projected stochastic subgradient descent reaches globalϵ\\epsilon\-suboptimality withO​\(ϵ−2\)O\(\\epsilon^\{\-2\}\)sample complexity\. Third, in the online policy\-induced setting, the objective is weakly convex and admitsO~​\(ϵ−2\)\\widetilde\{O\}\(\\epsilon^\{\-2\}\)Moreau\-envelope stationarity\. Empirically, Robust PL largely preserves performance under clean labels, while improving robustness when the training rankings are corrupted, especially under severe top\-rank corruption\. In online alignment, it makesK=4K=4candidate expansion more reliable under reward\-model\-generated rankings and improves both reward\-model and external GPT\-4 judge metrics\.

#### Limitations\.

Empirically, the robustness radiusρ\\rhomust be tuned\. Future work should study adaptive choices ofρ\\rho, richer ambiguity sets, and larger\-scale online alignment experiments\.

#### Broader impact\.

Robust listwise alignment can reduce the influence of noisy or inconsistent rankings on LLM behavior\. The techniques studied are primarily methodological and analytical in nature\. We do not foresee any immediate negative societal impact arising from this work\.

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## Appendix AAdditional Related Work

A related line of work studies noisy or corrupted pairwise preference labels in offline DPO\-style training\. cDPO\[Mitchell,[2023](https://arxiv.org/html/2607.01715#bib.bib24)\]and rDPO\[Chowdhuryet al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib25)\]introduce correction mechanisms for binary preference flipping under pairwise BT/DPO supervision\. These methods are offline fixed\-pair approaches: they correct noisy chosen–rejected labels in a static preference dataset, rather than addressing the online policy\-induced setting where candidate lists are sampled from the current policy\. Their loss forms are also different from ours: cDPO/rDPO correct binary BT/DPO losses, whereas our pointwise\-TV formulation optimizes a worst\-case PL loss over full ranking labels\.

Another related line of work studies distributionally robust preference optimization under perturbations of the empirical data distribution, prompt distribution, or preference distribution\. For example,Wuet al\.\[[2025](https://arxiv.org/html/2607.01715#bib.bib8)\]formulate distributionally robust pairwise DPO under perturbations of the dataset distribution;Mandalet al\.\[[2025](https://arxiv.org/html/2607.01715#bib.bib5)\]study TV\-based ambiguity over the joint training distribution, including the prompt distribution, for pairwise DPO and policy optimization;Xuet al\.\[[2026](https://arxiv.org/html/2607.01715#bib.bib6)\]consider Wasserstein\- and KL\-based ambiguity around the empirical preference distribution; andLiet al\.\[[2026](https://arxiv.org/html/2607.01715#bib.bib7)\]analyze an online oracle\-robust alignment setting\.

In contrast, we study the listwise Plackett–Luce \(PL\) setting and robustify a different object: the conditional ranking\-label distribution given a candidate list\. This distinction is substantive\. WhenK≥3K\\geq 3, the resulting robust correction is intrinsically listwise: it involves a PL max\-gap over permutations and admits an efficient ascending\-sort solution, rather than reducing to a collection of independent pairwise BT corrections\. Our framework therefore complements prior robust pairwise DPO methods while extending robustness analysis from BT preferences to PL rankings, covering both the offline fixed\-list setting and the online listwise alignment setting\.

## Appendix BNotation and conventions

Throughout the appendix,∥⋅∥\\\|\\cdot\\\|denotes the Euclidean norm on the parameter space and∥⋅∥op\\\|\\cdot\\\|\_\{\\mathrm\{op\}\}denotes the spectral norm\. We writeSKS\_\{K\}for the symmetric group onKKsymbols and∂Cf\\partial\_\{C\}ffor the Clarke subdifferential offf\. For convex functions,∂Cf\\partial\_\{C\}freduces to the usual convex subdifferential\. For weakly convex nonsmooth functions, we use the Clarke subdifferential together with the standard Moreau\-envelope calculus for proper lower semicontinuous weakly convex objectives\. Throughout,ℓPL\\ell\_\{\\mathrm\{PL\}\}is defined in \([2](https://arxiv.org/html/2607.01715#S2.E2)\) andℓrob\\ell\_\{\\mathrm\{rob\}\}in \([4](https://arxiv.org/html/2607.01715#S3.E4)\)\.

#### Convention check \(β\\beta\)\.

In the online analysis we use the unscaled log\-ratio score

sθ​\(x,y\):=log⁡πθ​\(y∣x\)πref​\(y∣x\)\.s\_\{\\theta\}\(x,y\):=\\log\\frac\{\\pi\_\{\\theta\}\(y\\mid x\)\}\{\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\}\.Some DPO conventions instead use the scaled score

gθ​\(x,y\):=β​sθ​\(x,y\)\.g\_\{\\theta\}\(x,y\):=\\beta s\_\{\\theta\}\(x,y\)\.All results below are stated for the unscaled convention\. Under the scaled convention, the stagewise score gaps are multiplied byβ\\beta, gradients byβ\\beta, and Hessians byβ2\\beta^\{2\}\. Accordingly, the constants become

CL=K​\(log⁡K\+2​β​D​Bψ\),CG=2​β​K​Bψ,CH=β2​K​Bψ2,C\_\{L\}=K\(\\log K\+2\\beta DB\_\{\\psi\}\),\\qquad C\_\{G\}=2\\beta KB\_\{\\psi\},\\qquad C\_\{H\}=\\beta^\{2\}KB\_\{\\psi\}^\{2\},and

κ=K2​β2​Bψ2​\(8\+log⁡K\+2​β​D​Bψ\)\.\\kappa=K^\{2\}\\beta^\{2\}B\_\{\\psi\}^\{2\}\\bigl\(8\+\\log K\+2\\beta DB\_\{\\psi\}\\bigr\)\.

## Appendix CFull proof of the pointwise TV decomposition \([Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1)\)

###### Proof\.

Writeℓ​\(σ\):=ℓPL​\(θ;x,y1:K,σ\)\\ell\(\\sigma\):=\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)\. We solvemaxP∈Δ​\(SK\)​∑σP​\(σ\)​ℓ​\(σ\)\\max\_\{P\\in\\Delta\(S\_\{K\}\)\}\\sum\_\{\\sigma\}P\(\\sigma\)\\ell\(\\sigma\)subject toTV​\(P,δσ⋆\)≤ρ\\mathrm\{TV\}\(P,\\delta\_\{\\sigma^\{\\star\}\}\)\\leq\\rho\.

Expanding the TV distance and usingδσ⋆​\(σ⋆\)=1\\delta\_\{\\sigma^\{\\star\}\}\(\\sigma^\{\\star\}\)=1andδσ⋆​\(σ\)=0\\delta\_\{\\sigma^\{\\star\}\}\(\\sigma\)=0forσ≠σ⋆\\sigma\\neq\\sigma^\{\\star\},

TV​\(P,δσ⋆\)\\displaystyle\\mathrm\{TV\}\(P,\\delta\_\{\\sigma^\{\\star\}\}\)=12​\(\|P​\(σ⋆\)−1\|\+∑σ≠σ⋆P​\(σ\)\)\\displaystyle=\\tfrac\{1\}\{2\}\\big\(\|P\(\\sigma^\{\\star\}\)\-1\|\+\\sum\_\{\\sigma\\neq\\sigma^\{\\star\}\}P\(\\sigma\)\\big\)=12​\(\(1−P​\(σ⋆\)\)\+\(1−P​\(σ⋆\)\)\)=1−P​\(σ⋆\)\.\\displaystyle=\\tfrac\{1\}\{2\}\\big\(\(1\-P\(\\sigma^\{\\star\}\)\)\+\(1\-P\(\\sigma^\{\\star\}\)\)\\big\)=1\-P\(\\sigma^\{\\star\}\)\.HenceTV≤ρ⇔P​\(σ⋆\)≥1−ρ\\mathrm\{TV\}\\leq\\rho\\iff P\(\\sigma^\{\\star\}\)\\geq 1\-\\rho\. Letϵ:=1−P​\(σ⋆\)∈\[0,ρ\]\\epsilon:=1\-P\(\\sigma^\{\\star\}\)\\in\[0,\\rho\]\. Then∑σ≠σ⋆P​\(σ\)=ϵ\\sum\_\{\\sigma\\neq\\sigma^\{\\star\}\}P\(\\sigma\)=\\epsilonand

𝔼σ∼P​\[ℓ​\(σ\)\]=\(1−ϵ\)​ℓ​\(σ⋆\)\+∑σ≠σ⋆P​\(σ\)​ℓ​\(σ\)\.\\mathbb\{E\}\_\{\\sigma\\sim P\}\[\\ell\(\\sigma\)\]=\(1\-\\epsilon\)\\ell\(\\sigma^\{\\star\}\)\+\\sum\_\{\\sigma\\neq\\sigma^\{\\star\}\}P\(\\sigma\)\\ell\(\\sigma\)\.For fixedϵ\\epsilon, the adversary maximizes by allocating all massϵ\\epsilontoarg​maxσ≠σ⋆⁡ℓ​\(σ\)\\operatorname\*\{arg\\,max\}\_\{\\sigma\\neq\\sigma^\{\\star\}\}\\ell\(\\sigma\), giving

𝔼σ∼P​\[ℓ​\(σ\)\]≤\(1−ϵ\)​ℓ​\(σ⋆\)\+ϵ​maxσ∈SK⁡ℓ​\(σ\)=ℓ​\(σ⋆\)\+ϵ⋅\(maxσ⁡ℓ​\(σ\)−ℓ​\(σ⋆\)\)\.\\mathbb\{E\}\_\{\\sigma\\sim P\}\[\\ell\(\\sigma\)\]\\leq\(1\-\\epsilon\)\\ell\(\\sigma^\{\\star\}\)\+\\epsilon\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\(\\sigma\)=\\ell\(\\sigma^\{\\star\}\)\+\\epsilon\\cdot\\Big\(\\max\_\{\\sigma\}\\ell\(\\sigma\)\-\\ell\(\\sigma^\{\\star\}\)\\Big\)\.Sincemaxσ⁡ℓ​\(σ\)≥ℓ​\(σ⋆\)\\max\_\{\\sigma\}\\ell\(\\sigma\)\\geq\\ell\(\\sigma^\{\\star\}\)always \(asσ⋆∈SK\\sigma^\{\\star\}\\in S\_\{K\}\), the parenthetical is≥0\\geq 0, so the linear function ofϵ\\epsilonis monotone non\-decreasing and is maximized atϵ=ρ\\epsilon=\\rho\. Substituting yields \([5](https://arxiv.org/html/2607.01715#S3.E5)\)\. ∎

## Appendix DFull proof of the worst\-case sorting theorem \([Theorem˜1](https://arxiv.org/html/2607.01715#Thmtheorem1)\)

###### Proof\.

Writegi:=gθ​\(x,yi\)g\_\{i\}:=g\_\{\\theta\}\(x,y\_\{i\}\)\. From \([2](https://arxiv.org/html/2607.01715#S2.E2)\),

ℓPL​\(θ;x,y1:K,σ\)=−∑i=1Kgσi\+∑i=1Klog⁡\(∑j=iKegσj\)\.\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma\)=\-\\sum\_\{i=1\}^\{K\}g\_\{\\sigma\_\{i\}\}\+\\sum\_\{i=1\}^\{K\}\\log\\\!\\Big\(\\sum\_\{j=i\}^\{K\}e^\{g\_\{\\sigma\_\{j\}\}\}\\Big\)\.The first sum−∑igσi\-\\sum\_\{i\}g\_\{\\sigma\_\{i\}\}is permutation\-invariant; maximizingℓPL\\ell\_\{\\mathrm\{PL\}\}inσ\\sigmais equivalent to maximizingF​\(σ\)=∑i=1Klog​∑j=iKegσjF\(\\sigma\)=\\sum\_\{i=1\}^\{K\}\\log\\sum\_\{j=i\}^\{K\}e^\{g\_\{\\sigma\_\{j\}\}\}\.

Supposeσ\\sigmahas an adjacent inversion at positiontt:a:=gσt\>b:=gσt\+1a:=g\_\{\\sigma\_\{t\}\}\>b:=g\_\{\\sigma\_\{t\+1\}\}\. SetR:=∑j≥t\+2egσj≥0R:=\\sum\_\{j\\geq t\+2\}e^\{g\_\{\\sigma\_\{j\}\}\}\\geq 0\. Only thett\-th and\(t\+1\)\(t\+1\)\-st suffix terms can change under the swap of positionst,t\+1t,t\+1: all later suffixes contain the same multiset and are unchanged, all earlier suffixes contain\{a,b\}\\\{a,b\\\}jointly and are also unchanged\. Thett\-th term islog⁡\(ea\+eb\+R\)\\log\(e^\{a\}\+e^\{b\}\+R\)before the swap andlog⁡\(eb\+ea\+R\)\\log\(e^\{b\}\+e^\{a\}\+R\)after the swap, hence equal\. The\(t\+1\)\(t\+1\)\-st term changes fromlog⁡\(eb\+R\)\\log\(e^\{b\}\+R\)before the swap tolog⁡\(ea\+R\)\\log\(e^\{a\}\+R\)after the swap\. Sincea\>ba\>bandR≥0R\\geq 0,log⁡\(ea\+R\)\>log⁡\(eb\+R\)\\log\(e^\{a\}\+R\)\>\\log\(e^\{b\}\+R\), soFFstrictly increases\.

By repeated adjacent swaps that fix local inversions, every permutation can be transformed into the unique inversion\-free permutation, the nondecreasing\-score order\. At each stepFFstrictly increases \(modulo equal scores, where the swap leavesFFunchanged\)\. Hence the ascending\-score permutation attains the maximum\. With ties, any consistent within\-group ordering achieves the same maximum\. Computing such aσworst\\sigma\_\{\\mathrm\{worst\}\}requires only sortingKKscores, which costsO​\(K​log⁡K\)O\(K\\log K\)\. ∎

## Appendix EOffline theory: full proofs

### E\.1Convexity \([Proposition˜1](https://arxiv.org/html/2607.01715#Thmproposition1)\)

###### Proof\.

Under[Assumption˜1](https://arxiv.org/html/2607.01715#Thmassumption1), the induced PL score is affine inθ\\theta\. In particular, up to a prompt\-dependent additive term that cancels in the PL loss, we may write

sθ​\(x,y\)=\(θ−θref\)⊤​ψ​\(x,y\)\.s\_\{\\theta\}\(x,y\)=\(\\theta\-\\theta\_\{\\rm ref\}\)^\{\\top\}\\psi\(x,y\)\.For a fixed rankingσ∈SK\\sigma\\in S\_\{K\}, the PL loss is

ℓPL​\(θ;x,Y,σ\)=−∑i=1Ksθ​\(x,yσi\)\+∑i=1Klog​∑j=iKexp⁡\(sθ​\(x,yσj\)\)\.\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)=\-\\sum\_\{i=1\}^\{K\}s\_\{\\theta\}\(x,y\_\{\\sigma\_\{i\}\}\)\+\\sum\_\{i=1\}^\{K\}\\log\\sum\_\{j=i\}^\{K\}\\exp\\big\(s\_\{\\theta\}\(x,y\_\{\\sigma\_\{j\}\}\)\\big\)\.The first term is affine inθ\\theta\. Each term in the second sum is a log\-sum\-exp of affine functions ofθ\\theta, and is therefore convex\. Hence

θ↦ℓPL​\(θ;x,Y,σ\)\\theta\\mapsto\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)is convex for every fixed\(x,Y,σ\)\(x,Y,\\sigma\)\.

By[Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1), the robust loss admits the decomposition

ℓrob​\(θ;x,Y,σ⋆\)=\(1−ρ\)​ℓPL​\(θ;x,Y,σ⋆\)\+ρ​maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\.\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\.The first term is convex, and the second term is a pointwise maximum of convex functions, hence convex\. Sinceρ∈\[0,1\]\\rho\\in\[0,1\],ℓrob​\(⋅;x,Y,σ⋆\)\\ell\_\{\\mathrm\{rob\}\}\(\\cdot;x,Y,\\sigma^\{\\star\}\)is convex\. Finally, the offline objective

Jrob​\(θ\)=𝔼ξ​\[ℓrob​\(θ;ξ\)\]J\_\{\\mathrm\{rob\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\xi\}\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;\\xi\)\]is an expectation of convex functions and therefore convex onΘ\\Theta\. ∎

### E\.2Subgradient bound \(used in[Theorem˜2](https://arxiv.org/html/2607.01715#Thmtheorem2)\)

###### Lemma E\.1\(Bounded sample subgradient\)\.

Suppose[Assumption˜1](https://arxiv.org/html/2607.01715#Thmassumption1)holds\. Then for any sampleξ=\(x,Y,σ⋆\)\\xi=\(x,Y,\\sigma^\{\\star\}\), anyθ∈Θ\\theta\\in\\Theta, and any

g∈∂θℓrob​\(θ;ξ\),g\\in\\partial\_\{\\theta\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;\\xi\),we have

‖g‖2≤2​K​Bψ\.\\\|g\\\|\_\{2\}\\leq 2KB\_\{\\psi\}\.

###### Proof\.

Fix\(x,Y,σ\)\(x,Y,\\sigma\), whereY=\(y1,…,yK\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\)\. Write

si:=sθ​\(x,yσi\),ψi:=ψ​\(x,yσi\)\.s\_\{i\}:=s\_\{\\theta\}\(x,y\_\{\\sigma\_\{i\}\}\),\\qquad\\psi\_\{i\}:=\\psi\(x,y\_\{\\sigma\_\{i\}\}\)\.The PL loss decomposes into stagewise terms:

ℓPL​\(θ;x,Y,σ\)=∑i=1K\[−si\+log​∑j=iKexp⁡\(sj\)\]\.\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)=\\sum\_\{i=1\}^\{K\}\\left\[\-s\_\{i\}\+\\log\\sum\_\{j=i\}^\{K\}\\exp\(s\_\{j\}\)\\right\]\.For each stageii, define the softmax weights

pj∣i​\(θ\):=exp⁡\(sj\)∑m=iKexp⁡\(sm\),j=i,…,K\.p\_\{j\\mid i\}\(\\theta\):=\\frac\{\\exp\(s\_\{j\}\)\}\{\\sum\_\{m=i\}^\{K\}\\exp\(s\_\{m\}\)\},\\qquad j=i,\\ldots,K\.Differentiating theii\-th stage gives

∇θℓi​\(θ\)=−ψi\+∑j=iKpj∣i​\(θ\)​ψj\.\\nabla\_\{\\theta\}\\ell\_\{i\}\(\\theta\)=\-\\psi\_\{i\}\+\\sum\_\{j=i\}^\{K\}p\_\{j\\mid i\}\(\\theta\)\\psi\_\{j\}\.Since the weightspj∣ip\_\{j\\mid i\}form a probability distribution and‖ψ​\(x,y\)‖≤Bψ\\\|\\psi\(x,y\)\\\|\\leq B\_\{\\psi\}, we have

‖∑j=iKpj∣i​\(θ\)​ψj‖≤∑j=iKpj∣i​\(θ\)​‖ψj‖≤Bψ\.\\left\\\|\\sum\_\{j=i\}^\{K\}p\_\{j\\mid i\}\(\\theta\)\\psi\_\{j\}\\right\\\|\\leq\\sum\_\{j=i\}^\{K\}p\_\{j\\mid i\}\(\\theta\)\\\|\\psi\_\{j\}\\\|\\leq B\_\{\\psi\}\.Therefore

‖∇θℓi​\(θ\)‖≤‖ψi‖\+‖∑j=iKpj∣i​\(θ\)​ψj‖≤2​Bψ\.\\\|\\nabla\_\{\\theta\}\\ell\_\{i\}\(\\theta\)\\\|\\leq\\\|\\psi\_\{i\}\\\|\+\\left\\\|\\sum\_\{j=i\}^\{K\}p\_\{j\\mid i\}\(\\theta\)\\psi\_\{j\}\\right\\\|\\leq 2B\_\{\\psi\}\.Summing overi=1,…,Ki=1,\\ldots,K, we obtain

‖∇θℓPL​\(θ;x,Y,σ\)‖≤∑i=1K‖∇θℓi​\(θ\)‖≤2​K​Bψ\.\\\|\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\\|\\leq\\sum\_\{i=1\}^\{K\}\\\|\\nabla\_\{\\theta\}\\ell\_\{i\}\(\\theta\)\\\|\\leq 2KB\_\{\\psi\}\.
Now consider the robust loss\. By[Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1),

ℓrob=\(1−ρ\)​ℓPL​\(σ⋆\)\+ρ​maxσ∈SK⁡ℓPL​\(σ\)\.\\ell\_\{\\mathrm\{rob\}\}=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\sigma^\{\\star\}\)\+\\rho\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\sigma\)\.The subdifferential of the finite maximum is the convex hull of active PL gradients\. Hence everyg∈∂θℓrob​\(θ;ξ\)g\\in\\partial\_\{\\theta\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;\\xi\)can be written as a convex combination of

∇θℓPL​\(θ;x,Y,σ⋆\)\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)and active gradients

∇θℓPL​\(θ;x,Y,σ\),σ∈arg​maxσ′∈SK⁡ℓPL​\(θ;x,Y,σ′\)\.\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\),\\qquad\\sigma\\in\\operatorname\*\{arg\\,max\}\_\{\\sigma^\{\\prime\}\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\prime\}\)\.Each such PL gradient has norm at most2​K​Bψ2KB\_\{\\psi\}\. Since convex combinations preserve the same norm bound, we conclude that

‖g‖2≤2​K​Bψ\.\\\|g\\\|\_\{2\}\\leq 2KB\_\{\\psi\}\.∎

### E\.3Subgradient rate \([Theorem˜2](https://arxiv.org/html/2607.01715#Thmtheorem2)\)

###### Proof\.

SinceΘ\\Thetais compact andJrobJ\_\{\\mathrm\{rob\}\}is convex and continuous, a minimizer exists\. Let

θ⋆∈arg​minθ∈Θ⁡Jrob​\(θ\)\.\\theta^\{\\star\}\\in\\operatorname\*\{arg\\,min\}\_\{\\theta\\in\\Theta\}J\_\{\\mathrm\{rob\}\}\(\\theta\)\.The projected stochastic subgradient update is

θt\+1=ΠΘ​\(θt−η​g^t\)\.\\theta\_\{t\+1\}=\\Pi\_\{\\Theta\}\(\\theta\_\{t\}\-\\eta\\widehat\{g\}\_\{t\}\)\.By non\-expansiveness of the Euclidean projection onto the closed convex setΘ\\Theta,

‖θt\+1−θ⋆‖2≤‖θt−η​g^t−θ⋆‖2\.\\\|\\theta\_\{t\+1\}\-\\theta^\{\\star\}\\\|^\{2\}\\leq\\\|\\theta\_\{t\}\-\\eta\\widehat\{g\}\_\{t\}\-\\theta^\{\\star\}\\\|^\{2\}\.Expanding the right\-hand side gives

