Traversing the Satisfaction-Diversity Frontier in Text-to-Image Diffusion

# Traversing the Satisfaction-Diversity Frontier in Text-to-Image Diffusion
Source: [https://arxiv.org/html/2610.02372](https://arxiv.org/html/2610.02372)
Siva Rajesh KasaAffiliation:AmazonSoumya RoyAffiliation:AmazonSumit NegiAffiliation:AmazonMubarak ShahAffiliation:University of Central Florida

###### Abstract

Text\-to\-image generation enables users to explore several images generated from the same prompt\. For these generated images to be useful, each one must reflect the user’s preferences, measured by a learned reward, and differ visually from the others to maintain diversity\. Existing methods are limited: they either address reward and diversity separately or combine them in one aggregate score, enabling high diversity to offset low rewards\. In this paper, we address these limitations by formulating generation as*satisficing*: every image \(candidate\) must satisfy a reward floor and the batch of images must satisfy a diversity cutoff\. The reward floor controls the balance between worst\-candidate reward and batch diversity; we show that varying this floor defines a Pareto frontier\. To traverse this frontier, we introduce SatisDive, a training\-free inference\-time method\. SatisDive uses a batch\-relative reward cutoff to distinguish lower\- from higher\-reward candidates, emphasizing reward improvement for candidates below the cutoff and diversity among candidates above it\. On Pick\-a\-Pic, at matched DreamSim, SatisDive improves worst\-candidate reward over FK steering by up to0\.430\.43with FLUX\.1\-dev as the base model and HPSv3 as the reward, and by up to0\.700\.70with SANA\-1\.6B as the base model and ImageReward as the reward\. More broadly, across their overlapping DreamSim ranges, SatisDive’s satisfaction–diversity curve Pareto\-dominates FK steering’s curve in each setting\.

## 1Introduction

An important goal of text\-to\-image generation is to give users a choice among several high\-quality candidate images\. For example, a user creating a poster illustration may compare images generated from the same prompt, choosing the lighting, background, or composition that best fits the poster\. Recent inference\-time methods increase diversity\([Singh et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib40)\)to provide a wider range of options, maximize a learned reward\([Singhal et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib41)\)to align generated images with user preferences, or select images by maximizing a linear combination of total image reward and batch diversity\([Parmar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib34)\)\.

However, these approaches can still limit the user’s viable choices\. Firstly, reward\-steering methods can concentrate sampling on higher\-reward images, making the returned images visually similar\([Singhal et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib41);[Kim et al\., 2025b](https://arxiv.org/html/2610.02372#bib.bib23);[Shekhar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib38)\)\. Secondly, reward\-free diversity methods make generated images more visually different, but do not explicitly account for learned reward\([Singh et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib40);[Kim et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib21);[Corso et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib3);[Vinograd et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib44)\)\. Thirdly, batch\-level reward–diversity objectives\([Parmar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib34)\)allow greater diversity or high rewards for other images to compensate for one image’s low reward\.

These limitations motivate controlling worst\-image reward through a reward floor, i\.e\., a minimum reward level required of each candidate\. As the reward floor increases, the batch can become less diverse since fewer images satisfy the floor\. We show that varying the reward floor defines a Pareto frontier between worst\-image reward and batch diversity, which we call the satisfaction–diversity frontier\. Along this frontier, we formulate image generation as*satisficing*\([Simon, 1956](https://arxiv.org/html/2610.02372#bib.bib39)\):*every generated image must satisfy a reward floor, and the batch must satisfy a diversity cutoff\.*Once a candidate clears the reward floor, further increases in reward do not improve satisfaction\. To this end, we introduce SatisDive, a training\-free inference\-time method\. As Figure[1](https://arxiv.org/html/2610.02372#S1.F1)shows, SatisDive generates visually distinct images above the reward floor, while FK steering\([Singhal et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib41)\), a strong reward\-steering baseline, returns nearly identical images\.

![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sana_qualitative_teaser_00252.png)Figure 1:SatisDive returns four visually distinct images above the reward floor, while FK steering returns four nearly identical images\.We use the SANA\-1\.6B\([Xie et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib47)\)flow matching model with ImageReward\([Xu et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib48)\)reward model\. On a Pick\-a\-Pic\([Kirstain et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib24)\)prompt depicting two colliding viruses, the minimum reward is0\.660\.66for FK steering and1\.251\.25for SatisDive, with reward floorρ=1\.19\\rho\{=\}1\.19\.Our objective traverses the satisfaction–diversity frontier through a per\-image reward penalty and a batch\-level diversity penalty\. Under the satisficing formulation, a visually different but low\-reward image should contribute less to batch diversity\. Thus, these penalties are complementary: images with rewards below a batch\-relative reward cutoff incur larger reward penalties and contribute less to the diversity penalty, and images with rewards above the cutoff incur smaller reward penalties but contribute more to the diversity penalty\. Because some candidates may remain low reward throughout denoising, we introduce late\-stage latent replacement, which replaces their latents with those of higher\-reward candidates\. Subsequent stochastic denoising and the diversity penalty allow the resulting trajectories to produce distinct images\.

Empirically, SatisDive improves worst\-candidate reward without sacrificing batch diversity: across their overlapping DreamSim ranges, SatisDive’s worst\-candidate reward is consistently higher than FK steering’s, thus our satisfaction–diversity curves Pareto\-dominate FK steering’s on both FLUX and SANA\. Our contributions are as follows\.

- •We show that different reward floors define a Pareto frontier between worst\-image reward and batch diversity\. Under our*satisficing*formulation, every image must satisfy the selected reward floor, and the batch must satisfy a diversity cutoff\.
- •We introduce SatisDive, a training\-free inference\-time method whose objective provably traverses this satisfaction–diversity frontier\. The objective uses a batch\-relative reward cutoff to emphasize reward improvement for lower\-reward candidates and diversity among higher\-reward candidates\.
- •At matched DreamSim, SatisDive improves worst\-image reward by up to0\.430\.43on FLUX and0\.700\.70on SANA\. Our satisfaction–diversity curve Pareto\-dominates the curves traced by FK steering on the FLUX\.1\-dev and SANA\-1\.6B base models\.

## 2Problem Formulation

A rectified\-flow text\-to\-image model\([Liu et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib31)\)generates an image from a promptccusing a learned velocity fieldvθ​\(xt,t,c\)v\_\{\\theta\}\(x\_\{t\},t,c\)\. Generation starts from Gaussian noisex1∼𝒩⁡\(0,I\)x\_\{1\}\\sim\\mathcal\{N\}\(0,I\)and integratesvθv\_\{\\theta\}fromt=1t=1tot=0t=0overTTdiscretized steps, producing the imagex0x\_\{0\}\. This formulation also covers other diffusion models such as DDPM\([Ho et al\., 2020](https://arxiv.org/html/2610.02372#bib.bib11)\), which differ in their noise schedule\([Lipman et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib30)\)\. Table[4](https://arxiv.org/html/2610.02372#A1.T4)in Appendix[A](https://arxiv.org/html/2610.02372#A1)summarizes the notation used throughout the paper\.