‖θt\+1−θ⋆‖2≤‖θt−θ⋆‖2−2​η​⟨g^t,θt−θ⋆⟩\+η2​‖g^t‖2\.\\\|\\theta\_\{t\+1\}\-\\theta^\{\\star\}\\\|^\{2\}\\leq\\\|\\theta\_\{t\}\-\\theta^\{\\star\}\\\|^\{2\}\-2\\eta\\langle\\widehat\{g\}\_\{t\},\\theta\_\{t\}\-\\theta^\{\\star\}\\rangle\+\\eta^\{2\}\\\|\\widehat\{g\}\_\{t\}\\\|^\{2\}\.Taking conditional expectation givenθt\\theta\_\{t\}, and writing

ht:=𝔼​\[g^t∣θt\]∈∂Jrob​\(θt\),h\_\{t\}:=\\mathbb\{E\}\[\\widehat\{g\}\_\{t\}\\mid\\theta\_\{t\}\]\\in\\partial J\_\{\\mathrm\{rob\}\}\(\\theta\_\{t\}\),we obtain

𝔼​\[‖θt\+1−θ⋆‖2∣θt\]≤‖θt−θ⋆‖2−2​η​⟨ht,θt−θ⋆⟩\+η2​𝔼​\[‖g^t‖2∣θt\]\.\\mathbb\{E\}\[\\\|\\theta\_\{t\+1\}\-\\theta^\{\\star\}\\\|^\{2\}\\mid\\theta\_\{t\}\]\\leq\\\|\\theta\_\{t\}\-\\theta^\{\\star\}\\\|^\{2\}\-2\\eta\\langle h\_\{t\},\\theta\_\{t\}\-\\theta^\{\\star\}\\rangle\+\\eta^\{2\}\\mathbb\{E\}\[\\\|\\widehat\{g\}\_\{t\}\\\|^\{2\}\\mid\\theta\_\{t\}\]\.By convexity ofJrobJ\_\{\\mathrm\{rob\}\},

Jrob​\(θt\)−Jrob​\(θ⋆\)≤⟨ht,θt−θ⋆⟩\.J\_\{\\mathrm\{rob\}\}\(\\theta\_\{t\}\)\-J\_\{\\mathrm\{rob\}\}\(\\theta^\{\\star\}\)\\leq\\langle h\_\{t\},\\theta\_\{t\}\-\\theta^\{\\star\}\\rangle\.Using the assumed mini\-batch second\-moment bound

𝔼​\[‖g^t‖2∣θt\]≤4​K2​Bψ2\+σg2Bs,\\mathbb\{E\}\[\\\|\\widehat\{g\}\_\{t\}\\\|^\{2\}\\mid\\theta\_\{t\}\]\\leq 4K^\{2\}B\_\{\\psi\}^\{2\}\+\\frac\{\\sigma\_\{g\}^\{2\}\}\{B\_\{s\}\},we get

2​η​\(Jrob​\(θt\)−Jrob​\(θ⋆\)\)≤‖θt−θ⋆‖2−𝔼​\[‖θt\+1−θ⋆‖2∣θt\]\+η2​\(4​K2​Bψ2\+σg2Bs\)\.2\\eta\\big\(J\_\{\\mathrm\{rob\}\}\(\\theta\_\{t\}\)\-J\_\{\\mathrm\{rob\}\}\(\\theta^\{\\star\}\)\\big\)\\leq\\\|\\theta\_\{t\}\-\\theta^\{\\star\}\\\|^\{2\}\-\\mathbb\{E\}\[\\\|\\theta\_\{t\+1\}\-\\theta^\{\\star\}\\\|^\{2\}\\mid\\theta\_\{t\}\]\+\\eta^\{2\}\\left\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\frac\{\\sigma\_\{g\}^\{2\}\}\{B\_\{s\}\}\\right\)\.Taking total expectation and summing overt=0,…,T−1t=0,\\ldots,T\-1, we obtain

2​η​∑t=0T−1𝔼​\[Jrob​\(θt\)−Jrob​\(θ⋆\)\]≤‖θ0−θ⋆‖2\+T​η2​\(4​K2​Bψ2\+σg2Bs\)\.2\\eta\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\theta\_\{t\}\)\-J\_\{\\mathrm\{rob\}\}\(\\theta^\{\\star\}\)\]\\leq\\\|\\theta\_\{0\}\-\\theta^\{\\star\}\\\|^\{2\}\+T\\eta^\{2\}\\left\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\frac\{\\sigma\_\{g\}^\{2\}\}\{B\_\{s\}\}\\right\)\.SinceΘ=\{θ:‖θ‖≤B\}\\Theta=\\\{\\theta:\\\|\\theta\\\|\\leq B\\\}, bothθ0\\theta\_\{0\}andθ⋆\\theta^\{\\star\}belong toΘ\\Theta, and hence

‖θ0−θ⋆‖≤2​B\.\\\|\\theta\_\{0\}\-\\theta^\{\\star\}\\\|\\leq 2B\.Therefore

1T​∑t=0T−1𝔼​\[Jrob​\(θt\)−Jrob​\(θ⋆\)\]≤2​B2η​T\+η2​\(4​K2​Bψ2\+σg2Bs\)\.\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\theta\_\{t\}\)\-J\_\{\\mathrm\{rob\}\}\(\\theta^\{\\star\}\)\]\\leq\\frac\{2B^\{2\}\}\{\\eta T\}\+\\frac\{\\eta\}\{2\}\\left\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\frac\{\\sigma\_\{g\}^\{2\}\}\{B\_\{s\}\}\\right\)\.By convexity ofJrobJ\_\{\\mathrm\{rob\}\}and Jensen’s inequality, for

θ¯T:=1T​∑t=0T−1θt,\\bar\{\\theta\}\_\{T\}:=\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\theta\_\{t\},we have

𝔼​\[Jrob​\(θ¯T\)\]≤1T​∑t=0T−1𝔼​\[Jrob​\(θt\)\]\.\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\bar\{\\theta\}\_\{T\}\)\]\\leq\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\theta\_\{t\}\)\]\.Hence

𝔼​\[Jrob​\(θ¯T\)\]−Jrob​\(θ⋆\)≤2​B2η​T\+η2​\(4​K2​Bψ2\+σg2Bs\)\.\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\bar\{\\theta\}\_\{T\}\)\]\-J\_\{\\mathrm\{rob\}\}\(\\theta^\{\\star\}\)\\leq\\frac\{2B^\{2\}\}\{\\eta T\}\+\\frac\{\\eta\}\{2\}\\left\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\frac\{\\sigma\_\{g\}^\{2\}\}\{B\_\{s\}\}\\right\)\.Choosing

η=2​BT​\(4​K2​Bψ2\+σg2/Bs\)\\eta=\\frac\{2B\}\{\\sqrt\{T\\left\(4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}\\right\)\}\}balances the two terms and gives

𝔼​\[Jrob​\(θ¯T\)\]−minθ∈Θ⁡Jrob​\(θ\)≤2​B​4​K2​Bψ2\+σg2/BsT\.\\mathbb\{E\}\[J\_\{\\mathrm\{rob\}\}\(\\bar\{\\theta\}\_\{T\}\)\]\-\\min\_\{\\theta\\in\\Theta\}J\_\{\\mathrm\{rob\}\}\(\\theta\)\\leq\\frac\{2B\\sqrt\{4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}\}\}\{\\sqrt\{T\}\}\.Solving forTTto achieveε\\varepsilon\-suboptimality gives

T=O​\(4​K2​Bψ2\+σg2/Bsε2\)\.T=O\\left\(\\frac\{4K^\{2\}B\_\{\\psi\}^\{2\}\+\\sigma\_\{g\}^\{2\}/B\_\{s\}\}\{\\varepsilon^\{2\}\}\\right\)\.With fixedBs=Θ​\(1\)B\_\{s\}=\\Theta\(1\)independent ofε\\varepsilon, the total sample complexity satisfies

Bs​T=O​\(ε−2\)\.B\_\{s\}T=O\(\\varepsilon^\{\-2\}\)\.∎

## Appendix FOnline theory: full proofs

The proof strategy in this section follows the weakly\-convex stochastic\-subgradient framework\[Liet al\.,[2026](https://arxiv.org/html/2607.01715#bib.bib7), Clarke,[1990](https://arxiv.org/html/2607.01715#bib.bib22)\]\. However, our listwise setting introduces an additional nonsmooth finite\-max structure: the robust correction involvesmaxσ∈𝔖K⁡ℓPL​\(θ;x,Y,σ\)\\max\_\{\\sigma\\in\\mathfrak\{S\}\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\. Consequently, beyond the standard score\-function correction for policy\-induced sampling, we must explicitly control the Clarke subdifferential of this finite maximum and fix a measurable tie\-breaking rule for nonunique worst\-case rankings\.

We use the notation of[Assumption˜2](https://arxiv.org/html/2607.01715#Thmassumption2):ψ\\psihas‖ψ‖≤Bψ\\\|\\psi\\\|\\leq B\_\{\\psi\},Θ\\Thetais closed, convex, bounded with diameterDD,πθ∝exp⁡\(θ⊤​ψ\)\\pi\_\{\\theta\}\\propto\\exp\(\\theta^\{\\top\}\\psi\)\. Recallsθ​\(x,y\):=log⁡\[πθ/πref\]=\(θ−θref\)⊤​ψ​\(x,y\)\+consts\_\{\\theta\}\(x,y\):=\\log\[\\pi\_\{\\theta\}/\\pi\_\{\\mathrm\{ref\}\}\]=\(\\theta\-\\theta\_\{\\mathrm\{ref\}\}\)^\{\\top\}\\psi\(x,y\)\+\\mathrm\{const\}andSθ​\(x,Y\):=∇θlog⁡Pπθ​\(Y∣x\)S\_\{\\theta\}\(x,Y\):=\\nabla\_\{\\theta\}\\log P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)\.

### F\.1Score\-function bounds and PL bounds

###### Lemma F\.2\(Score\-function bounds\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold,

∇θlog⁡πθ​\(y∣x\)=ψ​\(x,y\)−ψ¯θ​\(x\),ψ¯θ​\(x\):=𝔼y′∼πθ\(⋅∣x\)​\[ψ​\(x,y′\)\]\.\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\\mid x\)=\\psi\(x,y\)\-\\bar\{\\psi\}\_\{\\theta\}\(x\),\\qquad\\bar\{\\psi\}\_\{\\theta\}\(x\):=\\mathbb\{E\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]\.Consequently,

∥∇θlogπθ\(y∣x\)∥≤2Bψ,∥Sθ\(x,Y\)∥≤2KBψ,∥∇θSθ\(x,Y\)∥op≤KBψ2\.\\\|\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\\mid x\)\\\|\\leq 2B\_\{\\psi\},\\qquad\\\|S\_\{\\theta\}\(x,Y\)\\\|\\leq 2KB\_\{\\psi\},\\qquad\\\|\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\\\|\_\{\\mathrm\{op\}\}\\leq KB\_\{\\psi\}^\{2\}\.

###### Proof\.

Fixxxand write

Aθ​\(x\):=log​∑y′∈𝒴exp⁡\(θ⊤​ψ​\(x,y′\)\)\.A\_\{\\theta\}\(x\):=\\log\\sum\_\{y^\{\\prime\}\\in\\mathcal\{Y\}\}\\exp\(\\theta^\{\\top\}\\psi\(x,y^\{\\prime\}\)\)\.Under the log\-linear policy class,

log⁡πθ​\(y∣x\)=θ⊤​ψ​\(x,y\)−Aθ​\(x\)\.\\log\\pi\_\{\\theta\}\(y\\mid x\)=\\theta^\{\\top\}\\psi\(x,y\)\-A\_\{\\theta\}\(x\)\.Differentiating gives

∇θlog⁡πθ​\(y∣x\)=ψ​\(x,y\)−∇θAθ​\(x\)\.\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\\mid x\)=\\psi\(x,y\)\-\\nabla\_\{\\theta\}A\_\{\\theta\}\(x\)\.Moreover,

∇θAθ​\(x\)=∑y′∈𝒴exp⁡\(θ⊤​ψ​\(x,y′\)\)​ψ​\(x,y′\)∑y′∈𝒴exp⁡\(θ⊤​ψ​\(x,y′\)\)=𝔼y′∼πθ\(⋅∣x\)​\[ψ​\(x,y′\)\]=ψ¯θ​\(x\)\.\\nabla\_\{\\theta\}A\_\{\\theta\}\(x\)=\\frac\{\\sum\_\{y^\{\\prime\}\\in\\mathcal\{Y\}\}\\exp\(\\theta^\{\\top\}\\psi\(x,y^\{\\prime\}\)\)\\psi\(x,y^\{\\prime\}\)\}\{\\sum\_\{y^\{\\prime\}\\in\\mathcal\{Y\}\}\\exp\(\\theta^\{\\top\}\\psi\(x,y^\{\\prime\}\)\)\}=\\mathbb\{E\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]=\\bar\{\\psi\}\_\{\\theta\}\(x\)\.Therefore,

∇θlog⁡πθ​\(y∣x\)=ψ​\(x,y\)−ψ¯θ​\(x\)\.\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\\mid x\)=\\psi\(x,y\)\-\\bar\{\\psi\}\_\{\\theta\}\(x\)\.
Since‖ψ​\(x,y\)‖≤Bψ\\\|\\psi\(x,y\)\\\|\\leq B\_\{\\psi\}for every\(x,y\)\(x,y\), Jensen’s inequality gives

‖ψ¯θ​\(x\)‖=‖𝔼y′∼πθ\(⋅∣x\)​\[ψ​\(x,y′\)\]‖≤𝔼y′∼πθ\(⋅∣x\)​\[‖ψ​\(x,y′\)‖\]≤Bψ\.\\\|\\bar\{\\psi\}\_\{\\theta\}\(x\)\\\|=\\left\\\|\\mathbb\{E\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]\\right\\\|\\leq\\mathbb\{E\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\\|\\psi\(x,y^\{\\prime\}\)\\\|\]\\leq B\_\{\\psi\}\.Hence

∥∇θlogπθ\(y∣x\)∥≤∥ψ\(x,y\)∥\+∥ψ¯θ\(x\)∥≤2Bψ\.\\\|\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\\mid x\)\\\|\\leq\\\|\\psi\(x,y\)\\\|\+\\\|\\bar\{\\psi\}\_\{\\theta\}\(x\)\\\|\\leq 2B\_\{\\psi\}\.
Now letY=\(y1,…,yK\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\)\. Since the list is sampled iid fromπθ\(⋅∣x\)\\pi\_\{\\theta\}\(\\cdot\\mid x\),

Pπθ​\(Y∣x\)=∏i=1Kπθ​\(yi∣x\),P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)=\\prod\_\{i=1\}^\{K\}\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\),and therefore

Sθ​\(x,Y\):=∇θlog⁡Pπθ​\(Y∣x\)=∑i=1K∇θlog⁡πθ​\(yi∣x\)\.S\_\{\\theta\}\(x,Y\):=\\nabla\_\{\\theta\}\\log P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)=\\sum\_\{i=1\}^\{K\}\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)\.Using the bound above,

∥Sθ\(x,Y\)∥≤∑i=1K∥∇θlogπθ\(yi∣x\)∥≤2KBψ\.\\\|S\_\{\\theta\}\(x,Y\)\\\|\\leq\\sum\_\{i=1\}^\{K\}\\\|\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)\\\|\\leq 2KB\_\{\\psi\}\.
It remains to bound∇θSθ​\(x,Y\)\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\. Since

∇θlog⁡πθ​\(y∣x\)=ψ​\(x,y\)−ψ¯θ​\(x\),\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\\mid x\)=\\psi\(x,y\)\-\\bar\{\\psi\}\_\{\\theta\}\(x\),andψ​\(x,y\)\\psi\(x,y\)is independent ofθ\\theta,

∇θ2log⁡πθ​\(y∣x\)=−∇θψ¯θ​\(x\)\.\\nabla\_\{\\theta\}^\{2\}\\log\\pi\_\{\\theta\}\(y\\mid x\)=\-\\nabla\_\{\\theta\}\\bar\{\\psi\}\_\{\\theta\}\(x\)\.We compute∇θψ¯θ​\(x\)\\nabla\_\{\\theta\}\\bar\{\\psi\}\_\{\\theta\}\(x\)\. For any vectorv∈ℝdv\\in\\mathbb\{R\}^\{d\},

∇θ𝔼y′∼πθ\(⋅∣x\)​\[ψ​\(x,y′\)\]=𝔼y′∼πθ\(⋅∣x\)​\[\(ψ​\(x,y′\)−ψ¯θ​\(x\)\)​ψ​\(x,y′\)⊤\],\\nabla\_\{\\theta\}\\mathbb\{E\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]=\\mathbb\{E\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\(\\psi\(x,y^\{\\prime\}\)\-\\bar\{\\psi\}\_\{\\theta\}\(x\)\)\\psi\(x,y^\{\\prime\}\)^\{\\top\}\],which is the covariance matrix

Covy′∼πθ\(⋅∣x\)⁡\[ψ​\(x,y′\)\]=𝔼​\[\(ψ−ψ¯θ\)​\(ψ−ψ¯θ\)⊤\]\.\\operatorname\{Cov\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]=\\mathbb\{E\}\[\(\\psi\-\\bar\{\\psi\}\_\{\\theta\}\)\(\\psi\-\\bar\{\\psi\}\_\{\\theta\}\)^\{\\top\}\]\.Thus

∇θ2log⁡πθ​\(y∣x\)=−Covy′∼πθ\(⋅∣x\)⁡\[ψ​\(x,y′\)\]\.\\nabla\_\{\\theta\}^\{2\}\\log\\pi\_\{\\theta\}\(y\\mid x\)=\-\\operatorname\{Cov\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]\.Consequently,

∇θSθ​\(x,Y\)=∑i=1K∇θ2log⁡πθ​\(yi∣x\)=−K​Covy′∼πθ\(⋅∣x\)⁡\[ψ​\(x,y′\)\],\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)=\\sum\_\{i=1\}^\{K\}\\nabla\_\{\\theta\}^\{2\}\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)=\-K\\,\\operatorname\{Cov\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\],because the covariance term depends onxxandθ\\theta, but not on the particular sampled responseyiy\_\{i\}\.

Finally, for any unit vectorvv,

v⊤​Covy′∼πθ\(⋅∣x\)⁡\[ψ​\(x,y′\)\]​v=Vary′∼πθ\(⋅∣x\)⁡\(v⊤​ψ​\(x,y′\)\)≤𝔼​\[\(v⊤​ψ​\(x,y′\)\)2\]≤Bψ2\.v^\{\\top\}\\operatorname\{Cov\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]v=\\operatorname\{Var\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\(v^\{\\top\}\\psi\(x,y^\{\\prime\}\)\)\\leq\\mathbb\{E\}\[\(v^\{\\top\}\\psi\(x,y^\{\\prime\}\)\)^\{2\}\]\\leq B\_\{\\psi\}^\{2\}\.Therefore,

‖Covy′∼πθ\(⋅∣x\)⁡\[ψ​\(x,y′\)\]‖op≤Bψ2\.\\left\\\|\\operatorname\{Cov\}\_\{y^\{\\prime\}\\sim\\pi\_\{\\theta\}\(\\cdot\\mid x\)\}\[\\psi\(x,y^\{\\prime\}\)\]\\right\\\|\_\{\\mathrm\{op\}\}\\leq B\_\{\\psi\}^\{2\}\.Hence

‖∇θSθ​\(x,Y\)‖op≤K​Bψ2\.\\\|\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\\\|\_\{\\mathrm\{op\}\}\\leq KB\_\{\\psi\}^\{2\}\.∎

###### Lemma F\.3\(Convexity, magnitude, curvature ofℓPL\\ell\_\{\\mathrm\{PL\}\}\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold, for every fixed\(x,Y,σ\)\(x,Y,\\sigma\), the mapθ↦ℓPL​\(θ;x,Y,σ\)\\theta\\mapsto\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)is convex and hence Clarke regular\. Moreover, onΘ\\Theta,

0≤ℓPL​\(θ;x,Y,σ\)≤CL,‖∇θℓPL​\(θ;x,Y,σ\)‖≤CG,0⪯∇θ2ℓPL​\(θ;x,Y,σ\)⪯CH​I,0\\leq\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\leq C\_\{L\},\\qquad\\\|\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\\|\\leq C\_\{G\},\\qquad 0\\preceq\\nabla\_\{\\theta\}^\{2\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\preceq C\_\{H\}I,where

CL=K​\(log⁡K\+2​D​Bψ\),CG=2​K​Bψ,CH=K​Bψ2\.C\_\{L\}=K\(\\log K\+2DB\_\{\\psi\}\),\\qquad C\_\{G\}=2KB\_\{\\psi\},\\qquad C\_\{H\}=KB\_\{\\psi\}^\{2\}\.

###### Proof\.

Fix\(x,Y,σ\)\(x,Y,\\sigma\), whereY=\(y1,…,yK\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\)\. Recall that the induced affine score can be written as

sθ​\(x,y\)=\(θ−θref\)⊤​ψ​\(x,y\),s\_\{\\theta\}\(x,y\)=\(\\theta\-\\theta\_\{\\rm ref\}\)^\{\\top\}\\psi\(x,y\),up to an additive term depending only onxx, which cancels in the PL loss\. For notational simplicity, write

si:=sθ​\(x,yσi\),ψi:=ψ​\(x,yσi\)\.s\_\{i\}:=s\_\{\\theta\}\(x,y\_\{\\sigma\_\{i\}\}\),\\qquad\\psi\_\{i\}:=\\psi\(x,y\_\{\\sigma\_\{i\}\}\)\.The PL loss decomposes intoKKstagewise losses:

ℓPL​\(θ;x,Y,σ\)=∑i=1Kℓi​\(θ\),\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)=\\sum\_\{i=1\}^\{K\}\\ell\_\{i\}\(\\theta\),where

ℓi​\(θ\):=−si\+log​∑j=iKexp⁡\(sj\)\.\\ell\_\{i\}\(\\theta\):=\-s\_\{i\}\+\\log\\sum\_\{j=i\}^\{K\}\\exp\(s\_\{j\}\)\.

#### Convexity\.

For eachii, the term−si\-s\_\{i\}is affine inθ\\theta, and

θ↦log​∑j=iKexp⁡\(sθ​\(x,yσj\)\)\\theta\\mapsto\\log\\sum\_\{j=i\}^\{K\}\\exp\(s\_\{\\theta\}\(x,y\_\{\\sigma\_\{j\}\}\)\)is a log\-sum\-exp of affine functions, hence convex\. Therefore eachℓi\\ell\_\{i\}is convex, and soℓPL=∑i=1Kℓi\\ell\_\{\\mathrm\{PL\}\}=\\sum\_\{i=1\}^\{K\}\\ell\_\{i\}is convex\. Since it is finite\-valued and convex, it is Clarke regular\.