Letpθ​\(x0∣c\)p\_\{\\theta\}\(x\_\{0\}\\mid c\)denote the base model’s distribution\. We consider methods that maintainKKcandidates throughout a single sampling pass and generate a batch ofKKimages \(denotedx01:Kx\_\{0\}^\{1:K\}\)\. Given a rewardr⁡\(x0,c\)r\(x\_\{0\},c\), we first consider reward\-steering methods \(e\.g\., FK steering\([Singhal et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib41)\)\) that yield high\-reward images by targeting the tilted distributionpλp\_\{\\lambda\}:

pλ​\(x0∣c\)∝pθ​\(x0∣c\)​exp⁡\(λ​r​\(x0,c\)\),λ≥0,p\_\{\\lambda\}\(x\_\{0\}\\mid c\)\\propto p\_\{\\theta\}\(x\_\{0\}\\mid c\)\\,\\exp\\\!\\big\(\\lambda\\,r\(x\_\{0\},c\)\\big\),\\qquad\\lambda\\geq 0,\(1\)where largerλ\\lambdaincreases the relative probability of higher\-reward images\. These methods approximate Eq\.[1](https://arxiv.org/html/2610.02372#S2.E1)by resampling candidate latents at intermediate timesteps with selection probabilities determined by candidate rewards; these rewards are evaluated on the model’s estimatex^0\\hat\{x\}\_\{0\}of the final image, computed as:

x^0=xt−σt​vθ​\(xt,t,c\),\\hat\{x\}\_\{0\}=x\_\{t\}\-\\sigma\_\{t\}v\_\{\\theta\}\(x\_\{t\},t,c\),\(2\)whereσt\\sigma\_\{t\}is the noise level at timesteptt\. Since rectified flow is deterministic, these steering methods use the equivalent marginal\-preserving stochastic form\([Mark et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib33)\)\.

#### Limitations of current methods\.

The reward\-only tilt in Eq\.[1](https://arxiv.org/html/2610.02372#S2.E1)weights each candidate’s selection probability during resampling according to its reward, without considering how theKKimages differ\. Proposition[1](https://arxiv.org/html/2610.02372#Thmtheorem1)\(proof in Appendix[B](https://arxiv.org/html/2610.02372#A2)\) states what happens when this tilt is applied to the current candidates during resampling\.

###### Proposition 1\(Asλ→∞\\lambda\\to\\infty, resampling producesKKcopies of the highest\-reward candidate\)\.

At a resampling step with a unique highest\-reward candidate, increasingλ\\lambdaincreases its expected number of copies\. In the limit asλ→∞\\lambda\\to\\infty, the probability that allKKresampled candidates are copies of the highest\-reward candidate approaches one\.

Consequently, a greater reward tilt can reduce the number of distinct candidates retained after resampling and thereby reduce the visual diversity among the generated images\.

A second class of methods increases the*diversity*of generated images\. NegToMe\([Singh et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib40)\), for instance, increases the distance between the candidates’ features\. Being reward\-agnostic, these methods do not constrain the sampling distribution toward higher rewards\. Figure[2](https://arxiv.org/html/2610.02372#S2.F2)visualizes the limitations of reward\-steering and diversity methods in a toy 2D setting \(details in Appendix[C\.1](https://arxiv.org/html/2610.02372#A3.SS1)\)\.

![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/problem_grid_pareto_xy.png)Figure 2:In a toy setting, each baseline tends towards high diversity or high reward, not both\.Each method generatesK=4K\{=\}4candidates from a frozen flow model trained on a mixture of Gaussians \(details in Appendix[C\.1](https://arxiv.org/html/2610.02372#A3.SS1)\)\. Reward is bounded in\(0,1\]\(0,1\], and the midpointρ=0\.5\\rho=0\.5is a reward floor that every candidate must satisfy\.\(a\)To visualize clearly the coverage of high\-reward regions, candidates are 2D points on a Cartesianxx–yygrid rather than images\. We show one selected illustrative batch ofK=4K\{=\}4candidates per method\. DAS’s candidates are all aboveρ\\rhobut are not diverse; they cover only a single high\-reward region \(yellow circles\)\. Conversely, NegToMe’s candidates are diverse yet are all belowρ\\rho\(red circles\)\. In these batches, only ours has all candidates aboveρ\\rhoin distinct high\-reward regions\.\(b\)We plot worst\-candidate reward against diversity \(mean pairwise distance\) averaged over 200 runs for each method\. Methods in the shaded green region have average worst\-candidate rewards greater thanρ\\rho\. DAS achieves the highest reward but lowest diversity; NegToMe achieves the highest diversity but lowest reward\.A natural third alternative treats diversity as a second reward and samples from a joint tilt:

πjoint\(x01:K∣c\)∝∏k=1Kpθ\(x0k∣c\)exp\(λ∑k=1Kr\(x0k,c\)\+μD\(x01:K\)\),λ,μ≥0,\\pi\_\{\\mathrm\{joint\}\}\(x\_\{0\}^\{1:K\}\\mid c\)\\propto\\prod\_\{k=1\}^\{K\}p\_\{\\theta\}\(x\_\{0\}^\{k\}\\mid c\)\\,\\exp\\\!\\Big\(\\lambda\\sum\_\{k=1\}^\{K\}r\(x\_\{0\}^\{k\},c\)\+\\mu D\(x\_\{0\}^\{1:K\}\)\\Big\),\\qquad\\lambda,\\mu\\geq 0,\(3\)whereDDis a batch\-level diversity term andμ\\muweights it against rewardrr\. Because reward and diversity are aggregated, the joint objective can be gamed: forλ\>0\\lambda\>0, one unit ofDDoffsets a decrease ofμ/λ\\mu/\\lambdain total reward\. This exchange exists for every positiveμ\\mu, soπjoint\\pi\_\{\\mathrm\{joint\}\}never requires a candidate to be high\-reward\.

## 3Method

The approaches discussed in Section[2](https://arxiv.org/html/2610.02372#S2)do not jointly require a reward floor for every candidate and a minimum diversity for the whole batch\. We therefore formulate generation as*satisficing*\([Simon, 1956](https://arxiv.org/html/2610.02372#bib.bib39)\): every candidate must satisfy a reward floorρ\\rho, while the batch must satisfy a diversity requirement\. These requirements define the target distribution

π⋆\(x01:K∣c\)∝\(∏k=1Kpθ\(x0k∣c\)1\[r\(x0k,c\)≥ρ\]\)1\[D\(x01:K\)≥δ\],\\pi^\{\\star\}\(x\_\{0\}^\{1:K\}\\mid c\)\\;\\propto\\;\\Big\(\\prod\_\{k=1\}^\{K\}p\_\{\\theta\}\(x\_\{0\}^\{k\}\\mid c\)\\,\\mathbf\{1\}\[r\(x\_\{0\}^\{k\},c\)\\geq\\rho\]\\Big\)\\,\\mathbf\{1\}\[D\(x\_\{0\}^\{1:K\}\)\\geq\\delta\],\(4\)whereDDmeasures batch diversity,δ\\deltais the required diversity level, and each indicator equals one when its condition holds and zero otherwise\. Rejection sampling would drawKKcandidates independently frompθp\_\{\\theta\}and reject the entire batch if either requirement is violated\. Because the probability that allKKcandidates satisfy the reward floor decreases exponentially withKK, rejection sampling can be prohibitively expensive\.

We thus introduce SatisDive, which replaces both indicators with differentiable penalties\. Our training\-free approach rests on two mechanisms: \(1\) a satisficing objective that increases low rewards and diversifies the batch; and \(2\) late\-stage latent replacement, which replaces the latent of a low\-reward candidate with that of a higher\-reward candidate\. Section[3\.1](https://arxiv.org/html/2610.02372#S3.SS1)details the objective and late\-stage latent replacement and connects our objective toπ⋆\\pi^\{\\star\}, while Section[3\.2](https://arxiv.org/html/2610.02372#S3.SS2)proves that this objective traces the satisfaction–diversity Pareto frontier\.