#### Magnitude bound\.

For each stageii,

ℓi​\(θ\)=log​∑j=iKexp⁡\(sj−si\)\.\\ell\_\{i\}\(\\theta\)=\\log\\sum\_\{j=i\}^\{K\}\\exp\(s\_\{j\}\-s\_\{i\}\)\.Sinceθ∈Θ\\theta\\in\\Thetaand

D:=supθ∈Θ‖θ−θref‖,D:=\\sup\_\{\\theta\\in\\Theta\}\\\|\\theta\-\\theta\_\{\\rm ref\}\\\|,we have, for anyjj,

\|sj−si\|=\|\(θ−θref\)⊤​\(ψj−ψi\)\|≤‖θ−θref‖​‖ψj−ψi‖≤2​D​Bψ\.\|s\_\{j\}\-s\_\{i\}\|=\|\(\\theta\-\\theta\_\{\\rm ref\}\)^\{\\top\}\(\\psi\_\{j\}\-\\psi\_\{i\}\)\|\\leq\\\|\\theta\-\\theta\_\{\\rm ref\}\\\|\\,\\\|\\psi\_\{j\}\-\\psi\_\{i\}\\\|\\leq 2DB\_\{\\psi\}\.Also, the termj=ij=iequalsexp⁡\(si−si\)=1\\exp\(s\_\{i\}\-s\_\{i\}\)=1, so

ℓi​\(θ\)=log​∑j=iKexp⁡\(sj−si\)≥0\.\\ell\_\{i\}\(\\theta\)=\\log\\sum\_\{j=i\}^\{K\}\\exp\(s\_\{j\}\-s\_\{i\}\)\\geq 0\.For the upper bound,

ℓi​\(θ\)≤log​∑j=iKexp⁡\(2​D​Bψ\)≤log⁡K\+2​D​Bψ\.\\ell\_\{i\}\(\\theta\)\\leq\\log\\sum\_\{j=i\}^\{K\}\\exp\(2DB\_\{\\psi\}\)\\leq\\log K\+2DB\_\{\\psi\}\.Summing overi=1,…,Ki=1,\\ldots,Kgives

0≤ℓPL​\(θ;x,Y,σ\)≤K​\(log⁡K\+2​D​Bψ\)=CL\.0\\leq\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\leq K\(\\log K\+2DB\_\{\\psi\}\)=C\_\{L\}\.

#### Gradient bound\.

For each stageii, define the stagewise softmax weights

pj\|i​\(θ\):=exp⁡\(sj\)∑m=iKexp⁡\(sm\),j=i,…,K\.p\_\{j\|i\}\(\\theta\):=\\frac\{\\exp\(s\_\{j\}\)\}\{\\sum\_\{m=i\}^\{K\}\\exp\(s\_\{m\}\)\},\\qquad j=i,\\ldots,K\.Then

∇θℓi​\(θ\)=−ψi\+∑j=iKpj\|i​\(θ\)​ψj\.\\nabla\_\{\\theta\}\\ell\_\{i\}\(\\theta\)=\-\\psi\_\{i\}\+\\sum\_\{j=i\}^\{K\}p\_\{j\|i\}\(\\theta\)\\psi\_\{j\}\.Since the weightspj\|ip\_\{j\|i\}form a probability distribution over\{i,…,K\}\\\{i,\\ldots,K\\\},

‖∑j=iKpj\|i​\(θ\)​ψj‖≤∑j=iKpj\|i​\(θ\)​‖ψj‖≤Bψ\.\\left\\\|\\sum\_\{j=i\}^\{K\}p\_\{j\|i\}\(\\theta\)\\psi\_\{j\}\\right\\\|\\leq\\sum\_\{j=i\}^\{K\}p\_\{j\|i\}\(\\theta\)\\\|\\psi\_\{j\}\\\|\\leq B\_\{\\psi\}\.Thus

‖∇θℓi​\(θ\)‖≤‖ψi‖\+‖∑j=iKpj\|i​\(θ\)​ψj‖≤2​Bψ\.\\\|\\nabla\_\{\\theta\}\\ell\_\{i\}\(\\theta\)\\\|\\leq\\\|\\psi\_\{i\}\\\|\+\\left\\\|\\sum\_\{j=i\}^\{K\}p\_\{j\|i\}\(\\theta\)\\psi\_\{j\}\\right\\\|\\leq 2B\_\{\\psi\}\.Summing overiiyields

‖∇θℓPL​\(θ;x,Y,σ\)‖≤∑i=1K‖∇θℓi​\(θ\)‖≤2​K​Bψ=CG\.\\\|\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\\|\\leq\\sum\_\{i=1\}^\{K\}\\\|\\nabla\_\{\\theta\}\\ell\_\{i\}\(\\theta\)\\\|\\leq 2KB\_\{\\psi\}=C\_\{G\}\.

#### Hessian bound\.

For each stageii, differentiating the stagewise softmax gradient gives

∇θ2ℓi​\(θ\)=∑j=iKpj\|i​\(θ\)​\(ψj−ψ¯i\)​\(ψj−ψ¯i\)⊤,\\nabla\_\{\\theta\}^\{2\}\\ell\_\{i\}\(\\theta\)=\\sum\_\{j=i\}^\{K\}p\_\{j\|i\}\(\\theta\)\(\\psi\_\{j\}\-\\bar\{\\psi\}\_\{i\}\)\(\\psi\_\{j\}\-\\bar\{\\psi\}\_\{i\}\)^\{\\top\},where

ψ¯i:=∑j=iKpj\|i​\(θ\)​ψj\.\\bar\{\\psi\}\_\{i\}:=\\sum\_\{j=i\}^\{K\}p\_\{j\|i\}\(\\theta\)\\psi\_\{j\}\.Equivalently,

∇θ2ℓi​\(θ\)=Covj∼p⋅\|i⁡\[ψj\]\.\\nabla\_\{\\theta\}^\{2\}\\ell\_\{i\}\(\\theta\)=\\operatorname\{Cov\}\_\{j\\sim p\_\{\\cdot\|i\}\}\[\\psi\_\{j\}\]\.Therefore

∇θ2ℓi​\(θ\)⪰0\.\\nabla\_\{\\theta\}^\{2\}\\ell\_\{i\}\(\\theta\)\\succeq 0\.Moreover, for any unit vectorvv,

v⊤​∇θ2ℓi​\(θ\)​v=Varj∼p⋅\|i⁡\(v⊤​ψj\)≤𝔼j∼p⋅\|i​\[\(v⊤​ψj\)2\]≤Bψ2\.v^\{\\top\}\\nabla\_\{\\theta\}^\{2\}\\ell\_\{i\}\(\\theta\)v=\\operatorname\{Var\}\_\{j\\sim p\_\{\\cdot\|i\}\}\(v^\{\\top\}\\psi\_\{j\}\)\\leq\\mathbb\{E\}\_\{j\\sim p\_\{\\cdot\|i\}\}\[\(v^\{\\top\}\\psi\_\{j\}\)^\{2\}\]\\leq B\_\{\\psi\}^\{2\}\.Hence

0⪯∇θ2ℓi​\(θ\)⪯Bψ2​I\.0\\preceq\\nabla\_\{\\theta\}^\{2\}\\ell\_\{i\}\(\\theta\)\\preceq B\_\{\\psi\}^\{2\}I\.Summing overi=1,…,Ki=1,\\ldots,K, we obtain

0⪯∇θ2ℓPL​\(θ;x,Y,σ\)=∑i=1K∇θ2ℓi​\(θ\)⪯K​Bψ2​I=CH​I\.0\\preceq\\nabla\_\{\\theta\}^\{2\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)=\\sum\_\{i=1\}^\{K\}\\nabla\_\{\\theta\}^\{2\}\\ell\_\{i\}\(\\theta\)\\preceq KB\_\{\\psi\}^\{2\}I=C\_\{H\}I\.This completes the proof\. ∎

###### Corollary F\.1\(Convexity of fixed\-list robust loss\)\.

θ↦ℓrob​\(θ;x,Y,σ⋆\)\\theta\\mapsto\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)is convex\. DefiningLrob​\(θ;x,Y\):=𝔼σ⋆∼p⋆\(⋅∣x,Y\)​\[ℓrob​\(θ;x,Y,σ⋆\)\]L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\):=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)\}\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\],LrobL\_\{\\mathrm\{rob\}\}is convex with0≤Lrob≤CL0\\leq L\_\{\\mathrm\{rob\}\}\\leq C\_\{L\}and‖∂CLrob‖≤CG\\\|\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\\\|\\leq C\_\{G\}\.

### F\.2Smoothing and weak convexity \([Proposition˜2](https://arxiv.org/html/2607.01715#Thmproposition2)\)

###### Definition F\.1\(Smoothed objects\)\.

Forτ\>0\\tau\>0, letMτ​\(θ;x,Y\):=τ​log​∑σ∈SKexp⁡\(ℓPL​\(θ;x,Y,σ\)/τ\)M\_\{\\tau\}\(\\theta;x,Y\):=\\tau\\log\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\(\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)/\\tau\),ℓrobτ:=\(1−ρ\)​ℓPL​\(σ⋆\)\+ρ​Mτ\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}:=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\sigma^\{\\star\}\)\+\\rho M\_\{\\tau\}, andJrobon,τ​\(θ\):=𝔼x,Y,σ⋆​\[ℓrobτ\]J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\(\\theta\):=\\mathbb\{E\}\_\{x,Y,\\sigma^\{\\star\}\}\[\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}\]\.

###### Lemma F\.4\(Properties ofMτM\_\{\\tau\}\)\.

Forτ\>0\\tau\>0, define

Mτ​\(θ;x,Y\):=τ​log​∑σ∈SKexp⁡\(ℓPL​\(θ;x,Y,σ\)τ\)\.M\_\{\\tau\}\(\\theta;x,Y\):=\\tau\\log\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\}\{\\tau\}\\right\)\.Then:\(i\)

maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)≤Mτ​\(θ;x,Y\)≤maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\+τ​log⁡K\!\.\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\leq M\_\{\\tau\}\(\\theta;x,Y\)\\leq\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\+\\tau\\log K\!\.\(ii\)θ↦Mτ​\(θ;x,Y\)\\theta\\mapsto M\_\{\\tau\}\(\\theta;x,Y\)is convex andC∞C^\{\\infty\}\.

\(iii\)

‖∇θMτ​\(θ;x,Y\)‖≤CG,0⪯∇θ2Mτ​\(θ;x,Y\)⪯\(CH\+CG2τ\)​I\.\\\|\\nabla\_\{\\theta\}M\_\{\\tau\}\(\\theta;x,Y\)\\\|\\leq C\_\{G\},\\qquad 0\\preceq\\nabla\_\{\\theta\}^\{2\}M\_\{\\tau\}\(\\theta;x,Y\)\\preceq\\left\(C\_\{H\}\+\\frac\{C\_\{G\}^\{2\}\}\{\\tau\}\\right\)I\.

###### Proof\.

Fix\(x,Y\)\(x,Y\)and write

ℓσ​\(θ\):=ℓPL​\(θ;x,Y,σ\),σ∈SK\.\\ell\_\{\\sigma\}\(\\theta\):=\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\),\\qquad\\sigma\\in S\_\{K\}\.Then

Mτ​\(θ;x,Y\)=τ​log​∑σ∈SKexp⁡\(ℓσ​\(θ\)τ\)\.M\_\{\\tau\}\(\\theta;x,Y\)=\\tau\\log\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\theta\)\}\{\\tau\}\\right\)\.
\(i\) Let

m​\(θ\):=maxσ∈SK⁡ℓσ​\(θ\)\.m\(\\theta\):=\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\sigma\}\(\\theta\)\.Then

∑σ∈SKexp⁡\(ℓσ​\(θ\)τ\)=exp⁡\(m​\(θ\)τ\)​∑σ∈SKexp⁡\(ℓσ​\(θ\)−m​\(θ\)τ\)\.\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\theta\)\}\{\\tau\}\\right\)=\\exp\\left\(\\frac\{m\(\\theta\)\}\{\\tau\}\\right\)\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\theta\)\-m\(\\theta\)\}\{\\tau\}\\right\)\.Sinceℓσ​\(θ\)−m​\(θ\)≤0\\ell\_\{\\sigma\}\(\\theta\)\-m\(\\theta\)\\leq 0for everyσ\\sigma, and at least one maximizer attains zero, we have

1≤∑σ∈SKexp⁡\(ℓσ​\(θ\)−m​\(θ\)τ\)≤\|SK\|=K\!\.1\\leq\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\theta\)\-m\(\\theta\)\}\{\\tau\}\\right\)\\leq\|S\_\{K\}\|=K\!\.Takingτ​log⁡\(⋅\)\\tau\\log\(\\cdot\)gives

m​\(θ\)≤Mτ​\(θ;x,Y\)≤m​\(θ\)\+τ​log⁡K\!,m\(\\theta\)\\leq M\_\{\\tau\}\(\\theta;x,Y\)\\leq m\(\\theta\)\+\\tau\\log K\!,which proves \(i\)\.

\(ii\) By[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3), for everyσ∈SK\\sigma\\in S\_\{K\}, the map

θ↦ℓσ​\(θ\)\\theta\\mapsto\\ell\_\{\\sigma\}\(\\theta\)is convex andC∞C^\{\\infty\}\. SinceSKS\_\{K\}is finite,MτM\_\{\\tau\}is a finite log\-sum\-exp composition ofC∞C^\{\\infty\}functions, and hence isC∞C^\{\\infty\}\.

To prove convexity, take anyθ1,θ2\\theta\_\{1\},\\theta\_\{2\}andα∈\[0,1\]\\alpha\\in\[0,1\]\. Since eachℓσ\\ell\_\{\\sigma\}is convex,

ℓσ​\(α​θ1\+\(1−α\)​θ2\)≤α​ℓσ​\(θ1\)\+\(1−α\)​ℓσ​\(θ2\)\.\\ell\_\{\\sigma\}\(\\alpha\\theta\_\{1\}\+\(1\-\\alpha\)\\theta\_\{2\}\)\\leq\\alpha\\ell\_\{\\sigma\}\(\\theta\_\{1\}\)\+\(1\-\\alpha\)\\ell\_\{\\sigma\}\(\\theta\_\{2\}\)\.Therefore,

exp⁡\(ℓσ​\(α​θ1\+\(1−α\)​θ2\)τ\)≤exp⁡\(α​ℓσ​\(θ1\)τ\)​exp⁡\(\(1−α\)​ℓσ​\(θ2\)τ\)\.\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\alpha\\theta\_\{1\}\+\(1\-\\alpha\)\\theta\_\{2\}\)\}\{\\tau\}\\right\)\\leq\\exp\\left\(\\frac\{\\alpha\\ell\_\{\\sigma\}\(\\theta\_\{1\}\)\}\{\\tau\}\\right\)\\exp\\left\(\\frac\{\(1\-\\alpha\)\\ell\_\{\\sigma\}\(\\theta\_\{2\}\)\}\{\\tau\}\\right\)\.Summing overσ\\sigmaand applying Hölder’s inequality gives

∑σ∈SKexp⁡\(ℓσ​\(α​θ1\+\(1−α\)​θ2\)τ\)≤\[∑σ∈SKexp⁡\(ℓσ​\(θ1\)τ\)\]α​\[∑σ∈SKexp⁡\(ℓσ​\(θ2\)τ\)\]1−α\.\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\alpha\\theta\_\{1\}\+\(1\-\\alpha\)\\theta\_\{2\}\)\}\{\\tau\}\\right\)\\leq\\left\[\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\theta\_\{1\}\)\}\{\\tau\}\\right\)\\right\]^\{\\alpha\}\\left\[\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\sigma\}\(\\theta\_\{2\}\)\}\{\\tau\}\\right\)\\right\]^\{1\-\\alpha\}\.Takingτ​log⁡\(⋅\)\\tau\\log\(\\cdot\)yields

Mτ​\(α​θ1\+\(1−α\)​θ2\)≤α​Mτ​\(θ1\)\+\(1−α\)​Mτ​\(θ2\)\.M\_\{\\tau\}\(\\alpha\\theta\_\{1\}\+\(1\-\\alpha\)\\theta\_\{2\}\)\\leq\\alpha M\_\{\\tau\}\(\\theta\_\{1\}\)\+\(1\-\\alpha\)M\_\{\\tau\}\(\\theta\_\{2\}\)\.ThusMτM\_\{\\tau\}is convex\.

\(iii\) Define the softmax weights over rankings

wσ​\(θ\):=exp⁡\(ℓσ​\(θ\)/τ\)∑σ′∈SKexp⁡\(ℓσ′​\(θ\)/τ\)\.w\_\{\\sigma\}\(\\theta\):=\\frac\{\\exp\(\\ell\_\{\\sigma\}\(\\theta\)/\\tau\)\}\{\\sum\_\{\\sigma^\{\\prime\}\\in S\_\{K\}\}\\exp\(\\ell\_\{\\sigma^\{\\prime\}\}\(\\theta\)/\\tau\)\}\.Thenwσ​\(θ\)≥0w\_\{\\sigma\}\(\\theta\)\\geq 0and∑σ∈SKwσ​\(θ\)=1\\sum\_\{\\sigma\\in S\_\{K\}\}w\_\{\\sigma\}\(\\theta\)=1\. DifferentiatingMτM\_\{\\tau\}gives

∇θMτ​\(θ;x,Y\)=∑σ∈SKwσ​\(θ\)​∇θℓσ​\(θ\)\.\\nabla\_\{\\theta\}M\_\{\\tau\}\(\\theta;x,Y\)=\\sum\_\{\\sigma\\in S\_\{K\}\}w\_\{\\sigma\}\(\\theta\)\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\.By[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3),‖∇θℓσ​\(θ\)‖≤CG\\\|\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\\\|\\leq C\_\{G\}for everyσ\\sigma\. Hence

‖∇θMτ​\(θ;x,Y\)‖≤∑σ∈SKwσ​\(θ\)​‖∇θℓσ​\(θ\)‖≤CG\.\\\|\\nabla\_\{\\theta\}M\_\{\\tau\}\(\\theta;x,Y\)\\\|\\leq\\sum\_\{\\sigma\\in S\_\{K\}\}w\_\{\\sigma\}\(\\theta\)\\\|\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\\\|\\leq C\_\{G\}\.
For the Hessian, differentiating the gradient gives

∇θ2Mτ​\(θ;x,Y\)=∑σ∈SKwσ​\(θ\)​∇θ2ℓσ​\(θ\)\+1τ​\[∑σ∈SKwσ​\(θ\)​∇θℓσ​\(θ\)​∇θℓσ​\(θ\)⊤−∇θMτ​\(θ;x,Y\)​∇θMτ​\(θ;x,Y\)⊤\]\.\\nabla\_\{\\theta\}^\{2\}M\_\{\\tau\}\(\\theta;x,Y\)=\\sum\_\{\\sigma\\in S\_\{K\}\}w\_\{\\sigma\}\(\\theta\)\\nabla\_\{\\theta\}^\{2\}\\ell\_\{\\sigma\}\(\\theta\)\+\\frac\{1\}\{\\tau\}\\left\[\\sum\_\{\\sigma\\in S\_\{K\}\}w\_\{\\sigma\}\(\\theta\)\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)^\{\\top\}\-\\nabla\_\{\\theta\}M\_\{\\tau\}\(\\theta;x,Y\)\\nabla\_\{\\theta\}M\_\{\\tau\}\(\\theta;x,Y\)^\{\\top\}\\right\]\.The second bracket is the covariance matrix of the random vector∇θℓσ​\(θ\)\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)underσ∼w​\(θ\)\\sigma\\sim w\(\\theta\), and is therefore positive semidefinite\. Since each∇θ2ℓσ​\(θ\)⪰0\\nabla\_\{\\theta\}^\{2\}\\ell\_\{\\sigma\}\(\\theta\)\\succeq 0, we get

∇θ2Mτ​\(θ;x,Y\)⪰0\.\\nabla\_\{\\theta\}^\{2\}M\_\{\\tau\}\(\\theta;x,Y\)\\succeq 0\.
For the upper bound, by[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3),

∑σ∈SKwσ​\(θ\)​∇θ2ℓσ​\(θ\)⪯CH​I\.\\sum\_\{\\sigma\\in S\_\{K\}\}w\_\{\\sigma\}\(\\theta\)\\nabla\_\{\\theta\}^\{2\}\\ell\_\{\\sigma\}\(\\theta\)\\preceq C\_\{H\}I\.For the covariance term, for any unit vectorvv,

v⊤​\[∑σwσ​∇ℓσ​∇ℓσ⊤−∇Mτ​∇Mτ⊤\]​v=Varσ∼w​\(θ\)⁡\(v⊤​∇θℓσ​\(θ\)\)≤𝔼σ∼w​\(θ\)​\[\(v⊤​∇θℓσ​\(θ\)\)2\]\.v^\{\\top\}\\left\[\\sum\_\{\\sigma\}w\_\{\\sigma\}\\nabla\\ell\_\{\\sigma\}\\nabla\\ell\_\{\\sigma\}^\{\\top\}\-\\nabla M\_\{\\tau\}\\nabla M\_\{\\tau\}^\{\\top\}\\right\]v=\\operatorname\{Var\}\_\{\\sigma\\sim w\(\\theta\)\}\\left\(v^\{\\top\}\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\\right\)\\leq\\mathbb\{E\}\_\{\\sigma\\sim w\(\\theta\)\}\\left\[\(v^\{\\top\}\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\)^\{2\}\\right\]\.Using‖∇θℓσ​\(θ\)‖≤CG\\\|\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\\\|\\leq C\_\{G\}, we have

𝔼σ∼w​\(θ\)​\[\(v⊤​∇θℓσ​\(θ\)\)2\]≤CG2\.\\mathbb\{E\}\_\{\\sigma\\sim w\(\\theta\)\}\\left\[\(v^\{\\top\}\\nabla\_\{\\theta\}\\ell\_\{\\sigma\}\(\\theta\)\)^\{2\}\\right\]\\leq C\_\{G\}^\{2\}\.Therefore the covariance term is bounded byCG2​IC\_\{G\}^\{2\}I, and hence

∇θ2Mτ​\(θ;x,Y\)⪯\(CH\+CG2τ\)​I\.\\nabla\_\{\\theta\}^\{2\}M\_\{\\tau\}\(\\theta;x,Y\)\\preceq\\left\(C\_\{H\}\+\\frac\{C\_\{G\}^\{2\}\}\{\\tau\}\\right\)I\.This proves \(iii\)\. ∎

###### Proposition F\.1\(Smoothed online weak convexity\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold,Jrobon,τ∈C2J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\\in C^\{2\}an open neighborhood ofΘ\\Thetaand isκτ\\kappa^\{\\tau\}\-weakly convex withκτ≤8​K2​Bψ2\+\(CL\+ρ​τ​log⁡K\!\)⋅K​Bψ2\\kappa^\{\\tau\}\\leq 8K^\{2\}B\_\{\\psi\}^\{2\}\+\(C\_\{L\}\+\\rho\\tau\\log K\!\)\\cdot KB\_\{\\psi\}^\{2\}\.

###### Proof\.

Fixτ\>0\\tau\>0\. Recall that the smoothed online objective is

Jrobon,τ​\(θ\)=𝔼x∼𝒟x​∑Y∈𝒴KPπθ​\(Y∣x\)​Lτ​\(θ;x,Y\),J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\(\\theta\)=\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\_\{x\}\}\\sum\_\{Y\\in\\mathcal\{Y\}^\{K\}\}P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)L\_\{\\tau\}\(\\theta;x,Y\),where

Lτ​\(θ;x,Y\):=𝔼σ⋆∼p⋆\(⋅∣x,Y\)​\[ℓrobτ​\(θ;x,Y,σ⋆\)\]\.L\_\{\\tau\}\(\\theta;x,Y\):=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)\}\[\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\]\.Since𝒴\\mathcal\{Y\}andSKS\_\{K\}are finite, all sums overYYandσ⋆\\sigma^\{\\star\}are finite\. Moreover,

ℓrobτ​\(θ;x,Y,σ⋆\)=\(1−ρ\)​ℓPL​\(θ;x,Y,σ⋆\)\+ρ​Mτ​\(θ;x,Y\),\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho M\_\{\\tau\}\(\\theta;x,Y\),with

Mτ​\(θ;x,Y\):=τ​log​∑σ∈SKexp⁡\(ℓPL​\(θ;x,Y,σ\)τ\)\.M\_\{\\tau\}\(\\theta;x,Y\):=\\tau\\log\\sum\_\{\\sigma\\in S\_\{K\}\}\\exp\\left\(\\frac\{\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\}\{\\tau\}\\right\)\.By[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3), eachℓPL​\(⋅;x,Y,σ\)\\ell\_\{\\mathrm\{PL\}\}\(\\cdot;x,Y,\\sigma\)is convex andC∞C^\{\\infty\}inθ\\theta\. The log\-sum\-exp smoothingMτM\_\{\\tau\}is therefore convex andC∞C^\{\\infty\}\. Henceℓrobτ\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}andLτL\_\{\\tau\}are convex andC∞C^\{\\infty\}inθ\\theta\.

For each fixedxx, define

Hx​\(θ\):=∑Y∈𝒴KPπθ​\(Y∣x\)​Lτ​\(θ;x,Y\)\.H\_\{x\}\(\\theta\):=\\sum\_\{Y\\in\\mathcal\{Y\}^\{K\}\}P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)L\_\{\\tau\}\(\\theta;x,Y\)\.Because the log\-linear softmax policy isC∞C^\{\\infty\}and𝒴K\\mathcal\{Y\}^\{K\}is finite,HxH\_\{x\}isC2C^\{2\}\. The uniform bounds from[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3),[Lemma˜F\.4](https://arxiv.org/html/2607.01715#Thmlemma4), and[Lemma˜F\.2](https://arxiv.org/html/2607.01715#Thmlemma2)imply that the first and second derivatives ofHxH\_\{x\}are dominated uniformly overxxandθ∈Θ\\theta\\in\\Theta\. Therefore differentiation may be interchanged with the expectation overxx, and

Jrobon,τ​\(θ\)=𝔼x​\[Hx​\(θ\)\]J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\(\\theta\)=\\mathbb\{E\}\_\{x\}\[H\_\{x\}\(\\theta\)\]isC2C^\{2\}onΘ\\Theta\.

We now lower bound the Hessian\. ForY=\(y1,…,yK\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\), write

Pθ​\(Y∣x\):=Pπθ​\(Y∣x\)=∏i=1Kπθ​\(yi∣x\),P\_\{\\theta\}\(Y\\mid x\):=P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)=\\prod\_\{i=1\}^\{K\}\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\),and define the list score function

Sθ​\(x,Y\):=∇θlog⁡Pθ​\(Y∣x\)=∑i=1K∇θlog⁡πθ​\(yi∣x\)\.S\_\{\\theta\}\(x,Y\):=\\nabla\_\{\\theta\}\\log P\_\{\\theta\}\(Y\\mid x\)=\\sum\_\{i=1\}^\{K\}\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)\.Then

∇θPθ​\(Y∣x\)=Pθ​\(Y∣x\)​Sθ​\(x,Y\)\.\\nabla\_\{\\theta\}P\_\{\\theta\}\(Y\\mid x\)=P\_\{\\theta\}\(Y\\mid x\)S\_\{\\theta\}\(x,Y\)\.Differentiating once more gives

∇θ2Pθ​\(Y∣x\)=∇θ\(Pθ​\(Y∣x\)​Sθ​\(x,Y\)\)=Pθ​\(Y∣x\)​\[Sθ​\(x,Y\)​Sθ​\(x,Y\)⊤\+∇θSθ​\(x,Y\)\]\.\\nabla\_\{\\theta\}^\{2\}P\_\{\\theta\}\(Y\\mid x\)=\\nabla\_\{\\theta\}\\\!\\left\(P\_\{\\theta\}\(Y\\mid x\)S\_\{\\theta\}\(x,Y\)\\right\)=P\_\{\\theta\}\(Y\\mid x\)\\left\[S\_\{\\theta\}\(x,Y\)S\_\{\\theta\}\(x,Y\)^\{\\top\}\+\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\\right\]\.
Applying the product rule twice to

Hx​\(θ\)=∑Y∈𝒴KPθ​\(Y∣x\)​Lτ​\(θ;x,Y\),H\_\{x\}\(\\theta\)=\\sum\_\{Y\\in\\mathcal\{Y\}^\{K\}\}P\_\{\\theta\}\(Y\\mid x\)L\_\{\\tau\}\(\\theta;x,Y\),we obtain

∇θ2Hx​\(θ\)\\displaystyle\\nabla\_\{\\theta\}^\{2\}H\_\{x\}\(\\theta\)=∑Y∈𝒴KPθ\(Y∣x\)\[∇θ2Lτ​\(θ;x,Y\)⏟T1\\displaystyle=\\sum\_\{Y\\in\\mathcal\{Y\}^\{K\}\}P\_\{\\theta\}\(Y\\mid x\)\\bigg\[\\underbrace\{\\nabla\_\{\\theta\}^\{2\}L\_\{\\tau\}\(\\theta;x,Y\)\}\_\{T\_\{1\}\}\+Sθ​\(x,Y\)​∇θLτ​\(θ;x,Y\)⊤\+∇θLτ​\(θ;x,Y\)​Sθ​\(x,Y\)⊤⏟T2\\displaystyle\\qquad\+\\underbrace\{S\_\{\\theta\}\(x,Y\)\\nabla\_\{\\theta\}L\_\{\\tau\}\(\\theta;x,Y\)^\{\\top\}\+\\nabla\_\{\\theta\}L\_\{\\tau\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)^\{\\top\}\}\_\{T\_\{2\}\}\+Lτ​\(θ;x,Y\)​\(Sθ​\(x,Y\)​Sθ​\(x,Y\)⊤\+∇θSθ​\(x,Y\)\)⏟T3\]\.\\displaystyle\\qquad\+\\underbrace\{L\_\{\\tau\}\(\\theta;x,Y\)\\left\(S\_\{\\theta\}\(x,Y\)S\_\{\\theta\}\(x,Y\)^\{\\top\}\+\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\\right\)\}\_\{T\_\{3\}\}\\bigg\]\.\(F\.1\)
We bound the three terms from below\. SinceLτ​\(⋅;x,Y\)L\_\{\\tau\}\(\\cdot;x,Y\)is convex,

T1=∇θ2Lτ​\(θ;x,Y\)⪰0\.T\_\{1\}=\\nabla\_\{\\theta\}^\{2\}L\_\{\\tau\}\(\\theta;x,Y\)\\succeq 0\.For the cross termT2T\_\{2\}, for any unit vectorvv,

v⊤​T2​v=2​⟨v,Sθ​\(x,Y\)⟩​⟨v,∇θLτ​\(θ;x,Y\)⟩\.v^\{\\top\}T\_\{2\}v=2\\langle v,S\_\{\\theta\}\(x,Y\)\\rangle\\langle v,\\nabla\_\{\\theta\}L\_\{\\tau\}\(\\theta;x,Y\)\\rangle\.Therefore, by Cauchy–Schwarz,

v⊤​T2​v≥−2​‖Sθ​\(x,Y\)‖​‖∇θLτ​\(θ;x,Y\)‖\.v^\{\\top\}T\_\{2\}v\\geq\-2\\\|S\_\{\\theta\}\(x,Y\)\\\|\\\|\\nabla\_\{\\theta\}L\_\{\\tau\}\(\\theta;x,Y\)\\\|\.Equivalently,

T2⪰−2​‖Sθ​\(x,Y\)‖​‖∇θLτ​\(θ;x,Y\)‖​I\.T\_\{2\}\\succeq\-2\\\|S\_\{\\theta\}\(x,Y\)\\\|\\\|\\nabla\_\{\\theta\}L\_\{\\tau\}\(\\theta;x,Y\)\\\|I\.
ForT3T\_\{3\}, note thatLτ​\(θ;x,Y\)≥0L\_\{\\tau\}\(\\theta;x,Y\)\\geq 0and

Sθ​\(x,Y\)​Sθ​\(x,Y\)⊤⪰0\.S\_\{\\theta\}\(x,Y\)S\_\{\\theta\}\(x,Y\)^\{\\top\}\\succeq 0\.Hence

T3⪰Lτ​\(θ;x,Y\)​∇θSθ​\(x,Y\)\.T\_\{3\}\\succeq L\_\{\\tau\}\(\\theta;x,Y\)\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\.By[Lemma˜F\.2](https://arxiv.org/html/2607.01715#Thmlemma2),

‖∇θSθ​\(x,Y\)‖op≤K​Bψ2,\\\|\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\\\|\_\{\\mathrm\{op\}\}\\leq KB\_\{\\psi\}^\{2\},and thus

∇θSθ​\(x,Y\)⪰−K​Bψ2​I\.\\nabla\_\{\\theta\}S\_\{\\theta\}\(x,Y\)\\succeq\-KB\_\{\\psi\}^\{2\}I\.Therefore,

T3⪰−Lτ​\(θ;x,Y\)​K​Bψ2​I\.T\_\{3\}\\succeq\-L\_\{\\tau\}\(\\theta;x,Y\)KB\_\{\\psi\}^\{2\}I\.
Now use the uniform bounds

‖Sθ​\(x,Y\)‖≤2​K​Bψ,‖∇θLτ​\(θ;x,Y\)‖≤CG=2​K​Bψ,\\\|S\_\{\\theta\}\(x,Y\)\\\|\\leq 2KB\_\{\\psi\},\\qquad\\\|\\nabla\_\{\\theta\}L\_\{\\tau\}\(\\theta;x,Y\)\\\|\\leq C\_\{G\}=2KB\_\{\\psi\},and

0≤Lτ​\(θ;x,Y\)≤CL\+ρ​τ​log⁡K\!\.0\\leq L\_\{\\tau\}\(\\theta;x,Y\)\\leq C\_\{L\}\+\\rho\\tau\\log K\!\.The gradient bound follows because∇Mτ\\nabla M\_\{\\tau\}is a softmax\-weighted convex combination of the gradients∇ℓPL​\(θ;x,Y,σ\)\\nabla\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\), each of which has norm at mostCGC\_\{G\}\. The loss bound follows from

Mτ​\(θ;x,Y\)≤maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\+τ​log⁡K\!,M\_\{\\tau\}\(\\theta;x,Y\)\\leq\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\+\\tau\\log K\!,together with the uniform PL loss bound\.

Combining the bounds, for every unit vectorvv,

v⊤​∇θ2Hx​\(θ\)​v\\displaystyle v^\{\\top\}\\nabla\_\{\\theta\}^\{2\}H\_\{x\}\(\\theta\)v≥−2​\(2​K​Bψ\)​\(2​K​Bψ\)−\(CL\+ρ​τ​log⁡K\!\)​K​Bψ2\\displaystyle\\geq\-2\(2KB\_\{\\psi\}\)\(2KB\_\{\\psi\}\)\-\(C\_\{L\}\+\\rho\\tau\\log K\!\)KB\_\{\\psi\}^\{2\}=−8​K2​Bψ2−\(CL\+ρ​τ​log⁡K\!\)​K​Bψ2\.\\displaystyle=\-8K^\{2\}B\_\{\\psi\}^\{2\}\-\(C\_\{L\}\+\\rho\\tau\\log K\!\)KB\_\{\\psi\}^\{2\}\.\(F\.2\)Since the bound is uniform inYYandxx, summing overYYwith weightsPθ​\(Y∣x\)P\_\{\\theta\}\(Y\\mid x\)and taking expectation overxxpreserve the same lower bound\. Hence

∇θ2Jrobon,τ​\(θ\)⪰−\[8​K2​Bψ2\+\(CL\+ρ​τ​log⁡K\!\)​K​Bψ2\]​I\.\\nabla\_\{\\theta\}^\{2\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\(\\theta\)\\succeq\-\\left\[8K^\{2\}B\_\{\\psi\}^\{2\}\+\(C\_\{L\}\+\\rho\\tau\\log K\!\)KB\_\{\\psi\}^\{2\}\\right\]I\.ThereforeJrobon,τJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}isκτ\\kappa^\{\\tau\}\-weakly convex with

κτ≤8​K2​Bψ2\+\(CL\+ρ​τ​log⁡K\!\)​K​Bψ2\.\\kappa^\{\\tau\}\\leq 8K^\{2\}B\_\{\\psi\}^\{2\}\+\(C\_\{L\}\+\\rho\\tau\\log K\!\)KB\_\{\\psi\}^\{2\}\.Indeed, for aC2C^\{2\}function, the Hessian lower bound∇2f⪰−κ​I\\nabla^\{2\}f\\succeq\-\\kappa Iis equivalent to convexity off\+κ2∥⋅∥2f\+\\frac\{\\kappa\}\{2\}\\\|\\cdot\\\|^\{2\}, which is preciselyκ\\kappa\-weak convexity\. ∎

###### Proof of[Proposition˜2](https://arxiv.org/html/2607.01715#Thmproposition2)\.

Forτ\>0\\tau\>0, define

κ¯τ:=8​K2​Bψ2\+\(CL\+ρ​τ​log⁡K\!\)​K​Bψ2\.\\bar\{\\kappa\}\_\{\\tau\}:=8K^\{2\}B\_\{\\psi\}^\{2\}\+\(C\_\{L\}\+\\rho\\tau\\log K\!\)KB\_\{\\psi\}^\{2\}\.By[Proposition˜F\.1](https://arxiv.org/html/2607.01715#Thmproposition1a),Jrobon,τJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}isκ¯τ\\bar\{\\kappa\}\_\{\\tau\}\-weakly convex onΘ\\Theta\. Equivalently, the function

Φτ​\(θ\):=Jrobon,τ​\(θ\)\+κ¯τ2​‖θ‖2\\Phi\_\{\\tau\}\(\\theta\):=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\(\\theta\)\+\\frac\{\\bar\{\\kappa\}\_\{\\tau\}\}\{2\}\\\|\\theta\\\|^\{2\}is convex onΘ\\Theta\.

We first verify convergence of the smoothed objectives\. By the log\-sum\-exp bound in[Lemma˜F\.4](https://arxiv.org/html/2607.01715#Thmlemma4)\(i\),

0≤Mτ​\(θ;x,Y\)−M​\(θ;x,Y\)≤τ​log⁡K\!\.0\\leq M\_\{\\tau\}\(\\theta;x,Y\)\-M\(\\theta;x,Y\)\\leq\\tau\\log K\!\.Since

ℓrobτ=\(1−ρ\)​ℓPL​\(σ⋆\)\+ρ​Mτ,ℓrob=\(1−ρ\)​ℓPL​\(σ⋆\)\+ρ​M,\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\sigma^\{\\star\}\)\+\\rho M\_\{\\tau\},\\qquad\\ell\_\{\\mathrm\{rob\}\}=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\sigma^\{\\star\}\)\+\\rho M,we have, for every\(θ,x,Y,σ⋆\)\(\\theta,x,Y,\\sigma^\{\\star\}\),

0≤ℓrobτ​\(θ;x,Y,σ⋆\)−ℓrob​\(θ;x,Y,σ⋆\)≤ρ​τ​log⁡K\!\.0\\leq\\ell\_\{\\mathrm\{rob\}\}^\{\\tau\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\-\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\leq\\rho\\tau\\log K\!\.Taking expectation overxx,Y∼πθ⊗K\(⋅\|x\)Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\|x\), andσ⋆∼p⋆\(⋅\|x,Y\)\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\|x,Y\)gives the uniform bound

0≤Jrobon,τ​\(θ\)−Jrobon​\(θ\)≤ρ​τ​log⁡K\!,∀θ∈Θ\.0\\leq J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\(\\theta\)\-J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\\leq\\rho\\tau\\log K\!,\\qquad\\forall\\theta\\in\\Theta\.HenceJrobon,τ→JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\\to J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}uniformly onΘ\\Theta\.

Now define

κ:=limτ↓0κ¯τ=8​K2​Bψ2\+CL​K​Bψ2\.\\kappa:=\\lim\_\{\\tau\\downarrow 0\}\\bar\{\\kappa\}\_\{\\tau\}=8K^\{2\}B\_\{\\psi\}^\{2\}\+C\_\{L\}KB\_\{\\psi\}^\{2\}\.SinceCL=K​\(log⁡K\+2​D​Bψ\)C\_\{L\}=K\(\\log K\+2DB\_\{\\psi\}\), this becomes

κ=K2​Bψ2​\(8\+log⁡K\+2​D​Bψ\)\.\\kappa=K^\{2\}B\_\{\\psi\}^\{2\}\(8\+\\log K\+2DB\_\{\\psi\}\)\.
It remains to prove that

Φ​\(θ\):=Jrobon​\(θ\)\+κ2​‖θ‖2\\Phi\(\\theta\):=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\+\\frac\{\\kappa\}\{2\}\\\|\\theta\\\|^\{2\}is convex onΘ\\Theta\. Letθ1,θ2∈Θ\\theta\_\{1\},\\theta\_\{2\}\\in\\Thetaandα∈\[0,1\]\\alpha\\in\[0,1\]\. SinceΘ\\Thetais convex,

θα:=α​θ1\+\(1−α\)​θ2∈Θ\.\\theta\_\{\\alpha\}:=\\alpha\\theta\_\{1\}\+\(1\-\\alpha\)\\theta\_\{2\}\\in\\Theta\.For everyτ\>0\\tau\>0, convexity ofΦτ\\Phi\_\{\\tau\}gives

Φτ​\(θα\)≤α​Φτ​\(θ1\)\+\(1−α\)​Φτ​\(θ2\)\.\\Phi\_\{\\tau\}\(\\theta\_\{\\alpha\}\)\\leq\\alpha\\Phi\_\{\\tau\}\(\\theta\_\{1\}\)\+\(1\-\\alpha\)\\Phi\_\{\\tau\}\(\\theta\_\{2\}\)\.Takingτ↓0\\tau\\downarrow 0, using the uniform convergenceJrobon,τ→JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\},\\tau\}\\to J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}andκ¯τ→κ\\bar\{\\kappa\}\_\{\\tau\}\\to\\kappa, yields

Φ​\(θα\)≤α​Φ​\(θ1\)\+\(1−α\)​Φ​\(θ2\)\.\\Phi\(\\theta\_\{\\alpha\}\)\\leq\\alpha\\Phi\(\\theta\_\{1\}\)\+\(1\-\\alpha\)\\Phi\(\\theta\_\{2\}\)\.ThereforeΦ\\Phiis convex onΘ\\Theta\. Equivalently,JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}isκ\\kappa\-weakly convex onΘ\\Theta, with

κ=K2​Bψ2​\(8\+log⁡K\+2​D​Bψ\)\.\\kappa=K^\{2\}B\_\{\\psi\}^\{2\}\(8\+\\log K\+2DB\_\{\\psi\}\)\.∎

### F\.3Constrained Moreau envelope

ForF=Jrobon\+IΘF=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\+I\_\{\\Theta\}andλ^∈\(0,1/κ\)\\hat\{\\lambda\}\\in\(0,1/\\kappa\), defineFλ^​\(θ\)=minu∈Θ⁡\{Jrobon​\(u\)\+12​λ^​‖u−θ‖2\}F\_\{\\hat\{\\lambda\}\}\(\\theta\)=\\min\_\{u\\in\\Theta\}\\\{J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(u\)\+\\tfrac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\\\}andθ^​\(θ\)=proxλ^​F⁡\(θ\)∈Θ\\hat\{\\theta\}\(\\theta\)=\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}\(\\theta\)\\in\\Theta\.

###### Lemma F\.5\(Moreau\-envelope Properties\)\.

Let

F​\(θ\):=Jrobon​\(θ\)\+IΘ​\(θ\),F\(\\theta\):=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\+I\_\{\\Theta\}\(\\theta\),whereIΘI\_\{\\Theta\}is the indicator of the closed convex setΘ\\Theta\. SupposeJrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}isκ\\kappa\-weakly convex onΘ\\Theta, and assume thatFFis proper, lower semicontinuous, and bounded below\. Fix

λ^∈\(0,1/κ\)\.\\hat\{\\lambda\}\\in\(0,1/\\kappa\)\.Define the constrained Moreau envelope

Fλ^​\(θ\):=minu∈ℝd⁡\{F​\(u\)\+12​λ^​‖u−θ‖2\}=minu∈Θ⁡\{Jrobon​\(u\)\+12​λ^​‖u−θ‖2\},F\_\{\\hat\{\\lambda\}\}\(\\theta\):=\\min\_\{u\\in\\mathbb\{R\}^\{d\}\}\\left\\\{F\(u\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\\right\\\}=\\min\_\{u\\in\\Theta\}\\left\\\{J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(u\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\\right\\\},and denote the proximal point by

θ^​\(θ\):=proxλ^​F⁡\(θ\):=arg​minu∈ℝd⁡\{F​\(u\)\+12​λ^​‖u−θ‖2\}\.\\hat\{\\theta\}\(\\theta\):=\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}\(\\theta\):=\\operatorname\*\{arg\\,min\}\_\{u\\in\\mathbb\{R\}^\{d\}\}\\left\\\{F\(u\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\\right\\\}\.Then the following hold:

1. 1\.proxλ^​F\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}is single\-valued onℝd\\mathbb\{R\}^\{d\}\.
2. 2\.Fλ^∈C1F\_\{\\hat\{\\lambda\}\}\\in C^\{1\}, and ∇Fλ^​\(θ\)=λ^−1​\(θ−θ^​\(θ\)\)\.\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)=\\hat\{\\lambda\}^\{\-1\}\\left\(\\theta\-\\hat\{\\theta\}\(\\theta\)\\right\)\.
3. 3\.∇Fλ^\\nabla F\_\{\\hat\{\\lambda\}\}is Lipschitz with constant at most Lenv:=1λ^​\(1−κ​λ^\)\.L\_\{\\mathrm\{env\}\}:=\\frac\{1\}\{\\hat\{\\lambda\}\(1\-\\kappa\\hat\{\\lambda\}\)\}\.
4. 4\.*Near\-stationarity:* dist⁡\(0,∂CF​\(θ^​\(θ\)\)\)≤‖∇Fλ^​\(θ\)‖\.\\operatorname\{dist\}\\\!\\left\(0,\\partial\_\{C\}F\(\\hat\{\\theta\}\(\\theta\)\)\\right\)\\leq\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)\\\|\.

###### Proof\.

The Moreau\-envelope properties above are standard for proper lower semicontinuousκ\\kappa\-weakly convex functions with parameterλ^<1/κ\\hat\{\\lambda\}<1/\\kappa; see, e\.g\.,Davis and Drusvyatskiy \[[2019](https://arxiv.org/html/2607.01715#bib.bib20)\]andDrusvyatskiy and Lewis \[[2018](https://arxiv.org/html/2607.01715#bib.bib21)\]\.