### 3\.1Satisficing at Inference Time

Our objective𝒥\\mathcal\{J\}consists of a reward penalty and a diversity penalty, differentiable counterparts toπ⋆\\pi^\{\\star\}’s two indicators\. For a differentiable rewardr⁡\(⋅\)r\(\\cdot\), we minimize𝒥\\mathcal\{J\}over each candidate’s latentxtkx\_\{t\}^\{k\}\. At each step, SatisDive compares each candidate’s rewardrk=r⁡\(x^0k,c\)r\_\{k\}=r\(\\hat\{x\}\_\{0\}^\{k\},c\)against a batch\-relative reward cutoffτ\\tau:

τ=maxk⁡rk−Δ,Δ\>0,\\tau=\\max\_\{k\}r\_\{k\}\-\\Delta,\\qquad\\Delta\>0,\(5\)whereΔ\\Deltais a fixed tolerance\. Comparing againstτ\\taudistinguishes low\- from high\-reward candidates in the batch\. For candidatekk, we use the sigmoidqkq\_\{k\}as a smooth probability thatrk≥τr\_\{k\}\\geq\\tau:

qk=sigmoid⁡\(\(rk−τ\)/βr\),q\_\{k\}=\\operatorname\{sigmoid\}\\\!\\big\(\(r\_\{k\}\-\\tau\)/\\beta\_\{r\}\\big\),\(6\)where the temperatureβr\\beta\_\{r\}is computed once per run as the standard deviation of the batch’s rewards;qk→1q\_\{k\}\\rightarrow 1for rewards aboveτ\\tauandqk→0q\_\{k\}\\rightarrow 0belowτ\\tau\.

*Reward Penalty\.*We define a penalty for each candidatekkthat increases withτ−rk\\tau\-r\_\{k\}; lower\-reward candidates have larger penalties\. Summing over the batch gives𝒥r\\mathcal\{J\}\_\{r\}:

𝒥r≜−∑klogqk=∑ksoftplus\(\(τ−rk\)/βr\)\.\\mathcal\{J\}\_\{r\}\\triangleq\-\\sum\_\{k\}\\log q\_\{k\}=\\sum\_\{k\}\\operatorname\{softplus\}\\\!\\big\(\(\\tau\-r\_\{k\}\)/\\beta\_\{r\}\\big\)\.\(7\)Thus,𝒥r\\mathcal\{J\}\_\{r\}is the negative log\-likelihood that every candidate in the batch exceedsτ\\tau\.

*Diversity Penalty\.*We measure batch diversityDDas the weighted mean of the pairwise cosine distancesdi​jd\_\{ij\}between the pooled backbone features of candidatesiiandjj, each pair weighted byqi​qjq\_\{i\}q\_\{j\}\(Eq\.[6](https://arxiv.org/html/2610.02372#S3.E6)\)\. Consequently, pairs of higher\-reward candidates receive greater weight than pairs with lower\-reward candidates\. To account for the scale of pairwise distances in each run, we setδ\\deltato the median of thedi​jd\_\{ij\}, computed once per run\. At each step, the diversity penalty𝒥d\\mathcal\{J\}\_\{d\}is then:

𝒥d≜softplus⁡\(\(δ−D\)/βd\),whereD=∑i<jqi​qj​di​j∑i<jqi​qj,\\mathcal\{J\}\_\{d\}\\triangleq\\operatorname\{softplus\}\\\!\\big\(\(\\delta\-D\)/\\beta\_\{d\}\\big\),\\quad\\text\{where\}\\quad D=\\tfrac\{\\sum\_\{i<j\}q\_\{i\}q\_\{j\}\\,d\_\{ij\}\}\{\\sum\_\{i<j\}q\_\{i\}q\_\{j\}\},\(8\)andβd\\beta\_\{d\}is the diversity temperature\. Combining the two penalties yields our objective:

𝒥=𝒥r\+𝒥d=∑ksoftplus⁡\(\(τ−rk\)/βr\)\+softplus⁡\(\(δ−D\)/βd\)\.\\mathcal\{J\}=\\mathcal\{J\}\_\{r\}\+\\mathcal\{J\}\_\{d\}=\\sum\_\{k\}\\operatorname\{softplus\}\\\!\\big\(\(\\tau\-r\_\{k\}\)/\\beta\_\{r\}\\big\)\\;\+\\;\\operatorname\{softplus\}\\\!\\big\(\(\\delta\-D\)/\\beta\_\{d\}\\big\)\.\(9\)At intermediate denoising steps, we update the candidate latents using the gradients of𝒥d\\mathcal\{J\}\_\{d\}and𝒥r\\mathcal\{J\}\_\{r\}\(scaling specified in Appendix[C](https://arxiv.org/html/2610.02372#A3)\)\. The objective gives greater weight to increasing reward and diversity below their cutoffs than above them\. Proposition[2](https://arxiv.org/html/2610.02372#Thmtheorem2)\(proof in Appendix[B](https://arxiv.org/html/2610.02372#A2)\) connects the distribution induced by our objective toπ⋆\\pi^\{\\star\}\(Eq\.[4](https://arxiv.org/html/2610.02372#S3.E4)\) in the zero\-temperature limit\. We evaluate the finite\-step sampler empirically\.

###### Proposition 2\(The distribution induced by𝒥\\mathcal\{J\}converges toπ⋆\\pi^\{\\star\}\)\.

At any given timestep, letM=maxk⁡rkM=\\max\_\{k\}r\_\{k\}and setτ=ρ=M−Δ\\tau=\\rho=M\-\\Delta\. Here,DDinπ⋆\\pi^\{\\star\}is the unweighted mean pairwise distance, obtained from the weighted mean in Eq\.[8](https://arxiv.org/html/2610.02372#S3.E8)as all reward gates approach one\. Suppose that independently samplingKKcandidates from the base model has a nonzero chance of producing a batch withrk≥ρr\_\{k\}\\geq\\rhofor every candidatekkandD≥δD\\geq\\delta\(Eq\.[4](https://arxiv.org/html/2610.02372#S3.E4)\)\. Suppose also that the event thatrk=ρr\_\{k\}=\\rhofor somekk, and the event thatD=δD=\\delta, each have probability zero\. Then, asβr,βd→0\\beta\_\{r\},\\beta\_\{d\}\\to 0, the distribution induced by𝒥\\mathcal\{J\}converges toπ⋆\\pi^\{\\star\}\.

Late\-Stage Latent Replacement\.In practice, the gradient of𝒥\\mathcal\{J\}depends strongly on the structure of the latent space and the initial noisex1kx\_\{1\}^\{k\}, so candidates may remain in local reward optima throughout denoising\. We thus propose late\-stage latent replacement based on the intuition that, toward the end of denoising, it is preferable to copy from a known high\-reward candidate than to continue updating a candidate that remains low reward\. To see how reward may change over the remaining steps, we extrapolate each below\-τ\\taucandidate’s reward from the differencerk−rkprevr\_\{k\}\-r\_\{k\}^\{\\mathrm\{prev\}\}since the previous update\. If this candidate’s extrapolated reward is less thanτ\\tau, we replace its latent with that of a randomly chosen candidate aboveτ\\tau\. We apply late\-stage latent replacement only in the latter half of denoising, since an earlyx^0k\\hat\{x\}\_\{0\}^\{k\}estimate is a poor predictor of a candidate’s final reward\. After replacement, the two candidates become visually distinct by the end of denoising due to𝒥d\\mathcal\{J\}\_\{d\}and stochastic sampling\. Algorithm[1](https://arxiv.org/html/2610.02372#alg1)details our full procedure\.