By[Proposition˜2](https://arxiv.org/html/2607.01715#Thmproposition2),JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}isκ\\kappa\-weakly convex onΘ\\Theta\. SinceΘ\\Thetais closed and convex, the indicatorIΘI\_\{\\Theta\}is proper, lower semicontinuous, and convex\. Therefore

F=Jrobon\+IΘF=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\+I\_\{\\Theta\}is proper, lower semicontinuous, bounded below by assumption, andκ\\kappa\-weakly convex\. Here is the detailed statement:

We first prove single\-valuedness of the proximal map\. SinceFFisκ\\kappa\-weakly convex, the function

u↦F​\(u\)\+κ2​‖u‖2u\\mapsto F\(u\)\+\\frac\{\\kappa\}\{2\}\\\|u\\\|^\{2\}is convex\. For fixedθ\\theta, the proximal objective is

u↦F​\(u\)\+12​λ^​‖u−θ‖2\.u\\mapsto F\(u\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\.Adding and subtractingκ2​‖u‖2\\frac\{\\kappa\}\{2\}\\\|u\\\|^\{2\}, we can write it as

\(F​\(u\)\+κ2​‖u‖2\)\+12​\(1λ^−κ\)​‖u‖2−1λ^​⟨u,θ⟩\+12​λ^​‖θ‖2\.\\left\(F\(u\)\+\\frac\{\\kappa\}\{2\}\\\|u\\\|^\{2\}\\right\)\+\\frac\{1\}\{2\}\\left\(\\frac\{1\}\{\\hat\{\\lambda\}\}\-\\kappa\\right\)\\\|u\\\|^\{2\}\-\\frac\{1\}\{\\hat\{\\lambda\}\}\\langle u,\\theta\\rangle\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|\\theta\\\|^\{2\}\.The first term is convex, and the second quadratic term is strongly convex because

1λ^−κ\>0\.\\frac\{1\}\{\\hat\{\\lambda\}\}\-\\kappa\>0\.Hence the proximal objective is strongly convex\. Since it is also proper, lower semicontinuous, and coercive, it has a unique minimizer\. Thusproxλ^​F\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}is single\-valued\.

The standard Moreau\-envelope calculus for weakly convex functions then gives

Fλ^∈C1,∇Fλ^​\(θ\)=λ^−1​\(θ−proxλ^​F⁡\(θ\)\)=λ^−1​\(θ−θ^​\(θ\)\)\.F\_\{\\hat\{\\lambda\}\}\\in C^\{1\},\\qquad\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)=\\hat\{\\lambda\}^\{\-1\}\\left\(\\theta\-\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}\(\\theta\)\\right\)=\\hat\{\\lambda\}^\{\-1\}\\left\(\\theta\-\\hat\{\\theta\}\(\\theta\)\\right\)\.This proves the differentiability and gradient formula\.

The same weakly\-convex Moreau calculus gives the Lipschitz bound

‖∇Fλ^​\(θ\)−∇Fλ^​\(θ′\)‖≤1λ^​\(1−κ​λ^\)​‖θ−θ′‖,\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)\-\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta^\{\\prime\}\)\\\|\\leq\\frac\{1\}\{\\hat\{\\lambda\}\(1\-\\kappa\\hat\{\\lambda\}\)\}\\\|\\theta\-\\theta^\{\\prime\}\\\|,for allθ,θ′∈ℝd\\theta,\\theta^\{\\prime\}\\in\\mathbb\{R\}^\{d\}\. Hence∇Fλ^\\nabla F\_\{\\hat\{\\lambda\}\}is Lipschitz with constant at most

Lenv=1λ^​\(1−κ​λ^\)\.L\_\{\\mathrm\{env\}\}=\\frac\{1\}\{\\hat\{\\lambda\}\(1\-\\kappa\\hat\{\\lambda\}\)\}\.
It remains to prove the near\-stationarity claim\. Let

θ^:=θ^​\(θ\)=proxλ^​F⁡\(θ\)\.\\hat\{\\theta\}:=\\hat\{\\theta\}\(\\theta\)=\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}\(\\theta\)\.By the first\-order optimality condition for the proximal problem,

0∈∂CF​\(θ^\)\+1λ^​\(θ^−θ\)\.0\\in\\partial\_\{C\}F\(\\hat\{\\theta\}\)\+\\frac\{1\}\{\\hat\{\\lambda\}\}\(\\hat\{\\theta\}\-\\theta\)\.Equivalently,

1λ^​\(θ−θ^\)∈∂CF​\(θ^\)\.\\frac\{1\}\{\\hat\{\\lambda\}\}\(\\theta\-\\hat\{\\theta\}\)\\in\\partial\_\{C\}F\(\\hat\{\\theta\}\)\.Using the gradient formula,

∇Fλ^​\(θ\)=1λ^​\(θ−θ^\),\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)=\\frac\{1\}\{\\hat\{\\lambda\}\}\(\\theta\-\\hat\{\\theta\}\),we obtain

∇Fλ^​\(θ\)∈∂CF​\(θ^​\(θ\)\)\.\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)\\in\\partial\_\{C\}F\(\\hat\{\\theta\}\(\\theta\)\)\.Therefore,

dist⁡\(0,∂CF​\(θ^​\(θ\)\)\)≤‖∇Fλ^​\(θ\)‖\.\\operatorname\{dist\}\\\!\\left\(0,\\partial\_\{C\}F\(\\hat\{\\theta\}\(\\theta\)\)\\right\)\\leq\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\)\\\|\.This proves the lemma\. ∎

### F\.4Subdifferential calculus and oracle \([Lemma˜F\.9](https://arxiv.org/html/2607.01715#Thmlemma9)\)

###### Lemma F\.6\(Set\-valued Danskin\)\.

For every\(θ,x,Y\)\(\\theta,x,Y\), define

M​\(θ;x,Y\):=maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\.M\(\\theta;x,Y\):=\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\.ThenM​\(⋅;x,Y\)M\(\\cdot;x,Y\)is convex and Clarke regular\. Moreover,

∂CM​\(θ;x,Y\)=conv⁡\{∇θℓPL​\(θ;x,Y,σ\):σ∈arg​maxσ′∈SK⁡ℓPL​\(θ;x,Y,σ′\)\}\.\\partial\_\{C\}M\(\\theta;x,Y\)=\\operatorname\{conv\}\\left\\\{\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\):\\sigma\\in\\operatorname\*\{arg\\,max\}\_\{\\sigma^\{\\prime\}\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\prime\}\)\\right\\\}\.In particular,

∇θℓPL​\(θ;x,Y,σworst\)∈∂CM​\(θ;x,Y\)\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\_\{\\mathrm\{worst\}\}\)\\in\\partial\_\{C\}M\(\\theta;x,Y\)for anyσworst\\sigma\_\{\\mathrm\{worst\}\}selected by[Theorem˜1](https://arxiv.org/html/2607.01715#Thmtheorem1)\.

###### Proof\.

Fix\(x,Y\)\(x,Y\)throughout the proof and write

fσ​\(θ\):=ℓPL​\(θ;x,Y,σ\),σ∈SK\.f\_\{\\sigma\}\(\\theta\):=\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\),\\qquad\\sigma\\in S\_\{K\}\.By[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3), eachfσf\_\{\\sigma\}is convex and continuously differentiable inθ\\theta\. SinceSKS\_\{K\}is finite, the pointwise maximum

M​\(θ;x,Y\)=maxσ∈SK⁡fσ​\(θ\)M\(\\theta;x,Y\)=\\max\_\{\\sigma\\in S\_\{K\}\}f\_\{\\sigma\}\(\\theta\)is also convex\. Moreover, a finite maximum of continuously differentiable functions is locally Lipschitz\. Since every finite\-valued convex function is Clarke regular,M​\(⋅;x,Y\)M\(\\cdot;x,Y\)is Clarke regular, and its Clarke subdifferential coincides with its convex subdifferential:

∂CM​\(θ;x,Y\)=∂M​\(θ;x,Y\)\.\\partial\_\{C\}M\(\\theta;x,Y\)=\\partial M\(\\theta;x,Y\)\.
It remains to identify this subdifferential\. Define the active maximizer set

𝒜​\(θ;x,Y\):=arg​maxσ∈SK⁡fσ​\(θ\)\.\\mathcal\{A\}\(\\theta;x,Y\):=\\operatorname\*\{arg\\,max\}\_\{\\sigma\\in S\_\{K\}\}f\_\{\\sigma\}\(\\theta\)\.This set is nonempty becauseSKS\_\{K\}is finite\. We prove both inclusions\.

First, letσ∈𝒜​\(θ;x,Y\)\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\. Sincefσf\_\{\\sigma\}is convex and differentiable, for everyuu,

fσ​\(u\)≥fσ​\(θ\)\+⟨∇θfσ​\(θ\),u−θ⟩\.f\_\{\\sigma\}\(u\)\\geq f\_\{\\sigma\}\(\\theta\)\+\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),u\-\\theta\\right\\rangle\.Becauseσ\\sigmais active,fσ​\(θ\)=M​\(θ;x,Y\)f\_\{\\sigma\}\(\\theta\)=M\(\\theta;x,Y\)\. Also,M​\(u;x,Y\)≥fσ​\(u\)M\(u;x,Y\)\\geq f\_\{\\sigma\}\(u\)\. Hence

M​\(u;x,Y\)≥M​\(θ;x,Y\)\+⟨∇θfσ​\(θ\),u−θ⟩for all​u\.M\(u;x,Y\)\\geq M\(\\theta;x,Y\)\+\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),u\-\\theta\\right\\rangle\\qquad\\text\{for all \}u\.Therefore,

∇θfσ​\(θ\)∈∂M​\(θ;x,Y\)\.\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\)\\in\\partial M\(\\theta;x,Y\)\.Since∂M​\(θ;x,Y\)\\partial M\(\\theta;x,Y\)is convex, we obtain

conv⁡\{∇θfσ​\(θ\):σ∈𝒜​\(θ;x,Y\)\}⊆∂M​\(θ;x,Y\)\.\\operatorname\{conv\}\\left\\\{\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\):\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\\right\\\}\\subseteq\\partial M\(\\theta;x,Y\)\.
We now prove the reverse inclusion\. For any directiondd, the one\-sided directional derivative ofMMatθ\\thetasatisfies

M′​\(θ;d\)=limt↓0M​\(θ\+t​d;x,Y\)−M​\(θ;x,Y\)t\.M^\{\\prime\}\(\\theta;d\)=\\lim\_\{t\\downarrow 0\}\\frac\{M\(\\theta\+td;x,Y\)\-M\(\\theta;x,Y\)\}\{t\}\.We claim that

M′​\(θ;d\)=maxσ∈𝒜​\(θ;x,Y\)⁡⟨∇θfσ​\(θ\),d⟩\.M^\{\\prime\}\(\\theta;d\)=\\max\_\{\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\}\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),d\\right\\rangle\.To see this, note first that for every activeσ∈𝒜​\(θ;x,Y\)\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\),

M​\(θ\+t​d;x,Y\)≥fσ​\(θ\+t​d\),M\(\\theta\+td;x,Y\)\\geq f\_\{\\sigma\}\(\\theta\+td\),and therefore

lim inft↓0M​\(θ\+t​d;x,Y\)−M​\(θ;x,Y\)t≥⟨∇θfσ​\(θ\),d⟩\.\\liminf\_\{t\\downarrow 0\}\\frac\{M\(\\theta\+td;x,Y\)\-M\(\\theta;x,Y\)\}\{t\}\\geq\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),d\\right\\rangle\.Taking the maximum over activeσ\\sigmagives the lower bound\.

For the upper bound, choose for eacht\>0t\>0a maximizer

σt∈arg​maxσ∈SK⁡fσ​\(θ\+t​d\)\.\\sigma\_\{t\}\\in\\operatorname\*\{arg\\,max\}\_\{\\sigma\\in S\_\{K\}\}f\_\{\\sigma\}\(\\theta\+td\)\.SinceSKS\_\{K\}is finite, along any sequencetn↓0t\_\{n\}\\downarrow 0there is a subsequence, still denotedtnt\_\{n\}, such thatσtn=σ¯\\sigma\_\{t\_\{n\}\}=\\bar\{\\sigma\}is constant\. By continuity,

fσ¯​\(θ\)=limn→∞fσtn​\(θ\+tn​d\)=limn→∞M​\(θ\+tn​d;x,Y\)=M​\(θ;x,Y\),f\_\{\\bar\{\\sigma\}\}\(\\theta\)=\\lim\_\{n\\to\\infty\}f\_\{\\sigma\_\{t\_\{n\}\}\}\(\\theta\+t\_\{n\}d\)=\\lim\_\{n\\to\\infty\}M\(\\theta\+t\_\{n\}d;x,Y\)=M\(\\theta;x,Y\),soσ¯∈𝒜​\(θ;x,Y\)\\bar\{\\sigma\}\\in\\mathcal\{A\}\(\\theta;x,Y\)\. Hence, along this subsequence,

limn→∞M​\(θ\+tn​d;x,Y\)−M​\(θ;x,Y\)tn=⟨∇θfσ¯​\(θ\),d⟩≤maxσ∈𝒜​\(θ;x,Y\)⁡⟨∇θfσ​\(θ\),d⟩\.\\lim\_\{n\\to\\infty\}\\frac\{M\(\\theta\+t\_\{n\}d;x,Y\)\-M\(\\theta;x,Y\)\}\{t\_\{n\}\}=\\left\\langle\\nabla\_\{\\theta\}f\_\{\\bar\{\\sigma\}\}\(\\theta\),d\\right\\rangle\\leq\\max\_\{\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\}\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),d\\right\\rangle\.Since this argument applies to every vanishing sequencetn↓0t\_\{n\}\\downarrow 0, the claimed directional derivative formula follows\.

Now letv∈∂M​\(θ;x,Y\)v\\in\\partial M\(\\theta;x,Y\)\. By the characterization of the convex subdifferential through directional derivatives,

⟨v,d⟩≤M′​\(θ;d\)for every direction​d\.\\langle v,d\\rangle\\leq M^\{\\prime\}\(\\theta;d\)\\qquad\\text\{for every direction \}d\.Using the formula above,

⟨v,d⟩≤maxσ∈𝒜​\(θ;x,Y\)⁡⟨∇θfσ​\(θ\),d⟩for every​d\.\\langle v,d\\rangle\\leq\\max\_\{\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\}\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),d\\right\\rangle\\qquad\\text\{for every \}d\.We show that this implies

v∈conv⁡\{∇θfσ​\(θ\):σ∈𝒜​\(θ;x,Y\)\}\.v\\in\\operatorname\{conv\}\\left\\\{\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\):\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\\right\\\}\.Indeed, ifvvwere not in this closed convex hull, then by the finite\-dimensional separating hyperplane theorem there would exist a directionddsuch that

⟨v,d⟩\>maxσ∈𝒜​\(θ;x,Y\)⁡⟨∇θfσ​\(θ\),d⟩,\\langle v,d\\rangle\>\\max\_\{\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\}\\left\\langle\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\),d\\right\\rangle,contradicting the previous inequality\. Therefore,

∂M​\(θ;x,Y\)⊆conv⁡\{∇θfσ​\(θ\):σ∈𝒜​\(θ;x,Y\)\}\.\\partial M\(\\theta;x,Y\)\\subseteq\\operatorname\{conv\}\\left\\\{\\nabla\_\{\\theta\}f\_\{\\sigma\}\(\\theta\):\\sigma\\in\\mathcal\{A\}\(\\theta;x,Y\)\\right\\\}\.Combining the two inclusions gives

∂CM​\(θ;x,Y\)=∂M​\(θ;x,Y\)=conv⁡\{∇θℓPL​\(θ;x,Y,σ\):σ∈arg​maxσ′∈SK⁡ℓPL​\(θ;x,Y,σ′\)\}\.\\partial\_\{C\}M\(\\theta;x,Y\)=\\partial M\(\\theta;x,Y\)=\\operatorname\{conv\}\\left\\\{\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\):\\sigma\\in\\operatorname\*\{arg\\,max\}\_\{\\sigma^\{\\prime\}\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\prime\}\)\\right\\\}\.
Finally, by[Theorem˜1](https://arxiv.org/html/2607.01715#Thmtheorem1), any selected worst\-case rankingσworst\\sigma\_\{\\mathrm\{worst\}\}belongs to the active maximizer set\. Therefore its gradient is one of the active gradients and hence belongs to∂CM​\(θ;x,Y\)\\partial\_\{C\}M\(\\theta;x,Y\)\. ∎

###### Lemma F\.7\(Clarke regularity and product rule\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold, the following statements hold\.

\(a\)For every\(x,Y,σ⋆\)\(x,Y,\\sigma^\{\\star\}\),ℓrob​\(⋅;x,Y,σ⋆\)\\ell\_\{\\mathrm\{rob\}\}\(\\cdot;x,Y,\\sigma^\{\\star\}\)is convex and Clarke regular\. Moreover,

∂Cℓrob​\(θ;x,Y,σ⋆\)=\(1−ρ\)​∇θℓPL​\(θ;x,Y,σ⋆\)\+ρ​∂CM​\(θ;x,Y\),\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\,\\partial\_\{C\}M\(\\theta;x,Y\),where

M​\(θ;x,Y\):=maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\.M\(\\theta;x,Y\):=\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\.
\(b\)For every\(x,Y\)\(x,Y\),

Lrob​\(θ;x,Y\):=𝔼σ⋆∼p⋆\(⋅∣x,Y\)​\[ℓrob​\(θ;x,Y,σ⋆\)\]L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\):=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)\}\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\]is convex and Clarke regular\. Moreover,

∂CLrob​\(θ;x,Y\)=𝔼σ⋆∼p⋆\(⋅∣x,Y\)​\[∂Cℓrob​\(θ;x,Y,σ⋆\)\]\.\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)\}\[\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\]\.
\(c\)For every\(x,Y\)\(x,Y\), the map

θ↦Pπθ​\(Y∣x\)​Lrob​\(θ;x,Y\)\\theta\\mapsto P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)is Clarke regular, and

∂C\[Pπθ​\(Y∣x\)​Lrob​\(θ;x,Y\)\]=Pπθ​\(Y∣x\)​∂CLrob​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​∇θPπθ​\(Y∣x\)\.\\partial\_\{C\}\\left\[P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\right\]=P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\nabla\_\{\\theta\}P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)\.

###### Proof\.

We prove the three claims separately\.

\(a\) Fix\(x,Y,σ⋆\)\(x,Y,\\sigma^\{\\star\}\)\. By the robust\-TV decomposition,

ℓrob​\(θ;x,Y,σ⋆\)=\(1−ρ\)​ℓPL​\(θ;x,Y,σ⋆\)\+ρ​M​\(θ;x,Y\),\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho M\(\\theta;x,Y\),where

M​\(θ;x,Y\)=maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\.M\(\\theta;x,Y\)=\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\.By[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3), for everyσ∈SK\\sigma\\in S\_\{K\}, the map

θ↦ℓPL​\(θ;x,Y,σ\)\\theta\\mapsto\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)is convex and continuously differentiable\. ThereforeℓPL​\(⋅;x,Y,σ⋆\)\\ell\_\{\\mathrm\{PL\}\}\(\\cdot;x,Y,\\sigma^\{\\star\}\)is convex and Clarke regular\. By[Lemma˜F\.6](https://arxiv.org/html/2607.01715#Thmlemma6),M​\(⋅;x,Y\)M\(\\cdot;x,Y\)is also convex and Clarke regular\. Sinceρ∈\[0,1\]\\rho\\in\[0,1\], the functionℓrob​\(⋅;x,Y,σ⋆\)\\ell\_\{\\mathrm\{rob\}\}\(\\cdot;x,Y,\\sigma^\{\\star\}\)is a nonnegative linear combination of convex functions, and hence is convex\. Since it is finite\-valued and convex on the parameter space, it is Clarke regular\.

It remains to compute the subdifferential\. The convex subdifferential sum rule gives

∂ℓrob​\(θ;x,Y,σ⋆\)=\(1−ρ\)​∂ℓPL​\(θ;x,Y,σ⋆\)\+ρ​∂M​\(θ;x,Y\)\.\\partial\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\partial\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\,\\partial M\(\\theta;x,Y\)\.BecauseℓPL​\(⋅;x,Y,σ⋆\)\\ell\_\{\\mathrm\{PL\}\}\(\\cdot;x,Y,\\sigma^\{\\star\}\)is differentiable,

∂ℓPL​\(θ;x,Y,σ⋆\)=\{∇θℓPL​\(θ;x,Y,σ⋆\)\}\.\\partial\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\\\{\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\\}\.Moreover, since all the functions involved are convex and Clarke regular, their Clarke subdifferentials coincide with their convex subdifferentials\. Hence

∂Cℓrob​\(θ;x,Y,σ⋆\)=\(1−ρ\)​∇θℓPL​\(θ;x,Y,σ⋆\)\+ρ​∂CM​\(θ;x,Y\)\.\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\,\\partial\_\{C\}M\(\\theta;x,Y\)\.This proves \(a\)\.

\(b\) Fix\(x,Y\)\(x,Y\)\. SinceSKS\_\{K\}is finite, the conditional expectation overσ⋆∼p⋆\(⋅∣x,Y\)\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)is a finite weighted sum:

Lrob​\(θ;x,Y\)=∑σ⋆∈SKp⋆​\(σ⋆∣x,Y\)​ℓrob​\(θ;x,Y,σ⋆\)\.L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)=\\sum\_\{\\sigma^\{\\star\}\\in S\_\{K\}\}p^\{\\star\}\(\\sigma^\{\\star\}\\mid x,Y\)\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\.By part \(a\), each summand is convex and Clarke regular\. The weightsp⋆​\(σ⋆∣x,Y\)p^\{\\star\}\(\\sigma^\{\\star\}\\mid x,Y\)are nonnegative, sum to one, and are independent ofθ\\theta\. ThereforeLrob​\(⋅;x,Y\)L\_\{\\mathrm\{rob\}\}\(\\cdot;x,Y\)is convex and finite\-valued, hence Clarke regular\.

For the subdifferential, the finite convex\-sum rule yields

∂Lrob​\(θ;x,Y\)=∑σ⋆∈SKp⋆​\(σ⋆∣x,Y\)​∂ℓrob​\(θ;x,Y,σ⋆\)\.\\partial L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)=\\sum\_\{\\sigma^\{\\star\}\\in S\_\{K\}\}p^\{\\star\}\(\\sigma^\{\\star\}\\mid x,Y\)\\partial\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\.Equivalently,

∂CLrob​\(θ;x,Y\)=∑σ⋆∈SKp⋆​\(σ⋆∣x,Y\)​∂Cℓrob​\(θ;x,Y,σ⋆\)\.\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)=\\sum\_\{\\sigma^\{\\star\}\\in S\_\{K\}\}p^\{\\star\}\(\\sigma^\{\\star\}\\mid x,Y\)\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\.Writing the finite weighted sum in expectation notation gives

∂CLrob​\(θ;x,Y\)=𝔼σ⋆∼p⋆\(⋅∣x,Y\)​\[∂Cℓrob​\(θ;x,Y,σ⋆\)\]\.\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)\}\[\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\]\.This is the finite\-dimensional Aumann identity in the present setting\. Since the label spaceSKS\_\{K\}is finite and the subgradients are uniformly bounded by[Lemma˜F\.3](https://arxiv.org/html/2607.01715#Thmlemma3), measurability and integrability are automatic\. This proves \(b\)\.

\(c\) Fix\(x,Y\)\(x,Y\)and define

q​\(θ\):=Pπθ​\(Y∣x\),L​\(θ\):=Lrob​\(θ;x,Y\)\.q\(\\theta\):=P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\),\\qquad L\(\\theta\):=L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\.Under[Assumption˜1](https://arxiv.org/html/2607.01715#Thmassumption1),qqis continuously differentiable inθ\\theta\. In fact, because the softmax policy assigns strictly positive probability to every response in the finite response set,

By part \(b\),LLis convex and Clarke regular\. Moreover,L​\(θ\)≥0L\(\\theta\)\\geq 0, sinceℓPL≥0\\ell\_\{\\mathrm\{PL\}\}\\geq 0andℓrob\\ell\_\{\\mathrm\{rob\}\}is a convex combination of nonnegative PL losses\.

We first verify Clarke regularity of the productq​LqL\. For any directiondd, sinceqqisC1C^\{1\}andLLis directionally differentiable and Clarke regular,

\(q​L\)′​\(θ;d\)=⟨∇q​\(θ\),d⟩​L​\(θ\)\+q​\(θ\)​L′​\(θ;d\)\.\(qL\)^\{\\prime\}\(\\theta;d\)=\\langle\\nabla q\(\\theta\),d\\rangle L\(\\theta\)\+q\(\\theta\)L^\{\\prime\}\(\\theta;d\)\.BecauseLLis Clarke regular,

L′​\(θ;d\)=maxv∈∂CL​\(θ\)⁡⟨v,d⟩\.L^\{\\prime\}\(\\theta;d\)=\\max\_\{v\\in\\partial\_\{C\}L\(\\theta\)\}\\langle v,d\\rangle\.Therefore

\(q​L\)′​\(θ;d\)=maxv∈∂CL​\(θ\)⁡⟨L​\(θ\)​∇q​\(θ\)\+q​\(θ\)​v,d⟩\.\(qL\)^\{\\prime\}\(\\theta;d\)=\\max\_\{v\\in\\partial\_\{C\}L\(\\theta\)\}\\left\\langle L\(\\theta\)\\nabla q\(\\theta\)\+q\(\\theta\)v,d\\right\\rangle\.The right\-hand side is the support function of the compact convex set

L​\(θ\)​∇q​\(θ\)\+q​\(θ\)​∂CL​\(θ\)\.L\(\\theta\)\\nabla q\(\\theta\)\+q\(\\theta\)\\partial\_\{C\}L\(\\theta\)\.Hence the Clarke directional derivative ofq​LqLagrees with its ordinary directional derivative, soq​LqLis Clarke regular\. Its Clarke subdifferential is exactly the set whose support function appears above:

∂C\(q​L\)​\(θ\)=L​\(θ\)​∇q​\(θ\)\+q​\(θ\)​∂CL​\(θ\)\.\\partial\_\{C\}\(qL\)\(\\theta\)=L\(\\theta\)\\nabla q\(\\theta\)\+q\(\\theta\)\\partial\_\{C\}L\(\\theta\)\.Substituting back

q​\(θ\)=Pπθ​\(Y∣x\),L​\(θ\)=Lrob​\(θ;x,Y\),q\(\\theta\)=P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\),\\qquad L\(\\theta\)=L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\),we obtain

∂C\[Pπθ​\(Y∣x\)​Lrob​\(θ;x,Y\)\]=Pπθ​\(Y∣x\)​∂CLrob​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​∇θPπθ​\(Y∣x\)\.\\partial\_\{C\}\\left\[P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\right\]=P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\nabla\_\{\\theta\}P\_\{\\pi\_\{\\theta\}\}\(Y\\mid x\)\.This proves \(c\)\. ∎

###### Lemma F\.8\(Score\-function identity\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold,JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}is locally Lipschitz onΘ\\Thetawith constant≤CG\+2​K​Bψ​CL\\leq C\_\{G\}\+2KB\_\{\\psi\}C\_\{L\}, and

∂CJrobon​\(θ\)⊇𝔼x​𝔼Y∼πθ⊗K​\[∂CLrob​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]\.\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\\;\\supseteq\\;\\mathbb\{E\}\_\{x\}\\,\\mathbb\{E\}\_\{Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\}\\big\[\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\big\]\.\(F\.3\)

###### Proof\.

Fixθ∈Θ\\theta\\in\\Theta\. The set𝒴\\mathcal\{Y\}is finite, and hence

Jrobon​\(θ\)=𝔼x∼𝒟x​\[H​\(θ;x\)\],J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)=\\mathbb\{E\}\_\{x\\sim\\mathcal\{D\}\_\{x\}\}\\left\[H\(\\theta;x\)\\right\],where

H​\(θ;x\):=∑Y∈𝒴KPθ​\(Y∣x\)​Lrob​\(θ;x,Y\),Pθ​\(Y∣x\):=πθ⊗K​\(Y∣x\)\.H\(\\theta;x\):=\\sum\_\{Y\\in\\mathcal\{Y\}^\{K\}\}P\_\{\\theta\}\(Y\\mid x\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\),\\qquad P\_\{\\theta\}\(Y\\mid x\):=\\pi\_\{\\theta\}^\{\\otimes K\}\(Y\\mid x\)\.WritingY=\(y1,…,yK\)Y=\(y\_\{1\},\\ldots,y\_\{K\}\), the policy\-induced list probability is

Pθ​\(Y∣x\)=∏i=1Kπθ​\(yi∣x\)\.P\_\{\\theta\}\(Y\\mid x\)=\\prod\_\{i=1\}^\{K\}\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)\.Therefore,

∇θlogPθ\(Y∣x\)=∑i=1K∇θlogπθ\(yi∣x\)=:Sθ\(x,Y\)\.\\nabla\_\{\\theta\}\\log P\_\{\\theta\}\(Y\\mid x\)=\\sum\_\{i=1\}^\{K\}\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)=:S\_\{\\theta\}\(x,Y\)\.SincePθ​\(Y∣x\)P\_\{\\theta\}\(Y\\mid x\)is continuously differentiable inθ\\theta, we obtain the score\-function identity

∇θPθ​\(Y∣x\)=Pθ​\(Y∣x\)​Sθ​\(x,Y\)\.\\nabla\_\{\\theta\}P\_\{\\theta\}\(Y\\mid x\)=P\_\{\\theta\}\(Y\\mid x\)S\_\{\\theta\}\(x,Y\)\.\(F\.4\)
We first establish the subdifferential inclusion\. FixxxandYY\. By construction,Lrob​\(θ;x,Y\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)is locally Lipschitz and Clarke regular inθ\\theta\. Let

V​\(θ;x,Y\)∈∂CLrob​\(θ;x,Y\)V\(\\theta;x,Y\)\\in\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)be an arbitrary measurable selection\. SincePθ​\(Y∣x\)P\_\{\\theta\}\(Y\\mid x\)isC1C^\{1\}, the Clarke product rule in[Lemma˜F\.7](https://arxiv.org/html/2607.01715#Thmlemma7)\(c\) gives

Pθ​\(Y∣x\)​V​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​∇θPθ​\(Y∣x\)∈∂C\(Pθ​\(Y∣x\)​Lrob​\(θ;x,Y\)\)\.P\_\{\\theta\}\(Y\\mid x\)V\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\nabla\_\{\\theta\}P\_\{\\theta\}\(Y\\mid x\)\\in\\partial\_\{C\}\\Big\(P\_\{\\theta\}\(Y\\mid x\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\Big\)\.Using \([F\.4](https://arxiv.org/html/2607.01715#A6.E4)\), this becomes

Pθ​\(Y∣x\)​\[V​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]∈∂C\(Pθ​\(Y∣x\)​Lrob​\(θ;x,Y\)\)\.P\_\{\\theta\}\(Y\\mid x\)\\Big\[V\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\Big\]\\in\\partial\_\{C\}\\Big\(P\_\{\\theta\}\(Y\\mid x\)L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\Big\)\.Since𝒴K\\mathcal\{Y\}^\{K\}is finite, we may sum overYY\. By the Clarke sum rule,

∑Y∈𝒴KPθ​\(Y∣x\)​\[V​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]∈∂CH​\(θ;x\)\.\\sum\_\{Y\\in\\mathcal\{Y\}^\{K\}\}P\_\{\\theta\}\(Y\\mid x\)\\Big\[V\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\Big\]\\in\\partial\_\{C\}H\(\\theta;x\)\.Equivalently,

𝔼Y∼πθ⊗K\(⋅∣x\)​\[V​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]∈∂CH​\(θ;x\)\.\\mathbb\{E\}\_\{Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\\mid x\)\}\\Big\[V\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\Big\]\\in\\partial\_\{C\}H\(\\theta;x\)\.
Now take expectation overx∼𝒟xx\\sim\\mathcal\{D\}\_\{x\}\. The selection above is measurable by the deterministic tie\-breaking rule used in the definition of the worst\-case ranking, and it is integrable by the uniform bounds on∂CLrob\\partial\_\{C\}L\_\{\\mathrm\{rob\}\},LrobL\_\{\\mathrm\{rob\}\}, andSθS\_\{\\theta\}\. Hence the Aumann expectation rule yields

𝔼x​𝔼Y∼πθ⊗K\(⋅∣x\)​\[V​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]∈∂CJrobon​\(θ\)\.\\mathbb\{E\}\_\{x\}\\mathbb\{E\}\_\{Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\\mid x\)\}\\Big\[V\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\Big\]\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\.SinceV​\(θ;x,Y\)V\(\\theta;x,Y\)was an arbitrary measurable selection from∂CLrob​\(θ;x,Y\)\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\), this proves the set\-valued inclusion

∂CJrobon​\(θ\)⊇𝔼x​𝔼Y∼πθ⊗K​\[∂CLrob​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]\.\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\\;\\supseteq\\;\\mathbb\{E\}\_\{x\}\\,\\mathbb\{E\}\_\{Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\}\\big\[\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\big\]\.
It remains to verify the stated Lipschitz bound\. By the PL gradient bound and the robust\-loss construction,

supV∈∂CLrob​\(θ;x,Y\)‖V‖≤CG\.\\sup\_\{V\\in\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\}\\\|V\\\|\\leq C\_\{G\}\.Moreover,

0≤Lrob​\(θ;x,Y\)≤CL,‖Sθ​\(x,Y\)‖≤2​K​Bψ\.0\\leq L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\\leq C\_\{L\},\\qquad\\\|S\_\{\\theta\}\(x,Y\)\\\|\\leq 2KB\_\{\\psi\}\.Therefore, every vector in the set

∂CLrob​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)has norm at most

CG\+2​K​Bψ​CL\.C\_\{G\}\+2KB\_\{\\psi\}C\_\{L\}\.Averaging overY∼πθ⊗K\(⋅∣x\)Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\\mid x\)and then overx∼𝒟xx\\sim\\mathcal\{D\}\_\{x\}preserves this bound\. Hence every element constructed in the right\-hand side of \([F\.3](https://arxiv.org/html/2607.01715#A6.E3)\) has norm at mostCG\+2​K​Bψ​CLC\_\{G\}\+2KB\_\{\\psi\}C\_\{L\}\. SinceJrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}is locally Lipschitz and its Clarke subgradients are uniformly bounded by this quantity,JrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}is locally Lipschitz onΘ\\Thetawith constant at most

CG\+2​K​Bψ​CL\.C\_\{G\}\+2KB\_\{\\psi\}C\_\{L\}\.The same argument holds on any bounded neighborhood ofΘ\\Theta, withDDinCLC\_\{L\}replaced by the corresponding radius on that neighborhood\. ∎

###### Definition F\.2\(Measurable selector and oracle\)\.

Fix any deterministic linear order⪯\\preceqonSKS\_\{K\}and letσsel​\(θ;x,Y\):=min⪯​arg​maxσ⁡ℓPL​\(θ;x,Y,σ\)\\sigma^\{\\mathrm\{sel\}\}\(\\theta;x,Y\):=\\min\_\{\\preceq\}\\operatorname\*\{arg\\,max\}\_\{\\sigma\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\), a Borel\-measurable selection computable by ascending sort with⪯\\preceqas tie\-breaking\. The per\-sample oracle is \([10](https://arxiv.org/html/2607.01715#S5.E10)\)\.

###### Lemma F\.9\(Oracle properties\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold:\(a\)G1∈∂Cℓrob​\(θ;x,Y,σ⋆\)G\_\{1\}\\in\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)for every\(θ,x,Y,σ⋆\)\(\\theta,x,Y,\\sigma^\{\\star\}\)\.\(b\)𝔼​\[G​\(θ;Z\)∣θ\]∈∂CJrobon​\(θ\)\\mathbb\{E\}\[G\(\\theta;Z\)\\mid\\theta\]\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\.\(c\)𝔼​\[‖G‖2∣θ\]≤Gtot2\\mathbb\{E\}\[\\\|G\\\|^\{2\}\\mid\\theta\]\\leq G\_\{\\mathrm\{tot\}\}^\{2\}, whereGtot2G\_\{\\mathrm\{tot\}\}^\{2\}is defined below\.

Gtot2:=2​CG2\+2​\(2​K​Bψ​CL\)2,CG:=2​K​Bψ,CL:=K​\(log⁡K\+2​D​Bψ\)\.G\_\{\\mathrm\{tot\}\}^\{2\}:=2C\_\{G\}^\{2\}\+2\(2KB\_\{\\psi\}C\_\{L\}\)^\{2\},\\qquad C\_\{G\}:=2KB\_\{\\psi\},\\qquad C\_\{L\}:=K\(\\log K\+2DB\_\{\\psi\}\)\.\(F\.5\)

###### Proof\.

Fixθ∈Θ\\theta\\in\\Theta\. Throughout the proof, all expectations are conditional on the current parameterθ\\thetaunless otherwise stated\. Recall that

G​\(θ;Z\)=G1​\(θ;Z\)\+G2​\(θ;Z\),G\(\\theta;Z\)=G\_\{1\}\(\\theta;Z\)\+G\_\{2\}\(\\theta;Z\),where

G1​\(θ;Z\):=\(1−ρ\)​∇θℓPL​\(θ;x,Y,σ⋆\)\+ρ​∇θℓPL​\(θ;x,Y,σsel\),G\_\{1\}\(\\theta;Z\):=\(1\-\\rho\)\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\mathrm\{sel\}\}\),and

G2​\(θ;Z\):=ℓrob​\(θ;x,Y,σ⋆\)​Sθ​\(x,Y\)\.G\_\{2\}\(\\theta;Z\):=\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)S\_\{\\theta\}\(x,Y\)\.
\(a\) Fix\(x,Y,σ⋆\)\(x,Y,\\sigma^\{\\star\}\)\. Define the pointwise worst\-case PL loss

M​\(θ;x,Y\):=maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\.M\(\\theta;x,Y\):=\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\.Then

ℓrob​\(θ;x,Y,σ⋆\)=\(1−ρ\)​ℓPL​\(θ;x,Y,σ⋆\)\+ρ​M​\(θ;x,Y\)\.\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)=\(1\-\\rho\)\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho M\(\\theta;x,Y\)\.SinceSKS\_\{K\}is finite and, for eachσ∈SK\\sigma\\in S\_\{K\},θ↦ℓPL​\(θ;x,Y,σ\)\\theta\\mapsto\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)is continuously differentiable, the functionM​\(θ;x,Y\)M\(\\theta;x,Y\)is locally Lipschitz and Clarke regular\. By the finite\-max Danskin theorem in[Lemma˜F\.6](https://arxiv.org/html/2607.01715#Thmlemma6),

∂CM​\(θ;x,Y\)=conv⁡\{∇θℓPL​\(θ;x,Y,σ\):σ∈arg⁡maxσ~∈SK⁡ℓPL​\(θ;x,Y,σ~\)\}\.\\partial\_\{C\}M\(\\theta;x,Y\)=\\operatorname\{conv\}\\left\\\{\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\):\\sigma\\in\\arg\\max\_\{\\tilde\{\\sigma\}\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\tilde\{\\sigma\}\)\\right\\\}\.By construction,σsel\\sigma^\{\\mathrm\{sel\}\}is selected from the active maximizer set:

σsel∈arg⁡maxσ∈SK⁡ℓPL​\(θ;x,Y,σ\)\.\\sigma^\{\\mathrm\{sel\}\}\\in\\arg\\max\_\{\\sigma\\in S\_\{K\}\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\.Therefore,

∇θℓPL​\(θ;x,Y,σsel\)∈∂CM​\(θ;x,Y\)\.\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\mathrm\{sel\}\}\)\\in\\partial\_\{C\}M\(\\theta;x,Y\)\.Using the Clarke sum rule in[Lemma˜F\.7](https://arxiv.org/html/2607.01715#Thmlemma7)\(a\), we obtain

\(1−ρ\)​∇θℓPL​\(θ;x,Y,σ⋆\)\+ρ​∇θℓPL​\(θ;x,Y,σsel\)∈∂Cℓrob​\(θ;x,Y,σ⋆\)\.\(1\-\\rho\)\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\+\\rho\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\mathrm\{sel\}\}\)\\in\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\.The left\-hand side is exactlyG1​\(θ;Z\)G\_\{1\}\(\\theta;Z\)\. Hence

G1​\(θ;Z\)∈∂Cℓrob​\(θ;x,Y,σ⋆\)\.G\_\{1\}\(\\theta;Z\)\\in\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\.This argument also covers ties: when the active maximizer set is not a singleton, the deterministic tie\-breaking rule selects one active maximizer, and every active gradient is a valid element of the Clarke subdifferential of the finite maximum\.

\(b\) For fixed\(x,Y\)\(x,Y\), define the conditional robust loss

Lrob​\(θ;x,Y\):=𝔼σ⋆∼p⋆\(⋅\|x,Y\)​\[ℓrob​\(θ;x,Y,σ⋆\)\]\.L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\):=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\\sim p^\{\\star\}\(\\cdot\|x,Y\)\}\\left\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\right\]\.SinceSKS\_\{K\}is finite, the expectation overσ⋆\\sigma^\{\\star\}is a finite sum\. Moreover, by part \(a\), for every possibleσ⋆\\sigma^\{\\star\},

G1​\(θ;x,Y,σ⋆\)∈∂Cℓrob​\(θ;x,Y,σ⋆\)\.G\_\{1\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\in\\partial\_\{C\}\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\.Therefore, by the Aumann expectation rule in[Lemma˜F\.7](https://arxiv.org/html/2607.01715#Thmlemma7)\(b\),

H\(θ;x,Y\):=𝔼σ⋆\[G1\(θ;Z\)\|x,Y,θ\]∈∂CLrob\(θ;x,Y\)\.H\(\\theta;x,Y\):=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\}\\left\[G\_\{1\}\(\\theta;Z\)\\,\\middle\|\\,x,Y,\\theta\\right\]\\in\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\.The measurability ofHHfollows from the deterministic tie\-breaking rule used to defineσsel\\sigma^\{\\mathrm\{sel\}\}, and integrability follows from the uniform gradient bound\.

Next, sinceSθ​\(x,Y\)S\_\{\\theta\}\(x,Y\)depends only on the policy\-generated list\(x,Y\)\(x,Y\)and not on the oracle labelσ⋆\\sigma^\{\\star\}, we have

𝔼σ⋆\[G2\(θ;Z\)\|x,Y,θ\]=𝔼σ⋆\[ℓrob\(θ;x,Y,σ⋆\)\|x,Y,θ\]Sθ\(x,Y\)=Lrob\(θ;x,Y\)Sθ\(x,Y\)\.\\mathbb\{E\}\_\{\\sigma^\{\\star\}\}\\left\[G\_\{2\}\(\\theta;Z\)\\,\\middle\|\\,x,Y,\\theta\\right\]=\\mathbb\{E\}\_\{\\sigma^\{\\star\}\}\\left\[\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\,\\middle\|\\,x,Y,\\theta\\right\]S\_\{\\theta\}\(x,Y\)=L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\.Combining the two conditional expectations gives

𝔼σ⋆\[G\(θ;Z\)\|x,Y,θ\]=H\(θ;x,Y\)\+Lrob\(θ;x,Y\)Sθ\(x,Y\),\\mathbb\{E\}\_\{\\sigma^\{\\star\}\}\\left\[G\(\\theta;Z\)\\,\\middle\|\\,x,Y,\\theta\\right\]=H\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\),with

H​\(θ;x,Y\)∈∂CLrob​\(θ;x,Y\)\.H\(\\theta;x,Y\)\\in\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)\.
Now take expectation overx∼𝒟xx\\sim\\mathcal\{D\}\_\{x\}andY∼πθ⊗K\(⋅\|x\)Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\(\\cdot\|x\)\. We obtain

𝔼​\[G​\(θ;Z\)∣θ\]=𝔼x,Y​\[H​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\],\\mathbb\{E\}\[G\(\\theta;Z\)\\mid\\theta\]=\\mathbb\{E\}\_\{x,Y\}\\left\[H\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\right\],whereH​\(θ;x,Y\)∈∂CLrob​\(θ;x,Y\)H\(\\theta;x,Y\)\\in\\partial\_\{C\}L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)is a measurable selection\. By the score\-function identity in[Lemma˜F\.8](https://arxiv.org/html/2607.01715#Thmlemma8),

𝔼x,Y​\[H​\(θ;x,Y\)\+Lrob​\(θ;x,Y\)​Sθ​\(x,Y\)\]∈∂CJrobon​\(θ\)\.\\mathbb\{E\}\_\{x,Y\}\\left\[H\(\\theta;x,Y\)\+L\_\{\\mathrm\{rob\}\}\(\\theta;x,Y\)S\_\{\\theta\}\(x,Y\)\\right\]\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\.Therefore,

𝔼​\[G​\(θ;Z\)∣θ\]∈∂CJrobon​\(θ\)\.\\mathbb\{E\}\[G\(\\theta;Z\)\\mid\\theta\]\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\.
\(c\) By the PL gradient bound, for every rankingσ∈SK\\sigma\\in S\_\{K\},

‖∇θℓPL​\(θ;x,Y,σ\)‖≤CG\.\\\|\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\\\|\\leq C\_\{G\}\.SinceG1G\_\{1\}is a convex combination of two PL gradients, we have

‖G1‖≤\(1−ρ\)​‖∇θℓPL​\(θ;x,Y,σ⋆\)‖\+ρ​‖∇θℓPL​\(θ;x,Y,σsel\)‖≤\(1−ρ\)​CG\+ρ​CG=CG\.\\\|G\_\{1\}\\\|\\leq\(1\-\\rho\)\\\|\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\\|\+\\rho\\\|\\nabla\_\{\\theta\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma^\{\\mathrm\{sel\}\}\)\\\|\\leq\(1\-\\rho\)C\_\{G\}\+\\rho C\_\{G\}=C\_\{G\}\.Moreover, by the uniform loss bound and the score\-function bound,

0≤ℓrob​\(θ;x,Y,σ⋆\)≤CL,‖Sθ​\(x,Y\)‖≤2​K​Bψ\.0\\leq\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)\\leq C\_\{L\},\\qquad\\\|S\_\{\\theta\}\(x,Y\)\\\|\\leq 2KB\_\{\\psi\}\.Hence