Algorithm 1SatisDive\.1:base model

pθp\_\{\\theta\}, prompt

cc, reward

rr, batch size

KK, steps

TT, update steps

𝒢\\mathcal\{G\}, tolerance

Δ\\Delta
2:sample

x1,…,xK∼𝒩⁡\(0,I\)x^\{1\},\\dots,x^\{K\}\\sim\\mathcal\{N\}\(0,I\)
3:fordenoising step

s=1,…,Ts=1,\\ldots,Tdo

4:if

s∈𝒢s\\in\\mathcal\{G\}then

5:for allcandidates

kkdo

6:Predict clean image

x^0k\\hat\{x\}\_\{0\}^\{k\}⊳\\trianglerightEq\. \([2](https://arxiv.org/html/2610.02372#S2.E2)\)

7:Compute reward

rk=r⁡\(x^0k,c\)r\_\{k\}=r\(\\hat\{x\}\_\{0\}^\{k\},c\)
8:Compute reward cutoff

τ=maxk⁡rk−Δ\\tau=\\max\_\{k\}r\_\{k\}\-\\Delta⊳\\trianglerightEq\. \([5](https://arxiv.org/html/2610.02372#S3.E5)\)

9:Using

τ\\tau, compute

𝒥d\\mathcal\{J\}\_\{d\}and

𝒥r\\mathcal\{J\}\_\{r\}; update each

xkx^\{k\}using their gradients⊳\\trianglerightEq\. \([9](https://arxiv.org/html/2610.02372#S3.E9)\)

10:if

s\>T/2s\>T/2and

rkprevr\_\{k\}^\{\\mathrm\{prev\}\}exists for every

kkthen

11:

m=\|\{u∈𝒢:u\>s\}\|m=\\lvert\\\{u\\in\\mathcal\{G\}:u\>s\\\}\\rvert⊳\\trianglerightmmis the number of remaining update steps

12:

r~k=rk\+m⋅max⁡\(rk−rkprev,0\)\\tilde\{r\}\_\{k\}=r\_\{k\}\+m\\cdot\\max\(r\_\{k\}\-r\_\{k\}^\{\\mathrm\{prev\}\},0\), for every

kk⊳\\trianglerightGet extrapolated rewardr~k\\tilde\{r\}\_\{k\}

13:for all

kksuch that

rk<τr\_\{k\}<\\tauand

r~k<τ\\tilde\{r\}\_\{k\}<\\taudo

14:Sample

j∼𝒰⁡\(\{i:ri≥τ\}\)j\\sim\\mathcal\{U\}\(\\\{i:r\_\{i\}\\geq\\tau\\\}\)
15:Replace latent

xkx^\{k\}with

xjx^\{j\}⊳\\trianglerightLate\-stage latent replacement

16:

rkprev←rkr\_\{k\}^\{\\mathrm\{prev\}\}\\leftarrow r\_\{k\}, for every

kk⊳\\trianglerightStore rewards for the next update

17:Take next sampling step

18:return

x01:Kx\_\{0\}^\{1:K\}

### 3\.2SatisDive’s Objective Traces the Satisfaction–Diversity Pareto Frontier

By sweepingΔ\\Delta, SatisDive’s objective traces a Pareto frontier between reward and diversity\. We now characterize how the reward constraint inπ⋆\\pi^\{\\star\}\(Eq\.[4](https://arxiv.org/html/2610.02372#S3.E4)\) changes as the reward floorρ\\rhovaries\. LetAρ=\{x:r⁡\(x,c\)≥ρ\}A\_\{\\rho\}=\\\{x:r\(x,c\)\\geq\\rho\\\}be the set of images that satisfy the floor, and letpρp\_\{\\rho\}be the base output distribution restricted toAρA\_\{\\rho\}, preserving relative probabilities within this set\. Because the reward constraint applies separately to each candidate, we rewriteπ⋆\\pi^\{\\star\}as a product ofKKreward\-restricted distributions and the batch diversity indicator:

π⋆\(x1:K∣c\)∝\(∏k=1Kpρ\(xk\)\)1\[D\(x1:K\)≥δ\],pρ\(x\)∝pθ\(x∣c\)1\[x∈Aρ\]\.\\pi^\{\\star\}\(x^\{1:K\}\\mid c\)\\propto\\Big\(\\prod\_\{k=1\}^\{K\}p\_\{\\rho\}\(x^\{k\}\)\\Big\)\\,\\mathbf\{1\}\[D\(x^\{1:K\}\)\\geq\\delta\],\\qquad p\_\{\\rho\}\(x\)\\propto p\_\{\\theta\}\(x\\mid c\)\\,\\mathbf\{1\}\[x\\in A\_\{\\rho\}\]\.\(10\)We denote\{pρ\}\\\{p\_\{\\rho\}\\\}as the family of distributions obtained by varying the reward floorρ\\rho\. For each floor, Lemma[5](https://arxiv.org/html/2610.02372#Thmtheorem5)\(Appendix[B](https://arxiv.org/html/2610.02372#A2)\) shows two properties ofpρp\_\{\\rho\}\. First, among all distributions that assign probability one toAρA\_\{\\rho\},pρp\_\{\\rho\}uniquely minimizes theKL\\mathrm\{KL\}divergence to the base model\. Second, the support of every such distribution with finiteKL\\mathrm\{KL\}is contained in the support ofpρp\_\{\\rho\}\. Here, satisfaction means the highest reward floor satisfied with probability one\. Theorem[3](https://arxiv.org/html/2610.02372#Thmtheorem3)\(proof in Appendix[B](https://arxiv.org/html/2610.02372#A2)\) establishes\{pρ\}\\\{p\_\{\\rho\}\\\}as the satisfaction–diversity Pareto frontier\.

###### Theorem 3\(The family\{pρ\}\\\{p\_\{\\rho\}\\\}forms the satisfaction–diversity Pareto frontier asρ\\rhovaries\)\.

Consider reward floorsρ\\rhofor whichpθ​\(Aρ∣c\)\>0p\_\{\\theta\}\(A\_\{\\rho\}\\mid c\)\>0, and suppose diversity is nonincreasing under support restriction, strictly so for proper restrictions\. Every candidate drawn frompρp\_\{\\rho\}has reward at leastρ\\rho\. For any two such reward floorsρ′\\rho^\{\\prime\}andρ\\rhowithρ′≥ρ\\rho^\{\\prime\}\\geq\\rho,pρ′p\_\{\\rho^\{\\prime\}\}has diversity at most that ofpρp\_\{\\rho\}\. For eachρ\\rho, no distribution overAρA\_\{\\rho\}at finite KL topθp\_\{\\theta\}is more diverse thanpρp\_\{\\rho\}\. Therefore,\{pρ\}\\\{p\_\{\\rho\}\\\}forms the satisfaction–diversity Pareto frontier among distributions with finite guaranteed reward and finite KL topθp\_\{\\theta\}\.

The theorem applies to diversity measures satisfying the stated support\-restriction assumption; our experiments evaluate the practical satisfaction–diversity tradeoff using mean pairwise DreamSim\. Reward steering instead targets the tilted distributionspλp\_\{\\lambda\}\. For finiteλ\\lambdaand any reward floorρ\\rhowith0<pθ​\(Aρ∣c\)<10<p\_\{\\theta\}\(A\_\{\\rho\}\\mid c\)<1, Corollary[4](https://arxiv.org/html/2610.02372#Thmtheorem4)\(Appendix[B](https://arxiv.org/html/2610.02372#A2)\) shows thatpλp\_\{\\lambda\}assigns positive probability to images belowρ\\rho, unlikepρp\_\{\\rho\}\. Thus, varyingλ\\lambdatraces\{pλ\}\\\{p\_\{\\lambda\}\\\}rather than the satisfaction–diversity Pareto frontier established by\{pρ\}\\\{p\_\{\\rho\}\\\}\.

## 4Experiments

Models, datasets, and baselines\.We show results with three frozen base models: FLUX\.1\-dev\([Labs, 2024](https://arxiv.org/html/2610.02372#bib.bib26)\), SANA\-1\.6B\([Xie et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib47)\), and Stable Diffusion 1\.5 \(SD1\.5\)\([Rombach et al\., 2022](https://arxiv.org/html/2610.02372#bib.bib35)\)\. We use two reward models, HPSv3\([Ma et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib32)\)and ImageReward\([Xu et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib48)\), and evaluate on the Pick\-a\-Pic\([Kirstain et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib24)\)and HPDv2\([Wu et al\., 2023b](https://arxiv.org/html/2610.02372#bib.bib46)\)datasets\. Our main baselines fall into two categories: 1\) reward\-steering methods, namely Feynman–Kac \(FK\) steering\([Singhal et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib41)\), VASR\([Shekhar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib38)\), and DAS\([Kim et al\., 2025b](https://arxiv.org/html/2610.02372#bib.bib23)\); and 2\) diversity methods, namely Negative Token Merging \(NegToMe\)\([Singh et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib40)\), which repels candidates in feature space\. We also evaluate FK\+NegToMe, which combines FK steering with NegToMe’s diversity mechanism\. All methods use the same seeds and returnK=4K\{=\}4images \(i\.e\., candidates\) per prompt\. We report SGI\([Parmar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib34)\)separately in Appendix[D\.9](https://arxiv.org/html/2610.02372#A4.SS9)because it prunes a much larger pool of128128candidates\. Baselines use their recommended settings; sampling details are provided in Appendix[C](https://arxiv.org/html/2610.02372#A3)\.

Metrics\.We evaluate images on two axes: reward and diversity\. On the reward axis, every method is scored against a reward floorρ\\rho\. Rather than choosingρ\\rhoarbitrarily, we set it to the base model’s median reward over all prompts\. We report worst\-candidate rewardmink⁡rk\\min\_\{k\}r\_\{k\}, averaged over prompts, since a batch satisfies the floorρ\\rhoprecisely whenmink⁡rk≥ρ\\min\_\{k\}r\_\{k\}\\geq\\rho; we also report mean reward\. We define the satisfaction rateSATρ\\mathrm\{SAT\}\_\{\\rho\}as the fraction of generated candidates whose reward is at leastρ\\rho:

SATρ=1N⋅K∑i=1N⋅K\[ri≥ρ\],\\mathrm\{SAT\}\_\{\\rho\}\\;=\\;\\frac\{1\}\{N\\cdot K\}\\sum\_\{i=1\}^\{N\\cdot K\}\\mathbf\{1\}\\\!\\left\[\\,r\_\{i\}\\geq\\rho\\,\\right\],\(11\)whererir\_\{i\}is the reward of theii\-th generated candidate and𝟏​\[⋅\]\\mathbf\{1\}\[\\cdot\]is the indicator function\.NNis the number of prompts, withKKcandidates per prompt\. Sinceρ\\rhois the median reward of the base model’s outputs,SATρ=0\.5\\mathrm\{SAT\}\_\{\\rho\}\\\!=\\\!0\.5indicates parity with the base model\. On the diversity axis, each batch is measured by the mean pairwise DreamSim\([Fu et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib6)\)and LPIPS\([Zhang et al\., 2018](https://arxiv.org/html/2610.02372#bib.bib53)\)among itsKKcandidates\. We additionally report PickScore\([Kirstain et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib24)\)as a held\-out reward, which indicates whether reward gains reflect a broader improvement in preference\.

### 4\.1SatisDive Traverses the Satisfaction–Diversity Frontier

Sweeping the toleranceΔ\\Deltaof our satisficing objective \(Eq\.[9](https://arxiv.org/html/2610.02372#S3.E9)\) traces a frontier curve between worst\-candidate reward and diversity\. As shown in Figure[3](https://arxiv.org/html/2610.02372#S4.F3)on FLUX and SANA, baseline methods tend to exhibit either high worst\-candidate reward but low diversity \(DAS, VASR\), or low worst\-candidate reward and high diversity \(NegToMe\)\. Although varying FK steering’s reward\-tilt parameterλ\\lambdaalso traces a reward–diversity curve, SatisDive’s curve Pareto\-dominates\([Hwang and Masud, 2012](https://arxiv.org/html/2610.02372#bib.bib14)\)FK steering’s over their common DreamSim ranges\. At matched DreamSim, SatisDive has higher worst\-candidate reward at every overlapping point, with improvements of up to0\.430\.43on FLUX using HPSv3 and0\.700\.70on SANA using ImageReward\.

![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sana_mink_pickapic.png)

Figure 3:SatisDive achieves better worst\-candidate reward at matched diversity versus FK steering\.We plot mean worst\-candidate rewardmink⁡rk\\min\_\{k\}r\_\{k\}against mean pairwise DreamSim, on Pick\-a\-Pic\. We sweep SatisDive and FK steering on two settings\.\(a\)FLUX\.1\-dev is the base model and HPSv3 is the reward, withΔ\\Deltafrom0\.00010\.0001to33andλ\\lambdafrom0\.50\.5to22\.\(b\)SANA\-1\.6B is the base model and ImageReward is the reward, withΔ\\Deltafrom0\.10\.1to1\.51\.5andλ\\lambdafrom0\.50\.5to4040\. The shaded band represents the worst\-candidate reward gap between the two curves at the same diversity, marked atΔ=1\.5\\Delta\{=\}1\.5in \(a\) where it is≈0\.43\\approx\\\!0\.43on FLUX HPSv3, and atΔ=0\.5\\Delta\{=\}0\.5in \(b\) where it is≈0\.70\\approx\\\!0\.70on SANA ImageReward\. We note that the top right region, representing high reward and high diversity, is ideal\. DAS and VASR collapse to diversity below0\.060\.06in both settings, and NegToMe is the most diverse but its worst candidate on average falls below the base model’s\. SatisDive traces a better empirical frontier than FK steering between these extremes\.Table 1:On Pick\-a\-Pic, SatisDive is competitive on satisfaction rate without collapsing diversity\.We use FLUX\.1\-dev with HPSv3 as the reward\. DAS has the highest satisfaction rate at0\.6120\.612but a mean pairwise DreamSim of only0\.0590\.059, while NegToMe is the most diverse at0\.8850\.885and has the lowest satisfaction rate at0\.2760\.276\. SatisDive attains a satisfaction rate of0\.5520\.552at a DreamSim of0\.5550\.555, and matches base FLUX on both held\-out rewards\. We useρ=9\.66\\rho\{=\}9\.66;±\\pmdenotes 95% CIs\.Table 2:On HPDv2, SatisDive is competitive on satisfaction rate without collapsing diversity\.We use FLUX\.1\-dev with HPSv3 as the reward\. SatisDive attains a satisfaction rate of0\.5530\.553at a mean pairwise DreamSim of0\.5670\.567, against DAS’s0\.5800\.580at0\.0820\.082, and matches base FLUX on both held\-out rewards\.SATρ\\mathrm\{SAT\}\_\{\\rho\}uses floorρ=13\.93\\rho\{=\}13\.93\.±\\pmdenotes 95% confidence intervals\.On Pick\-a\-Pic, Table[1](https://arxiv.org/html/2610.02372#S4.T1)shows that SatisDive improves mean HPSv3 reward over base FLUX \(10\.0310\.03against9\.659\.65\)\. Although DAS and VASR achieve slightly higher mean rewards \(10\.2610\.26and10\.1710\.17compared to SatisDive’s10\.0310\.03\), both are markedly less diverse than SatisDive\. The same satisfaction–diversity pattern holds on HPDv2 in Table[2](https://arxiv.org/html/2610.02372#S4.T2), where SatisDive achieves a competitive satisfaction rate and mean HPSv3 reward with notably higher diversity than the reward\-steering methods\. In particular, adding NegToMe to FK steering increases diversity but reduces the satisfaction rate below the base model’s0\.50\.5, whereas SatisDive maintains above\-base satisfaction at near\-base diversity\. Appendix[D](https://arxiv.org/html/2610.02372#A4)evaluates SatisDive across additional settings and provides ablation and computational analyses\.