‖G2‖=‖ℓrob​\(θ;x,Y,σ⋆\)​Sθ​\(x,Y\)‖≤2​K​Bψ​CL\.\\\|G\_\{2\}\\\|=\\\|\\ell\_\{\\mathrm\{rob\}\}\(\\theta;x,Y,\\sigma^\{\\star\}\)S\_\{\\theta\}\(x,Y\)\\\|\\leq 2KB\_\{\\psi\}C\_\{L\}\.Using the elementary inequality

‖a\+b‖2≤2​‖a‖2\+2​‖b‖2,\\\|a\+b\\\|^\{2\}\\leq 2\\\|a\\\|^\{2\}\+2\\\|b\\\|^\{2\},we obtain the pointwise bound

‖G​\(θ;Z\)‖2≤2​‖G1‖2\+2​‖G2‖2≤2​CG2\+2​\(2​K​Bψ​CL\)2=Gtot2\.\\\|G\(\\theta;Z\)\\\|^\{2\}\\leq 2\\\|G\_\{1\}\\\|^\{2\}\+2\\\|G\_\{2\}\\\|^\{2\}\\leq 2C\_\{G\}^\{2\}\+2\(2KB\_\{\\psi\}C\_\{L\}\)^\{2\}=G\_\{\\mathrm\{tot\}\}^\{2\}\.Taking conditional expectation givenθ\\thetapreserves the bound:

𝔼​\[‖G​\(θ;Z\)‖2∣θ\]≤Gtot2\.\\mathbb\{E\}\[\\\|G\(\\theta;Z\)\\\|^\{2\}\\mid\\theta\]\\leq G\_\{\\mathrm\{tot\}\}^\{2\}\.This proves \(c\)\. ∎

### F\.5One\-step descent and convergence \([Theorem˜3](https://arxiv.org/html/2607.01715#Thmtheorem3),[Corollary˜1](https://arxiv.org/html/2607.01715#Thmcorollary1)\)

###### Lemma F\.10\(One\-step Moreau\-envelope descent\)\.

Suppose[Assumptions˜1](https://arxiv.org/html/2607.01715#Thmassumption1)and[2](https://arxiv.org/html/2607.01715#Thmassumption2)hold, the conditions of[Proposition˜2](https://arxiv.org/html/2607.01715#Thmproposition2),[Lemma˜F\.5](https://arxiv.org/html/2607.01715#Thmlemma5), and[Lemma˜F\.9](https://arxiv.org/html/2607.01715#Thmlemma9)hold\. Letλ^∈\(0,1/κ\)\\hat\{\\lambda\}\\in\(0,1/\\kappa\)\. Consider the iterates of[Algorithm˜3](https://arxiv.org/html/2607.01715#alg3),

G^t=1Bs​∑i=1BsG​\(θt;Zi\),θt\+1=ΠΘ​\(θt−η​G^t\),\\widehat\{G\}\_\{t\}=\\frac\{1\}\{B\_\{s\}\}\\sum\_\{i=1\}^\{B\_\{s\}\}G\(\\theta\_\{t\};Z\_\{i\}\),\\qquad\\theta\_\{t\+1\}=\\Pi\_\{\\Theta\}\(\\theta\_\{t\}\-\\eta\\widehat\{G\}\_\{t\}\),whereZi=\(xi,Yi,σi⋆\)Z\_\{i\}=\(x\_\{i\},Y\_\{i\},\\sigma\_\{i\}^\{\\star\}\)are sampled as in the algorithm\. Then, for everyη\>0\\eta\>0,

𝔼​\[Fλ^​\(θt\+1\)\|θt\]≤Fλ^​\(θt\)−η​\(1−κ​λ^\)2​‖∇Fλ^​\(θt\)‖2\+η2​Gtot22​λ^\.\\mathbb\{E\}\\big\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\\,\\big\|\\,\\theta\_\{t\}\\big\]\\leq F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\-\\frac\{\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\}\{2\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\+\\frac\{\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.\(F\.6\)In fact, the stronger bound with coefficientη​\(1−κ​λ^\)\\eta\(1\-\\kappa\\hat\{\\lambda\}\)instead ofη​\(1−κ​λ^\)/2\\eta\(1\-\\kappa\\hat\{\\lambda\}\)/2also holds\.

###### Proof\.

We first record the mini\-batch oracle properties\. Conditional onθt\\theta\_\{t\}, the samplesZ1,…,ZBsZ\_\{1\},\\ldots,Z\_\{B\_\{s\}\}are iid from the policy\-induced online sampling distribution\. By[Lemma˜F\.9](https://arxiv.org/html/2607.01715#Thmlemma9)\(b\),

𝔼​\[G​\(θt;Zi\)∣θt\]∈∂CJrobon​\(θt\)for every​i\.\\mathbb\{E\}\[G\(\\theta\_\{t\};Z\_\{i\}\)\\mid\\theta\_\{t\}\]\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\_\{t\}\)\\qquad\\text\{for every \}i\.Since the Clarke subdifferential is convex,

G¯t:=𝔼​\[G^t∣θt\]=1Bs​∑i=1Bs𝔼​\[G​\(θt;Zi\)∣θt\]∈∂CJrobon​\(θt\)\.\\bar\{G\}\_\{t\}:=\\mathbb\{E\}\[\\widehat\{G\}\_\{t\}\\mid\\theta\_\{t\}\]=\\frac\{1\}\{B\_\{s\}\}\\sum\_\{i=1\}^\{B\_\{s\}\}\\mathbb\{E\}\[G\(\\theta\_\{t\};Z\_\{i\}\)\\mid\\theta\_\{t\}\]\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\_\{t\}\)\.Moreover, by Jensen’s inequality and[Lemma˜F\.9](https://arxiv.org/html/2607.01715#Thmlemma9)\(c\),

𝔼\[∥G^t∥2∣θt\]=𝔼\[∥1Bs∑i=1BsG\(θt;Zi\)∥2\|θt\]≤1Bs∑i=1Bs𝔼\[∥G\(θt;Zi\)∥2∣θt\]≤Gtot2\.\\mathbb\{E\}\[\\\|\\widehat\{G\}\_\{t\}\\\|^\{2\}\\mid\\theta\_\{t\}\]=\\mathbb\{E\}\\left\[\\left\\\|\\frac\{1\}\{B\_\{s\}\}\\sum\_\{i=1\}^\{B\_\{s\}\}G\(\\theta\_\{t\};Z\_\{i\}\)\\right\\\|^\{2\}\\,\\middle\|\\,\\theta\_\{t\}\\right\]\\leq\\frac\{1\}\{B\_\{s\}\}\\sum\_\{i=1\}^\{B\_\{s\}\}\\mathbb\{E\}\[\\\|G\(\\theta\_\{t\};Z\_\{i\}\)\\\|^\{2\}\\mid\\theta\_\{t\}\]\\leq G\_\{\\mathrm\{tot\}\}^\{2\}\.
Let

θ^t:=θ^\(θt\)=proxλ^​F\(θt\)\.\\hat\{\\theta\}\_\{t\}:=\\hat\{\\theta\}\(\\theta\_\{t\}\)=\\operatorname\{prox\}\_\{\\hat\{\\lambda\}F\}\(\\theta\_\{t\}\)\.Sinceθ^t∈Θ\\hat\{\\theta\}\_\{t\}\\in\\Theta, it is feasible for the constrained Moreau envelope atθt\+1\\theta\_\{t\+1\}\. Therefore,

Fλ^​\(θt\+1\)≤Jrobon​\(θ^t\)\+12​λ^​‖θ^t−θt\+1‖2\.F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\\leq J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\hat\{\\theta\}\_\{t\}\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|\\hat\{\\theta\}\_\{t\}\-\\theta\_\{t\+1\}\\\|^\{2\}\.By definition ofθ^t\\hat\{\\theta\}\_\{t\},

Fλ^​\(θt\)=Jrobon​\(θ^t\)\+12​λ^​‖θ^t−θt‖2\.F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\hat\{\\theta\}\_\{t\}\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|\\hat\{\\theta\}\_\{t\}\-\\theta\_\{t\}\\\|^\{2\}\.Subtracting yields

Fλ^​\(θt\+1\)−Fλ^​\(θt\)≤12​λ^​\(‖θ^t−θt\+1‖2−‖θ^t−θt‖2\)\.F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\-F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\leq\\frac\{1\}\{2\\hat\{\\lambda\}\}\\left\(\\\|\\hat\{\\theta\}\_\{t\}\-\\theta\_\{t\+1\}\\\|^\{2\}\-\\\|\\hat\{\\theta\}\_\{t\}\-\\theta\_\{t\}\\\|^\{2\}\\right\)\.\(F\.7\)
BecauseΘ\\Thetais closed and convex, the projectionΠΘ\\Pi\_\{\\Theta\}is nonexpansive\. Sinceθ^t∈Θ\\hat\{\\theta\}\_\{t\}\\in\\Theta,

‖θt\+1−θ^t‖2=‖ΠΘ​\(θt−η​G^t\)−ΠΘ​\(θ^t\)‖2≤‖θt−η​G^t−θ^t‖2\.\\\|\\theta\_\{t\+1\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}=\\\|\\Pi\_\{\\Theta\}\(\\theta\_\{t\}\-\\eta\\widehat\{G\}\_\{t\}\)\-\\Pi\_\{\\Theta\}\(\\hat\{\\theta\}\_\{t\}\)\\\|^\{2\}\\leq\\\|\\theta\_\{t\}\-\\eta\\widehat\{G\}\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}\.Expanding the right\-hand side,

‖θt\+1−θ^t‖2≤‖θt−θ^t‖2−2​η​⟨G^t,θt−θ^t⟩\+η2​‖G^t‖2\.\\\|\\theta\_\{t\+1\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}\\leq\\\|\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}\-2\\eta\\langle\\widehat\{G\}\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\+\\eta^\{2\}\\\|\\widehat\{G\}\_\{t\}\\\|^\{2\}\.\(F\.8\)Combining \([F\.7](https://arxiv.org/html/2607.01715#A6.E7)\) and \([F\.8](https://arxiv.org/html/2607.01715#A6.E8)\), then taking conditional expectation givenθt\\theta\_\{t\}, gives

𝔼​\[Fλ^​\(θt\+1\)−Fλ^​\(θt\)∣θt\]\\displaystyle\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\-F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\mid\\theta\_\{t\}\]≤−ηλ^​⟨G¯t,θt−θ^t⟩\+η22​λ^​𝔼​\[‖G^t‖2∣θt\]\\displaystyle\\leq\-\\frac\{\\eta\}\{\\hat\{\\lambda\}\}\\langle\\bar\{G\}\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\+\\frac\{\\eta^\{2\}\}\{2\\hat\{\\lambda\}\}\\mathbb\{E\}\[\\\|\\widehat\{G\}\_\{t\}\\\|^\{2\}\\mid\\theta\_\{t\}\]≤−ηλ^​⟨G¯t,θt−θ^t⟩\+η2​Gtot22​λ^\.\\displaystyle\\leq\-\\frac\{\\eta\}\{\\hat\{\\lambda\}\}\\langle\\bar\{G\}\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\+\\frac\{\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.\(F\.9\)
It remains to lower bound⟨G¯t,θt−θ^t⟩\\langle\\bar\{G\}\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\. The proximal point satisfies

θ^t∈arg​minu∈Θ⁡\{Jrobon​\(u\)\+12​λ^​‖u−θt‖2\}\.\\hat\{\\theta\}\_\{t\}\\in\\operatorname\*\{arg\\,min\}\_\{u\\in\\Theta\}\\left\\\{J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(u\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\_\{t\}\\\|^\{2\}\\right\\\}\.The first\-order optimality condition overΘ\\Thetagives some

ζt∈∂CJrobon​\(θ^t\),nt∈NΘ​\(θ^t\),\\zeta\_\{t\}\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\hat\{\\theta\}\_\{t\}\),\\qquad n\_\{t\}\\in N\_\{\\Theta\}\(\\hat\{\\theta\}\_\{t\}\),such that

0=ζt\+1λ^​\(θ^t−θt\)\+nt\.0=\\zeta\_\{t\}\+\\frac\{1\}\{\\hat\{\\lambda\}\}\(\\hat\{\\theta\}\_\{t\}\-\\theta\_\{t\}\)\+n\_\{t\}\.Using the normal\-cone convention

⟨nt,u−θ^t⟩≤0,∀u∈Θ,\\langle n\_\{t\},u\-\\hat\{\\theta\}\_\{t\}\\rangle\\leq 0,\\qquad\\forall u\\in\\Theta,and takingu=θt∈Θu=\\theta\_\{t\}\\in\\Theta, we get

⟨ζt\+1λ^​\(θ^t−θt\),θt−θ^t⟩=−⟨nt,θt−θ^t⟩≥0\.\\left\\langle\\zeta\_\{t\}\+\\frac\{1\}\{\\hat\{\\lambda\}\}\(\\hat\{\\theta\}\_\{t\}\-\\theta\_\{t\}\),\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\right\\rangle=\-\\langle n\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\\geq 0\.Therefore,

⟨ζt,θt−θ^t⟩≥1λ^​‖θt−θ^t‖2\.\\langle\\zeta\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\\geq\\frac\{1\}\{\\hat\{\\lambda\}\}\\\|\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}\.\(F\.10\)
Byκ\\kappa\-weak convexity ofJrobonJ\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}, the function

Φ​\(θ\):=Jrobon​\(θ\)\+κ2​‖θ‖2\\Phi\(\\theta\):=J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\)\+\\frac\{\\kappa\}\{2\}\\\|\\theta\\\|^\{2\}is convex onΘ\\Theta\. Since

G¯t∈∂CJrobon​\(θt\),ζt∈∂CJrobon​\(θ^t\),\\bar\{G\}\_\{t\}\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\theta\_\{t\}\),\\qquad\\zeta\_\{t\}\\in\\partial\_\{C\}J\_\{\\mathrm\{rob\}\}^\{\\mathrm\{on\}\}\(\\hat\{\\theta\}\_\{t\}\),we have

G¯t\+κ​θt∈∂CΦ​\(θt\),ζt\+κ​θ^t∈∂CΦ​\(θ^t\)\.\\bar\{G\}\_\{t\}\+\\kappa\\theta\_\{t\}\\in\\partial\_\{C\}\\Phi\(\\theta\_\{t\}\),\\qquad\\zeta\_\{t\}\+\\kappa\\hat\{\\theta\}\_\{t\}\\in\\partial\_\{C\}\\Phi\(\\hat\{\\theta\}\_\{t\}\)\.By monotonicity of the convex subdifferential ofΦ\\Phi,

⟨\(G¯t\+κ​θt\)−\(ζt\+κ​θ^t\),θt−θ^t⟩≥0\.\\left\\langle\(\\bar\{G\}\_\{t\}\+\\kappa\\theta\_\{t\}\)\-\(\\zeta\_\{t\}\+\\kappa\\hat\{\\theta\}\_\{t\}\),\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\right\\rangle\\geq 0\.Equivalently,

⟨G¯t−ζt,θt−θ^t⟩≥−κ​‖θt−θ^t‖2\.\\langle\\bar\{G\}\_\{t\}\-\\zeta\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\\geq\-\\kappa\\\|\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}\.\(F\.11\)Adding \([F\.10](https://arxiv.org/html/2607.01715#A6.E10)\) and \([F\.11](https://arxiv.org/html/2607.01715#A6.E11)\),

⟨G¯t,θt−θ^t⟩≥\(1λ^−κ\)​‖θt−θ^t‖2=1−κ​λ^λ^​‖θt−θ^t‖2\.\\langle\\bar\{G\}\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\\geq\\left\(\\frac\{1\}\{\\hat\{\\lambda\}\}\-\\kappa\\right\)\\\|\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}=\\frac\{1\-\\kappa\\hat\{\\lambda\}\}\{\\hat\{\\lambda\}\}\\\|\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}\.\(F\.12\)
By[Lemma˜F\.5](https://arxiv.org/html/2607.01715#Thmlemma5),

∇Fλ^​\(θt\)=λ^−1​\(θt−θ^t\)\.\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)=\\hat\{\\lambda\}^\{\-1\}\(\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\)\.Hence

‖θt−θ^t‖2=λ^2​‖∇Fλ^​\(θt\)‖2\.\\\|\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\\|^\{2\}=\\hat\{\\lambda\}^\{2\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\.Substituting into \([F\.12](https://arxiv.org/html/2607.01715#A6.E12)\) gives

⟨G¯t,θt−θ^t⟩≥\(1−κ​λ^\)​λ^​‖∇Fλ^​\(θt\)‖2\.\\langle\\bar\{G\}\_\{t\},\\theta\_\{t\}\-\\hat\{\\theta\}\_\{t\}\\rangle\\geq\(1\-\\kappa\\hat\{\\lambda\}\)\\hat\{\\lambda\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\.Plugging this into \([F\.9](https://arxiv.org/html/2607.01715#A6.E9)\), we obtain

𝔼​\[Fλ^​\(θt\+1\)−Fλ^​\(θt\)∣θt\]≤−η​\(1−κ​λ^\)​‖∇Fλ^​\(θt\)‖2\+η2​Gtot22​λ^\.\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\-F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\mid\\theta\_\{t\}\]\\leq\-\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\+\\frac\{\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.Since1−κ​λ^\>01\-\\kappa\\hat\{\\lambda\}\>0, this stronger bound implies \([F\.6](https://arxiv.org/html/2607.01715#A6.E6)\)\. ∎

###### Proof of[Theorem˜3](https://arxiv.org/html/2607.01715#Thmtheorem3)\.

IfΔ0=0\\Delta\_\{0\}=0, thenFλ^​\(θ0\)=FinfF\_\{\\hat\{\\lambda\}\}\(\\theta\_\{0\}\)=F\_\{\\inf\}, and the desired bound is trivial\. Hence assumeΔ0\>0\\Delta\_\{0\}\>0\. By[Lemma˜F\.10](https://arxiv.org/html/2607.01715#Thmlemma10), for everyt=0,…,T−1t=0,\\ldots,T\-1,

𝔼\[Fλ^\(θt\+1\)\|θt\]≤Fλ^\(θt\)−η​\(1−κ​λ^\)2∥∇Fλ^\(θt\)∥2\+η2​Gtot22​λ^\.\\mathbb\{E\}\\\!\\left\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\\,\\middle\|\\,\\theta\_\{t\}\\right\]\\leq F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\-\\frac\{\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\}\{2\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\+\\frac\{\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.Taking total expectation gives

𝔼​\[Fλ^​\(θt\+1\)\]≤𝔼​\[Fλ^​\(θt\)\]−η​\(1−κ​λ^\)2​𝔼​‖∇Fλ^​\(θt\)‖2\+η2​Gtot22​λ^\.\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\]\\leq\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\]\-\\frac\{\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\}\{2\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\+\\frac\{\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.Rearranging,

η​\(1−κ​λ^\)2​𝔼​‖∇Fλ^​\(θt\)‖2≤𝔼​\[Fλ^​\(θt\)\]−𝔼​\[Fλ^​\(θt\+1\)\]\+η2​Gtot22​λ^\.\\frac\{\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\}\{2\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\]\-\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\+1\}\)\]\+\\frac\{\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.Summing overt=0,…,T−1t=0,\\ldots,T\-1, we obtain

η​\(1−κ​λ^\)2​∑t=0T−1𝔼​‖∇Fλ^​\(θt\)‖2≤Fλ^​\(θ0\)−𝔼​\[Fλ^​\(θT\)\]\+T​η2​Gtot22​λ^\.\\frac\{\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\}\{2\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{0\}\)\-\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{T\}\)\]\+\\frac\{T\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.Since

Fλ^​\(θ\)=minu⁡\{F​\(u\)\+12​λ^​‖u−θ‖2\}≥infuF​\(u\)=Finf,F\_\{\\hat\{\\lambda\}\}\(\\theta\)=\\min\_\{u\}\\left\\\{F\(u\)\+\\frac\{1\}\{2\\hat\{\\lambda\}\}\\\|u\-\\theta\\\|^\{2\}\\right\\\}\\geq\\inf\_\{u\}F\(u\)=F\_\{\\inf\},we have

Fλ^​\(θ0\)−𝔼​\[Fλ^​\(θT\)\]≤Fλ^​\(θ0\)−Finf=Δ0\.F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{0\}\)\-\\mathbb\{E\}\[F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{T\}\)\]\\leq F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{0\}\)\-F\_\{\\inf\}=\\Delta\_\{0\}\.Therefore,

η​\(1−κ​λ^\)2​∑t=0T−1𝔼​‖∇Fλ^​\(θt\)‖2≤Δ0\+T​η2​Gtot22​λ^\.\\frac\{\\eta\(1\-\\kappa\\hat\{\\lambda\}\)\}\{2\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq\\Delta\_\{0\}\+\\frac\{T\\eta^\{2\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.Dividing byη​T\\eta Tgives

1−κ​λ^2⋅1T​∑t=0T−1𝔼​‖∇Fλ^​\(θt\)‖2≤Δ0η​T\+η​Gtot22​λ^\.\\frac\{1\-\\kappa\\hat\{\\lambda\}\}\{2\}\\cdot\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq\\frac\{\\Delta\_\{0\}\}\{\\eta T\}\+\\frac\{\\eta G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}\.With the choice

η=2​λ^​Δ0Gtot2​T,\\eta=\\sqrt\{\\frac\{2\\hat\{\\lambda\}\\Delta\_\{0\}\}\{G\_\{\\mathrm\{tot\}\}^\{2\}T\}\},the two terms on the right\-hand side are equal:

Δ0η​T=η​Gtot22​λ^=Δ0​Gtot22​λ^​T\.\\frac\{\\Delta\_\{0\}\}\{\\eta T\}=\\frac\{\\eta G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}\}=\\sqrt\{\\frac\{\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{2\\hat\{\\lambda\}T\}\}\.Hence

1−κ​λ^2⋅1T​∑t=0T−1𝔼​‖∇Fλ^​\(θt\)‖2≤2​Δ0​Gtot2λ^​T\.\\frac\{1\-\\kappa\\hat\{\\lambda\}\}\{2\}\\cdot\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\\leq\\sqrt\{\\frac\{2\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\hat\{\\lambda\}T\}\}\.LetR∼Uniform​\{0,…,T−1\}R\\sim\\mathrm\{Uniform\}\\\{0,\\ldots,T\-1\\\}be sampled independently of the algorithmic randomness\. Then

𝔼​‖∇Fλ^​\(θR\)‖2=1T​∑t=0T−1𝔼​‖∇Fλ^​\(θt\)‖2\.\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{R\}\)\\\|^\{2\}=\\frac\{1\}\{T\}\\sum\_\{t=0\}^\{T\-1\}\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{t\}\)\\\|^\{2\}\.Therefore,

𝔼​‖∇Fλ^​\(θR\)‖2≤21−κ​λ^​2​Δ0​Gtot2λ^​T,\\mathbb\{E\}\\\|\\nabla F\_\{\\hat\{\\lambda\}\}\(\\theta\_\{R\}\)\\\|^\{2\}\\leq\\frac\{2\}\{1\-\\kappa\\hat\{\\lambda\}\}\\sqrt\{\\frac\{2\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\hat\{\\lambda\}T\}\},which is exactly \([11](https://arxiv.org/html/2607.01715#S5.E11)\)\. ∎

###### Proof of[Corollary˜1](https://arxiv.org/html/2607.01715#Thmcorollary1)\.

By[Theorem˜3](https://arxiv.org/html/2607.01715#Thmtheorem3), it suffices to require

21−κ​λ^​2​Δ0​Gtot2λ^​T≤ε\.\\frac\{2\}\{1\-\\kappa\\hat\{\\lambda\}\}\\sqrt\{\\frac\{2\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\hat\{\\lambda\}T\}\}\\leq\\varepsilon\.Squaring both sides gives

4\(1−κ​λ^\)2⋅2​Δ0​Gtot2λ^​T≤ε2\.\\frac\{4\}\{\(1\-\\kappa\\hat\{\\lambda\}\)^\{2\}\}\\cdot\\frac\{2\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\hat\{\\lambda\}T\}\\leq\\varepsilon^\{2\}\.Equivalently,

T≥8​Δ0​Gtot2λ^​\(1−κ​λ^\)2​ε2\.T\\geq\\frac\{8\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\hat\{\\lambda\}\(1\-\\kappa\\hat\{\\lambda\}\)^\{2\}\\varepsilon^\{2\}\}\.This proves the first claim\.