#### Generalization to Other Base Models\.

To show SatisDive is not specific to one architecture or reward model, we extend to SANA\-1\.6B \(flow matching, Table[3](https://arxiv.org/html/2610.02372#S4.T3)\) and SD1\.5 \(UNet, Appendix[D](https://arxiv.org/html/2610.02372#A4)Table[11](https://arxiv.org/html/2610.02372#A4.T11)\) with ImageReward\([Xu et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib48)\)\. On SANA, SatisDive atΔ=0\.3\\Delta\\\!=\\\!0\.3achieves the highest worst\-candidate reward and satisfaction rate of any evaluated method, with1\.291\.29worst\-candidate reward versus VASR’s1\.231\.23and0\.7350\.735satisfaction rate versus VASR’s0\.6510\.651\. SatisDive also achieves a DreamSim of0\.5910\.591, versus VASR’s0\.0460\.046\.

Table 3:SatisDive exceeds every reward\-steering method on worst\-candidate reward and diversity on SANA\-1\.6B\.We use Pick\-a\-Pic with ImageReward; the reward floor isρ=1\.19\\rho\\\!=\\\!1\.19\. Our method is shown at threeΔ\\Deltasettings, tracing its satisfaction–diversity frontier as reflected in Figure[3](https://arxiv.org/html/2610.02372#S4.F3)b\. AtΔ=0\.3\\Delta=0\.3, SatisDive attains a worst\-candidate reward of1\.291\.29and DreamSim of0\.5910\.591, exceeding every reward\-steering method on both metrics\.±\\pmdenotes 95% confidence intervals\.We further test whether stopping reward maximization once a candidate satisfies the floor, without an explicit diversity term, is sufficient to preserve diversity\. We therefore cap DAS’s reward atρ\\rhowhile leaving its other settings unchanged\. This reward capping increases DAS’s DreamSim from0\.0160\.016to0\.0660\.066, but the resulting diversity remains far below SatisDive’s0\.6580\.658; the change in satisfaction rate is not statistically significant\. Thus, reward capping alone does not account for SatisDive’s diversity \(Appendix[D\.12](https://arxiv.org/html/2610.02372#A4.SS12)\)\.

### 4\.2Qualitative Results

![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sana_qualitative_pickapic_highlighted.png)Figure 4:On two Pick\-a\-Pic prompts, only SatisDive returns four visually distinct images that are all above the reward floor\.We use SANA\-1\.6B with ImageReward and matched starting seeds; the reward floor isρ=1\.19\\rho\{=\}1\.19\. The magenta border marks the worst\-reward candidate in each row, whose reward is reported on the right\. Base and NegToMe produce visually varied batches, but their worst\-reward candidates do not satisfy the reward floor\. NegToMe’s worst\-reward car image \(on the left\) is nearly black, and its worst\-reward fairy image \(on the right\) does not clearly show an owl\. FK steering’s images exceed the reward floor, but its four images repeat nearly the same composition\.Figure[4](https://arxiv.org/html/2610.02372#S4.F4)shows individual batches from the SANA evaluation summarized in Figure[3](https://arxiv.org/html/2610.02372#S4.F3)\. FK steering returns four images above the reward floor but repeats nearly the same composition, while Base and NegToMe produce more varied batches in which some images fall below the floor\. On both prompts, SatisDive is the only method that returns four above\-floor images with visibly different compositions\. Additional qualitative results appear in Appendix[E](https://arxiv.org/html/2610.02372#A5)\.

## 5Related Work

Reward\-driven sampling\.Existing methods increase reward through resampling\([Singhal et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib41);[Wu et al\., 2023a](https://arxiv.org/html/2610.02372#bib.bib45);[Kim et al\., 2025b](https://arxiv.org/html/2610.02372#bib.bib23);[Shekhar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib38)\), value\-based lookahead\([Li et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib28);[Jain et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib16)\), stochastic transitions\([Holderrieth et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib12)\), adaptive model evaluation\([Kim et al\., 2025a](https://arxiv.org/html/2610.02372#bib.bib22)\), or noise\-space optimization and regularization\([Hwang et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib15);[Tang et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib43);[Eyring et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib5);[Harrington et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib8);[Zhai et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib52)\)\. These methods do not explicitly optimize batch diversity and may collapse candidate lineages\([Shekhar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib38)\)\. CREPE\([He et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib9)\)uses replica exchange to preserve diversity under reward tilting \(comparison in Appendix[D\.8](https://arxiv.org/html/2610.02372#A4.SS8)\), while SGI\([Parmar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib34)\)selects a reward–diversity batch from a large candidate pool; neither imposes a reward floor on every candidate\.

Reward\-free diversity\.NegToMe\([Singh et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib40)\), CNO\([Kim et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib21)\), Particle Guidance\([Corso et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib3)\), and EDDY\([Vinograd et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib44)\)repel candidates in feature or prediction space\. Other methods vary conditioning, intermediate features, or initial noise\([Sadat et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib36);[Yadav et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib49);[Li et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib27)\), while general guidance modifies model predictions, attention, or guidance schedules\([Ho and Salimans, 2021](https://arxiv.org/html/2610.02372#bib.bib10);[Ahn et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib1);[Hong, 2024](https://arxiv.org/html/2610.02372#bib.bib13);[Karras et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib19);[Kynkäänniemi et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib25);[Yu et al\., 2023](https://arxiv.org/html/2610.02372#bib.bib51)\)\. These methods increase diversity without imposing a reward floor\.

Pareto and satisficing objectives\.AIG\([Jena et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib18)\)studies average reward against distributional coverage; PROUD\([Yao et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib50)\)defines Pareto objectives over per\-image properties; and ParetoSlider\([Golan et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib7)\), constrained diffusion alignment\([Khalafi et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib20)\), and SITAlign\([Chehade et al\., 2025](https://arxiv.org/html/2610.02372#bib.bib2)\)optimize expected rewards\. In contrast, our training\-free formulation imposes a reward floor on every candidate and a diversity requirement on the batch, producing a frontier between worst\-candidate reward and batch diversity\.

## 6Conclusion

SatisDive formulates image generation as satisficing: every candidate must satisfy a reward floor, and the batch must satisfy a diversity cutoff\. Varying the floor provably traces a satisfaction–diversity Pareto frontier\. On FLUX\.1\-dev with HPSv3 and SANA\-1\.6B with ImageReward, SatisDive’s curves Pareto\-dominate FK steering’s over their overlapping DreamSim ranges, improving worst\-candidate reward at matched DreamSim by up to0\.430\.43and0\.700\.70, respectively\. Extending the formulation to multiple reward floors remains future work\.

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###### Appendix Contents

1. [1Introduction](https://arxiv.org/html/2610.02372#S1)
2. [2Problem Formulation](https://arxiv.org/html/2610.02372#S2)
3. [3Method](https://arxiv.org/html/2610.02372#S3)1. [3\.1Satisficing at Inference Time](https://arxiv.org/html/2610.02372#S3.SS1) 2. [3\.2SatisDive’s Objective Traces the Satisfaction–Diversity Pareto Frontier](https://arxiv.org/html/2610.02372#S3.SS2)
4. [4Experiments](https://arxiv.org/html/2610.02372#S4)1. [4\.1SatisDive Traverses the Satisfaction–Diversity Frontier](https://arxiv.org/html/2610.02372#S4.SS1) 2. [4\.2Qualitative Results](https://arxiv.org/html/2610.02372#S4.SS2)
5. [5Related Work](https://arxiv.org/html/2610.02372#S5)
6. [6Conclusion](https://arxiv.org/html/2610.02372#S6)
7. [References](https://arxiv.org/html/2610.02372#bib)
8. [ANotation](https://arxiv.org/html/2610.02372#A1)
9. [BTheoretical Results and Proofs](https://arxiv.org/html/2610.02372#A2)1. [B\.1Reward Tilting](https://arxiv.org/html/2610.02372#A2.SS1) 2. [B\.2Reward\-Floor Frontier](https://arxiv.org/html/2610.02372#A2.SS2) 3. [B\.3Zero\-Temperature Limit of SatisDive’s Objective](https://arxiv.org/html/2610.02372#A2.SS3)
10. [CExperiment Settings](https://arxiv.org/html/2610.02372#A3)1. [C\.12D Toy Setup](https://arxiv.org/html/2610.02372#A3.SS1)
11. [DAdditional Experiments](https://arxiv.org/html/2610.02372#A4)1. [D\.1Effect of Increasing the Number of Candidates](https://arxiv.org/html/2610.02372#A4.SS1) 2. [D\.2Additional Results on DrawBench](https://arxiv.org/html/2610.02372#A4.SS2) 3. [D\.3Effect of a Higher Reward Floor](https://arxiv.org/html/2610.02372#A4.SS3) 4. [D\.4CMMD on MS\-COCO](https://arxiv.org/html/2610.02372#A4.SS4) 5. [D\.5Component Ablations](https://arxiv.org/html/2610.02372#A4.SS5) 6. [D\.6FLUX\.1\-dev with ImageReward](https://arxiv.org/html/2610.02372#A4.SS6) 7. [D\.7Stable Diffusion 1\.5 with ImageReward](https://arxiv.org/html/2610.02372#A4.SS7) 8. [D\.8Comparison with CREPE](https://arxiv.org/html/2610.02372#A4.SS8) 9. [D\.9Comparison with SGI](https://arxiv.org/html/2610.02372#A4.SS9) 10. [D\.10Inference\-Time Computation](https://arxiv.org/html/2610.02372#A4.SS10) 11. [D\.11Reward\-Temperature Robustness](https://arxiv.org/html/2610.02372#A4.SS11) 12. [D\.12Reward Capping without the Diversity Penalty](https://arxiv.org/html/2610.02372#A4.SS12)
12. [EAdditional Qualitative Comparisons](https://arxiv.org/html/2610.02372#A5)

## Appendix ANotation

We summarize the notation used throughout the paper\.

Table 4:Summary of Notation\.
## Appendix BTheoretical Results and Proofs

This section first analyzes reward tilting, then characterizes the reward\-floor frontier, and finally proves the zero\-temperature limit of SatisDive’s objective\.

Fix a promptcc\. Letp=pθ\(⋅∣c\)p=p\_\{\\theta\}\(\\cdot\\mid c\)denote the base\-model distribution, and writeR⁡\(x\)=r⁡\(x,c\)R\(x\)=r\(x,c\)for the reward\. Rewards are measurable and real\-valued\. All image distributions below have densities with respect to the same reference measure, denoted byd​xdxin integrals\. We use the same symbol for a distribution and its density\. For a measurable setEE, its probability isp⁡\(E\)=∫Ep⁡\(x\)​𝑑xp\(E\)=\\int\_\{E\}p\(x\)dx, andEcE^\{c\}denotes its complement\. An indicator𝟏E​\(x\)\\mathbf\{1\}\_\{E\}\(x\)is one onEEand zero elsewhere\.

We assume that the batch sizeK≥2K\\geq 2is a fixed finite integer and writex1:K=\(x1,…,xK\)x^\{1:K\}=\(x^\{1\},\\ldots,x^\{K\}\)\. Candidate indices satisfyk,ℓ∈\{1,…,K\}k,\\ell\\in\\\{1,\\ldots,K\\\}, and pair indices satisfy1≤i<j≤K1\\leq i<j\\leq K\. For a real reward floorρ\\rho, letAρ=\{x:R⁡\(x\)≥ρ\}A\_\{\\rho\}=\\\{x:R\(x\)\\geq\\rho\\\}be the set of images satisfying the floor\. Its base\-model probability isZρ=p⁡\(Aρ\)Z\_\{\\rho\}=p\(A\_\{\\rho\}\)\. A floor is feasible whenZρ\>0Z\_\{\\rho\}\>0\. For such a floor, the base distribution restricted toAρA\_\{\\rho\}is

pρ​\(x\)=p\(x\)𝟏\[x∈Aρ\]Zρ\.p\_\{\\rho\}\(x\)=\\frac\{p\(x\)\\mathbf\{1\}\[x\\in A\_\{\\rho\}\]\}\{Z\_\{\\rho\}\}\.\(12\)
### B\.1Reward Tilting

#### Precise conditions for Proposition[1](https://arxiv.org/html/2610.02372#Thmtheorem1)\.

At a given resampling step, fixKKcandidatesx1:Kx^\{1:K\}with rewardsrk=R⁡\(xk\)r\_\{k\}=R\(x^\{k\}\), and letj⋆j^\{\\star\}be the index of the unique highest\-reward candidate\. Letλ≥0\\lambda\\geq 0be the reward\-tilt strength andak\>0a\_\{k\}\>0a finite factor in candidatekk’s resampling mass that does not depend onλ\\lambda; setting everyak=1a\_\{k\}=1recovers the pure exponential reward weighting in Eq\.[1](https://arxiv.org/html/2610.02372#S2.E1)\. Define the normalized resampling mass of candidatekkas

ωk​\(λ\)=ak​eλ​rk∑ℓ=1Kaℓ​eλ​rℓ\.\\omega\_\{k\}\(\\lambda\)=\\frac\{a\_\{k\}e^\{\\lambda r\_\{k\}\}\}\{\\sum\_\{\\ell=1\}^\{K\}a\_\{\\ell\}e^\{\\lambda r\_\{\\ell\}\}\}\.LetNk∈\{0,…,K\}N\_\{k\}\\in\\\{0,\\ldots,K\\\}be the number of copies of candidatekkafter resampling\. We assume that resampling returnsKKcandidates from this fixed pool, so∑kNk=K\\sum\_\{k\}N\_\{k\}=K, and that𝔼⁡\[Nk\]=K​ωk​\(λ\)\\mathbb\{E\}\[N\_\{k\}\]=K\\omega\_\{k\}\(\\lambda\)for everykk\. We do not assume that the selections are independent\.

###### Proof of Proposition[1](https://arxiv.org/html/2610.02372#Thmtheorem1)\.

We first show that the highest\-reward candidate receives resampling mass tending to one\. We then bound the probability that any other candidate survives among theKKoutputs\. All probabilities and expectations concern only the randomness of resampling from the fixed pool above\.