Now set

λ^=12​κ\.\\hat\{\\lambda\}=\\frac\{1\}\{2\\kappa\}\.Then

1−κ​λ^=12,1\-\\kappa\\hat\{\\lambda\}=\\frac\{1\}\{2\},and hence

T≥8​Δ0​Gtot2\(1/\(2​κ\)\)​\(1/2\)2​ε2=64​κ​Δ0​Gtot2ε2\.T\\geq\\frac\{8\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\(1/\(2\\kappa\)\)\(1/2\)^\{2\}\\varepsilon^\{2\}\}=\\frac\{64\\kappa\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\varepsilon^\{2\}\}\.Therefore,

T=O​\(κ​Δ0​Gtot2ε2\)\.T=O\\left\(\\frac\{\\kappa\\Delta\_\{0\}G\_\{\\mathrm\{tot\}\}^\{2\}\}\{\\varepsilon^\{2\}\}\\right\)\.
It remains to substitute the explicit constants\. From \([9](https://arxiv.org/html/2607.01715#S5.E9)\),

κ=K2​Bψ2​\(8\+log⁡K\+2​D​Bψ\)=O~​\(K2​Bψ2​\(log⁡K\+D​Bψ\)\)\.\\kappa=K^\{2\}B\_\{\\psi\}^\{2\}\(8\+\\log K\+2DB\_\{\\psi\}\)=\\tilde\{O\}\\\!\\left\(K^\{2\}B\_\{\\psi\}^\{2\}\(\\log K\+DB\_\{\\psi\}\)\\right\)\.From \([F\.5](https://arxiv.org/html/2607.01715#A6.E5)\),

Gtot2=2​CG2\+2​\(2​K​Bψ​CL\)2,G\_\{\\mathrm\{tot\}\}^\{2\}=2C\_\{G\}^\{2\}\+2\(2KB\_\{\\psi\}C\_\{L\}\)^\{2\},where

CG=2​K​Bψ,CL=K​\(log⁡K\+2​D​Bψ\)\.C\_\{G\}=2KB\_\{\\psi\},\\qquad C\_\{L\}=K\(\\log K\+2DB\_\{\\psi\}\)\.Thus

2​CG2=O​\(K2​Bψ2\),2C\_\{G\}^\{2\}=O\(K^\{2\}B\_\{\\psi\}^\{2\}\),and

2​\(2​K​Bψ​CL\)2=O​\(K2​Bψ2⋅K2​\(log⁡K\+D​Bψ\)2\)=O​\(K4​Bψ2​\(log⁡K\+D​Bψ\)2\)\.2\(2KB\_\{\\psi\}C\_\{L\}\)^\{2\}=O\\\!\\left\(K^\{2\}B\_\{\\psi\}^\{2\}\\cdot K^\{2\}\(\\log K\+DB\_\{\\psi\}\)^\{2\}\\right\)=O\\\!\\left\(K^\{4\}B\_\{\\psi\}^\{2\}\(\\log K\+DB\_\{\\psi\}\)^\{2\}\\right\)\.The second term dominates for the relevant asymptotic regime, so

Gtot2=O​\(K4​Bψ2​\(log⁡K\+D​Bψ\)2\)\.G\_\{\\mathrm\{tot\}\}^\{2\}=O\\\!\\left\(K^\{4\}B\_\{\\psi\}^\{2\}\(\\log K\+DB\_\{\\psi\}\)^\{2\}\\right\)\.Combining the bounds gives

κ​Gtot2=O~​\(K6​Bψ4​\(log⁡K\+D​Bψ\)3\)\.\\kappa G\_\{\\mathrm\{tot\}\}^\{2\}=\\tilde\{O\}\\\!\\left\(K^\{6\}B\_\{\\psi\}^\{4\}\(\\log K\+DB\_\{\\psi\}\)^\{3\}\\right\)\.Therefore,

T=O~​\(K6​Bψ4​\(log⁡K\+D​Bψ\)3ε2\),T=\\tilde\{O\}\\\!\\left\(\\frac\{K^\{6\}B\_\{\\psi\}^\{4\}\(\\log K\+DB\_\{\\psi\}\)^\{3\}\}\{\\varepsilon^\{2\}\}\\right\),which proves \([12](https://arxiv.org/html/2607.01715#S5.E12)\)\. ∎

## Appendix GSAIL bilevel reduction \(used in[Section˜5\.2](https://arxiv.org/html/2607.01715#S5.SS2)\)

For online alignment with KL\-regularized RLHF, the lower\-level policy induced by a rewardrrisπr⋆​\(y∣x\)=πref​\(y∣x\)​exp⁡\(r​\(x,y\)/β\)/Zr​\(x\)\\pi\_\{r\}^\{\\star\}\(y\\mid x\)=\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\\exp\(r\(x,y\)/\\beta\)/Z\_\{r\}\(x\)\. Substituting into the PL likelihood \([1](https://arxiv.org/html/2607.01715#S2.E1)\) and noting thatlog⁡Zr​\(x\)\\log Z\_\{r\}\(x\)is a per\-prompt constant that cancels at every PL stage, the bilevel objectiveminr⁡𝔼x,Y∼\(πr⋆\)⊗K,σ∼p⋆​\[ℓPL​\(r;x,Y,σ\)\]\\min\_\{r\}\\,\\mathbb\{E\}\_\{x,Y\\sim\(\\pi\_\{r\}^\{\\star\}\)^\{\\otimes K\},\\sigma\\sim p^\{\\star\}\}\[\\ell\_\{\\mathrm\{PL\}\}\(r;x,Y,\\sigma\)\]subject to the lower\-level optimality is exactly equivalent to the single\-level objectiveminπ⁡𝔼x,Y∼π⊗K,σ∼p⋆​\[ℓPL​\(π;x,Y,σ\)\]\\min\_\{\\pi\}\\,\\mathbb\{E\}\_\{x,Y\\sim\\pi^\{\\otimes K\},\\sigma\\sim p^\{\\star\}\}\[\\ell\_\{\\mathrm\{PL\}\}\(\\pi;x,Y,\\sigma\)\], which is similar to SAIL\[Dinget al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib15)\]\. Parameterizingπ=πθ\\pi=\\pi\_\{\\theta\}withsθ​\(x,y\)=log⁡\[πθ​\(y∣x\)/πref​\(y∣x\)\]s\_\{\\theta\}\(x,y\)=\\log\[\\pi\_\{\\theta\}\(y\\mid x\)/\\pi\_\{\\mathrm\{ref\}\}\(y\\mid x\)\]recovers \([8](https://arxiv.org/html/2607.01715#S5.E8)\)\. AtK=2K=2this reduces exactly to the pairwise SAIL/DPO objective\.

#### Gradient\.

DefiningL​\(θ;x,Y\):=𝔼σ∼p⋆\(⋅∣x,Y\)​\[ℓPL​\(θ;x,Y,σ\)\]L\(\\theta;x,Y\):=\\mathbb\{E\}\_\{\\sigma\\sim p^\{\\star\}\(\\cdot\\mid x,Y\)\}\[\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,Y,\\sigma\)\], the score\-function gradient identity gives

∇θJ​\(θ\)=𝔼x​𝔼Y∼πθ⊗K​\[∇θL​\(θ;x,Y\)\+L​\(θ;x,Y\)​∑i=1K∇θlog⁡πθ​\(yi∣x\)\],\\nabla\_\{\\theta\}J\(\\theta\)=\\mathbb\{E\}\_\{x\}\\mathbb\{E\}\_\{Y\\sim\\pi\_\{\\theta\}^\{\\otimes K\}\}\\Big\[\\nabla\_\{\\theta\}L\(\\theta;x,Y\)\+L\(\\theta;x,Y\)\\sum\_\{i=1\}^\{K\}\\nabla\_\{\\theta\}\\log\\pi\_\{\\theta\}\(y\_\{i\}\\mid x\)\\Big\],\(G\.13\)which combined with[Lemma˜1](https://arxiv.org/html/2607.01715#Thmlemma1)produces the per\-sample stochastic oracle \([10](https://arxiv.org/html/2607.01715#S5.E10)\)\.

## Appendix HStagewise PL Hessian decomposition

###### Proposition H\.2\(Stagewise PL Hessian as a sum of conditional covariances\)\.

Fix\(x,y1:K,σ⋆\)\(x,y\_\{1:K\},\\sigma^\{\\star\}\)and letsθ​\(x,y\)=θ⊤​ϕ​\(x,y\)s\_\{\\theta\}\(x,y\)=\\theta^\{\\top\}\\phi\(x,y\)\. For each stagei=1,…,K−1i=1,\\dots,K\-1define the remaining setRi:=\{σi⋆,…,σK⋆\}R\_\{i\}:=\\\{\\sigma^\{\\star\}\_\{i\},\\dots,\\sigma^\{\\star\}\_\{K\}\\\}, the stagewise softmaxpiθ​\(j\)=esθ​\(x,yj\)/∑m∈Riesθ​\(x,ym\)p\_\{i\}^\{\\theta\}\(j\)=e^\{s\_\{\\theta\}\(x,y\_\{j\}\)\}/\\sum\_\{m\\in R\_\{i\}\}e^\{s\_\{\\theta\}\(x,y\_\{m\}\)\}, and the stagewise feature meanμi​\(θ\)=∑j∈Ripiθ​\(j\)​ϕ​\(x,yj\)\\mu\_\{i\}\(\\theta\)=\\sum\_\{j\\in R\_\{i\}\}p\_\{i\}^\{\\theta\}\(j\)\\phi\(x,y\_\{j\}\)\. Then

∇θ2ℓPL​\(θ;x,y1:K,σ⋆\)=∑i=1K−1Covj∼piθ​\[ϕ​\(x,yj\)\]⪰0\.\\nabla\_\{\\theta\}^\{2\}\\ell\_\{\\mathrm\{PL\}\}\(\\theta;x,y\_\{1:K\},\\sigma^\{\\star\}\)=\\sum\_\{i=1\}^\{K\-1\}\\mathrm\{Cov\}\_\{j\\sim p\_\{i\}^\{\\theta\}\}\[\\phi\(x,y\_\{j\}\)\]\\;\\succeq\\;0\.

###### Proof\.

WriteℓPL=∑i=1K−1ℓi\\ell\_\{\\mathrm\{PL\}\}=\\sum\_\{i=1\}^\{K\-1\}\\ell\_\{i\}whereℓi​\(θ\)=−θ⊤​ϕ​\(x,yσi⋆\)\+log​∑j∈Riexp⁡\(θ⊤​ϕ​\(x,yj\)\)\\ell\_\{i\}\(\\theta\)=\-\\theta^\{\\top\}\\phi\(x,y\_\{\\sigma^\{\\star\}\_\{i\}\}\)\+\\log\\sum\_\{j\\in R\_\{i\}\}\\exp\(\\theta^\{\\top\}\\phi\(x,y\_\{j\}\)\)is the negative log\-likelihood of theii\-th stagewise multinomial choice\. Differentiating gives∇ℓi=−ϕ​\(x,yσi⋆\)\+μi\\nabla\\ell\_\{i\}=\-\\phi\(x,y\_\{\\sigma^\{\\star\}\_\{i\}\}\)\+\\mu\_\{i\}and∇2ℓi=∑j∈Ripiθ​\(j\)​\(ϕ​\(x,yj\)−μi\)​\(ϕ​\(x,yj\)−μi\)⊤=Covj∼piθ​\[ϕ​\(x,yj\)\]⪰0\\nabla^\{2\}\\ell\_\{i\}=\\sum\_\{j\\in R\_\{i\}\}p\_\{i\}^\{\\theta\}\(j\)\(\\phi\(x,y\_\{j\}\)\-\\mu\_\{i\}\)\(\\phi\(x,y\_\{j\}\)\-\\mu\_\{i\}\)^\{\\top\}=\\mathrm\{Cov\}\_\{j\\sim p\_\{i\}^\{\\theta\}\}\[\\phi\(x,y\_\{j\}\)\]\\succeq 0\. Sum overii\. ∎

## Appendix IAdditional Experimental Results

This appendix contains additional experimental results omitted from the main text for space\. Appendix[I\.1](https://arxiv.org/html/2607.01715#A9.SS1)reports clean\-label external evaluation on RewardBench\. Appendix[I\.2](https://arxiv.org/html/2607.01715#A9.SS2)gives the hyperparameter sweeps used to select the pairwise robust\-DPO baselines in Table[2](https://arxiv.org/html/2607.01715#S6.T2)\. Appendix[I\.3](https://arxiv.org/html/2607.01715#A9.SS3)reports robustness\-radius sensitivity for Robust PL, including both noisy\-label and clean\-label settings\. Appendix[I\.4](https://arxiv.org/html/2607.01715#A9.SS4)provide full offline metrics on Qwen2\.5\.

### I\.1Clean\-label external evaluation

[Table˜I\.1](https://arxiv.org/html/2607.01715#A9.T1)evaluates clean\-label offline models on RewardBench\. The purpose is not to claim general benchmark dominance, but to check whether the robust correction damages external alignment quality\. The results support the main\-text claim that moderate robustification preserves clean\-label model quality\.

Table I\.1:External RewardBench evaluation for clean\-label offline models\. We report category\-level accuracy and the average score over Chat, Chat\-Hard, Safety, and Reasoning\.
### I\.2Hyperparameter Selection for Pairwise Robust\-DPO Baselines

For the pairwise robust\-DPO baselines in Table[2](https://arxiv.org/html/2607.01715#S6.T2), we select hyperparameters on a moderate noisy development condition: Qwen3\-0\.6B under near\-tie noise withϵ=0\.4\\epsilon=0\.4\. This condition is noisy enough to test robustness, but less extreme than top\-rankϵ=1\.0\\epsilon=1\.0, where loss\-level reweighting can directly amplify systematically corrupted pairs\. Both TV\-DR\-DPO and KLDPO are implemented in our pipeline following their loss\-level robust DPO objectives\. The selected values areρ=0\.10\\rho=0\.10for TV\-DR\-DPO andτ=1\.00\\tau=1\.00for KLDPO\.

#### Noisy development sweeps\.

Tables[I\.2](https://arxiv.org/html/2607.01715#A9.T2)and[I\.3](https://arxiv.org/html/2607.01715#A9.T3)report the hyperparameter sweeps used for selection\. For TV\-DR\-DPO,ρ=0\.10\\rho=0\.10gives the best Kendall’sτ\\tau\. For KLDPO,τ=1\.00\\tau=1\.00gives the best Kendall’sτ\\tauand NDCG, while smaller temperatures over\-concentrate on high\-loss samples\.

Table I\.2:Hyperparameter sweep for TV\-DR\-DPO on Qwen3\-0\.6B under near\-tie 0\.4 noise\.Table I\.3:Hyperparameter sweep for KLDPO on Qwen3\-0\.6B under near\-tie 0\.4 noise\.
#### Clean\-label sanity checks\.

The noisy development sweeps above are used for hyperparameter selection\. For completeness, Tables[I\.4](https://arxiv.org/html/2607.01715#A9.T4)and[I\.5](https://arxiv.org/html/2607.01715#A9.T5)report clean\-label sweeps for the same pairwise robust\-DPO baselines\. These clean sweeps are not used to select the main\-table hyperparameters\. Instead, they diagnose whether aggressive loss\-level reweighting harms performance when the observed labels are reliable\. The results show that large TV radii and small KL temperatures over\-concentrate on high\-loss samples and degrade clean ranking quality\.

Table I\.4:Clean\-label sanity sweep for TV\-DR\-DPO on Qwen3\-0\.6B\.Table I\.5:Clean\-label sanity sweep for KLDPO on Qwen3\-0\.6B\.Together, these sweeps show that the useful regime for pairwise robust\-DPO baselines is mild loss\-level reweighting\. More aggressive settings, such as largeρ\\rhofor TV\-DR\-DPO or smallτ\\taufor KLDPO, can overemphasize high\-loss comparisons and hurt ranking quality even when the training labels are clean\.

### I\.3Sensitivity of Robust PL to the Robustness Radius

To make the effect of the robustness strength explicit, we report full sweeps over the robustness coefficientρ\\rho\. Small positive values can stabilize learning under ranking\-label noise, while large values place excessive weight on the adversarial ranking in \([5](https://arxiv.org/html/2607.01715#S3.E5)\)\.

Table I\.6:Sensitivity analysis of robustness coefficientρ\\rhoon Qwen3\-0\.6B under near\-tie 0\.4 noise\.Table I\.7:Sensitivity analysis of robustness coefficientρ\\rhoon Qwen3\-0\.6B under clean labels\.
### I\.4Additional Qwen2\.5 offline fixed\-list results

[Table˜I\.9](https://arxiv.org/html/2607.01715#A9.T9)and[Table˜I\.10](https://arxiv.org/html/2607.01715#A9.T10)report full offline metrics on Qwen2\.5\. These results provide metric\-level support for the main\-text offline story: clean\-label performance is preserved, while robust PL becomes more useful when ranking labels are corrupted\.

Table I\.8:Sensitivity analysis of robustness coefficientρ\\rhoon Qwen2\.5\-0\.5B under top\-rank 0\.4 noise\.Table I\.9:Main results on Qwen2\.5\-0\.5B across clean and noisy settings\. Best results within each row are bold among available values\.Table I\.10:Results on Qwen2\.5\-7B under clean and noisy settings\.

## Appendix JExperimental Details

### J\.1Models, Data, and Evaluation Setup

#### Base Models\.

We report below the HuggingFace repositories of the base language models adopted throughout our experiments:

- •
- •

#### Datasets\.

#### Reward Model\.

For online reward\-based optimization and offline reward evaluation, we useopenbmb/Eurus\-RM\-7bas the frozen reward model throughout our RLHF experiments\. The reward model is used*as\-is*, without any additional fine\-tuning in our work:[https://huggingface\.co/openbmb/Eurus\-RM\-7b](https://huggingface.co/openbmb/Eurus-RM-7b)\.

All RLHF experiments were conducted on a cluster of 8 NVIDIA RTX 4090 GPUs\. The approximate training time is about 1 hour per model in the offline setting and about 3 hours per model in the online setting\.

### J\.2Prompt Templates

We describe below the prompt template used in our experiments for offline evaluation on the UltraFeedback dataset\[Cuiet al\.,[2024](https://arxiv.org/html/2607.01715#bib.bib23)\]\. Following the dataset authors, we adopt the official evaluation prompt template provided withUltraFeedback, which is also used during dataset construction\.

In our setting, the prompt is designed to elicit detailed and constructive feedback for a given model response, along with an overall quality score\. The evaluation focuses on multiple aspects of response quality, including helpfulness, truthfulness, honesty, and adherence to the given instruction\. We use this prompt template consistently across all methods to ensure a fair and controlled comparison\.

### J\.3Assets and licenses\.

We use only publicly available datasets and models\. Table[J\.11](https://arxiv.org/html/2607.01715#A10.T11)summarizes the main assets used in our experiments, together with their licenses or terms of use where applicable\.

Table J\.11:Main existing assets used in the experiments\.

## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: The main claims in the abstract and introduction accurately reflect the paper’s scope: a pointwise\-TV robust listwise PL objective, an exact worst\-case ranking computation, offline/online optimization theory, and empirical evaluation in offline and online LLM alignment settings\. The assumptions and scope of the theoretical results are stated in the theory sections, and the experimental claims are restricted to the reported UltraFeedback\-based settings\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: The paper discusses limitations in the conclusion, including the need to tune the robustness radiusρ\\rho, the restriction to the studied ambiguity model, and the need for larger\-scale online alignment experiments\. The theoretical sections also state the log\-linear and bounded\-domain assumptions under which the guarantees are proved\.
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12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
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23. Answer:\[Yes\]
24. Justification: The paper provides anonymized code repositories for the offline and online experiments and uses publicly available datasets and models\. The appendix describes the data, base models, reward model, and evaluation setup used in the experiments\.
25. Guidelines: - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: The paper specifies the base models, dataset, reward model, offline and online evaluation settings, ranking metrics, GPT\-as\-judge evaluation protocol, and robustness/noise settings\. Further implementation and prompt\-template details are provided in the appendix\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: We do not report multi\-seed error bars or formal statistical significance tests, due to the high computational cost of repeated LLM alignment runs\. Instead, we assess the robustness of the empirical conclusions across multiple model sizes, list sizes, robustness radii, ranking\-noise settings, and both offline fixed\-list and online policy\-induced settings\. The main claims are based on trends that are consistently observed across these experimental conditions rather than on a single isolated comparison\.
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: The paper reports that the RLHF experiments were conducted on a cluster of 8 NVIDIA RTX 4090 GPUs\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
44. Justification: The research conforms to the NeurIPS Code of Ethics\.
45. Guidelines: - •The answer\[N/A\]means that the authors have not reviewed the NeurIPS Code of Ethics\. - •If the authors answer\[No\], they should explain the special circumstances that require a deviation from the Code of Ethics\. - •The authors should make sure to preserve anonymity \(e\.g\., if there is a special consideration due to laws or regulations in their jurisdiction\)\.
46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[Yes\]
49. Justification: The paper includes a broader\-impact discussion\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification: The paper does not release new pretrained language models, high\-risk generative models, scraped datasets, or user\-facing systems that would require special safeguards\. The released assets are code for reproducing the proposed method and experiments\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: The paper credits the existing datasets, models, and reward models used in the experiments and provides URLs to them\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://arxiv.org/html/2607.01715v1/paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[Yes\]
64. Justification: The paper releases anonymized code repositories for the proposed offline and online experiments\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: The paper does not involve new crowdsourcing experiments or new research with human subjects\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: The paper does not conduct new human\-subject experiments or collect new human\-subject data\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[Yes\]
79. Justification: LLMs are central to the experimental setting: the paper fine\-tunes and evaluates open\-source language models and uses GPT\-as\-a\-judge evaluation as part of the reported empirical protocol\. The use of LLMs is described in the experiments and appendix; any use of LLMs for writing or formatting does not affect the scientific methodology\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.