Step 1: The highest\-reward candidate’s mass tends to one\.For anyℓ≠j⋆\\ell\\neq j^\{\\star\}, the shared normalization cancels:

ωℓ​\(λ\)ωj⋆​\(λ\)=aℓ​eλ​rℓaj⋆​eλ​rj⋆=aℓaj⋆​e−λ⁡\(rj⋆−rℓ\)⟶0\.\\frac\{\\omega\_\{\\ell\}\(\\lambda\)\}\{\\omega\_\{j^\{\\star\}\}\(\\lambda\)\}=\\frac\{a\_\{\\ell\}e^\{\\lambda r\_\{\\ell\}\}\}\{a\_\{j^\{\\star\}\}e^\{\\lambda r\_\{j^\{\\star\}\}\}\}=\\frac\{a\_\{\\ell\}\}\{a\_\{j^\{\\star\}\}\}e^\{\-\\lambda\(r\_\{j^\{\\star\}\}\-r\_\{\\ell\}\)\}\\longrightarrow 0\.\(13\)The limit holds becauserj⋆−rℓ\>0r\_\{j^\{\\star\}\}\-r\_\{\\ell\}\>0\. Dividing the numerator and denominator ofωj⋆\\omega\_\{j^\{\\star\}\}byaj⋆​eλ​rj⋆a\_\{j^\{\\star\}\}e^\{\\lambda r\_\{j^\{\\star\}\}\}gives

ωj⋆​\(λ\)=\(1\+∑ℓ≠j⋆aℓaj⋆​e−λ⁡\(rj⋆−rℓ\)\)−1⟶1\.\\omega\_\{j^\{\\star\}\}\(\\lambda\)=\\left\(1\+\\sum\_\{\\ell\\neq j^\{\\star\}\}\\frac\{a\_\{\\ell\}\}\{a\_\{j^\{\\star\}\}\}e^\{\-\\lambda\(r\_\{j^\{\\star\}\}\-r\_\{\\ell\}\)\}\\right\)^\{\-1\}\\longrightarrow 1\.Each summand decreases to zero, so the denominator decreases to one\. Its reciprocal and the expected copy countK​ωj⋆​\(λ\)K\\omega\_\{j^\{\\star\}\}\(\\lambda\)therefore increase\.

Step 2: AllKKoutputs are copies with probability tending to one\.Because resampling returns exactlyKKcandidates,Nj⋆<KN\_\{j^\{\\star\}\}<Kif and only if some other candidate survives\. Also,𝟏\[Nℓ≥1\]≤Nℓ\\mathbf\{1\}\[N\_\{\\ell\}\\geq 1\]\\leq N\_\{\\ell\}, so taking expectations givesPr\[Nℓ≥1\]≤𝔼\[Nℓ\]\\Pr\[N\_\{\\ell\}\\geq 1\]\\leq\\mathbb\{E\}\[N\_\{\\ell\}\]\. The union bound and the expected copy counts therefore give

Pr\[Nj⋆<K\]\\displaystyle\\Pr\[N\_\{j^\{\\star\}\}<K\]=Pr\[⋃ℓ≠j⋆\{Nℓ≥1\}\]≤∑ℓ≠j⋆Pr\[Nℓ≥1\]\\displaystyle=\\Pr\\\!\\left\[\\bigcup\_\{\\ell\\neq j^\{\\star\}\}\\\{N\_\{\\ell\}\\geq 1\\\}\\right\]\\leq\\sum\_\{\\ell\\neq j^\{\\star\}\}\\Pr\[N\_\{\\ell\}\\geq 1\]≤∑ℓ≠j⋆𝔼⁡\[Nℓ\]=K​∑ℓ≠j⋆ωℓ=K⁡\(1−ωj⋆\)⟶0\.\\displaystyle\\leq\\sum\_\{\\ell\\neq j^\{\\star\}\}\\mathbb\{E\}\[N\_\{\\ell\}\]=K\\sum\_\{\\ell\\neq j^\{\\star\}\}\\omega\_\{\\ell\}=K\(1\-\\omega\_\{j^\{\\star\}\}\)\\longrightarrow 0\.The last equality uses∑kωk=1\\sum\_\{k\}\\omega\_\{k\}=1, and the limit usesωj⋆→1\\omega\_\{j^\{\\star\}\}\\to 1\. Taking the complementary event givesPr\[Nj⋆=K\]=1−Pr\[Nj⋆<K\]→1\\Pr\[N\_\{j^\{\\star\}\}=K\]=1\-\\Pr\[N\_\{j^\{\\star\}\}<K\]\\to 1\.

Multinomial, Srinivasan Sampling Process \(SSP\), and systematic resampling satisfy the expectation condition\. The conclusion also holds if VASR\-MAX’s redirection rule is applied after this systematic resampling: when resampling returnsKKcopies of the highest\-reward candidate, the redirection rule does not change the result\. ∎

For the population\-level comparison, assumeRRis bounded above and letZ⁡\(λ\)=𝔼p​\[eλ​R\]Z\(\\lambda\)=\\mathbb\{E\}\_\{p\}\[e^\{\\lambda R\}\]\. The normalized tilt in Eq\. \([1](https://arxiv.org/html/2610.02372#S2.E1)\) ispλ​\(x\)=p⁡\(x\)​eλ​R​\(x\)/Z⁡\(λ\)p\_\{\\lambda\}\(x\)=p\(x\)e^\{\\lambda R\(x\)\}/Z\(\\lambda\)\.

###### Corollary 4\(A finite reward tilt does not enforce the reward floor\)\.

Fixρ\\rhosuch that0<p⁡\(Aρ\)<10<p\(A\_\{\\rho\}\)<1\. Thenpλ​\(Aρ\)<1p\_\{\\lambda\}\(A\_\{\\rho\}\)<1for every finiteλ≥0\\lambda\\geq 0\.

###### Proof\.

LetBBbe a finite upper bound forRR\. Since0<eλ​R≤eλ​B0<e^\{\\lambda R\}\\leq e^\{\\lambda B\}and∫p⁡\(x\)​𝑑x=1\\int p\(x\)dx=1, taking expectations gives0<Z⁡\(λ\)≤eλ​B<∞0<Z\(\\lambda\)\\leq e^\{\\lambda B\}<\\infty\. The tilt therefore preserves positive probability below the floor:

pλ​\(Aρc\)=∫Aρceλ​R​\(x\)​p​\(x\)​𝑑xZ⁡\(λ\)\>0,pλ​\(Aρ\)=1−pλ​\(Aρc\)<1\.p\_\{\\lambda\}\(A\_\{\\rho\}^\{c\}\)=\\frac\{\\int\_\{A\_\{\\rho\}^\{c\}\}e^\{\\lambda R\(x\)\}p\(x\)dx\}\{Z\(\\lambda\)\}\>0,\\qquad p\_\{\\lambda\}\(A\_\{\\rho\}\)=1\-p\_\{\\lambda\}\(A\_\{\\rho\}^\{c\}\)<1\.∎

### B\.2Reward\-Floor Frontier

In this subsection, support means the set on which a density is positive, with sets of zero base\-model probability ignored;supp⁡\(ν\)\\operatorname\{supp\}\(\\nu\)denotes the support of a distributionν\\nu\. A restriction is proper when it removes a set with positive base\-model probability\. The theorem’s diversity assumption compares distributions by inclusion of these supports, including distributions that assign different probabilities to the same support\. In particular, equal supports have equal diversity: each support is contained in the other, so both diversity inequalities hold\. This distribution\-level diversity is distinct from the per\-batch statisticDDused in the objective\.

We useKL\(ν∥p\)=∫ν\(x\)log\(ν\(x\)/p\(x\)\)dx\\mathrm\{KL\}\(\\nu\\\|p\)=\\int\\nu\(x\)\\log\(\\nu\(x\)/p\(x\)\)dx, with zero\-density contributions taken as zero\. This divergence is\+∞\+\\inftyifν\\nuassigns positive probability to any set of zeroppprobability\. Thus finite divergence requiresν\\nuto be supported withinpp\. All support comparisons and equalities between distributions below ignore sets of zero base\-model probability\.

The factorization in Eq\. \([10](https://arxiv.org/html/2610.02372#S3.E10)\) follows by substituting Eq\. \([12](https://arxiv.org/html/2610.02372#A2.E12)\) for each candidate:

∏k=1K\(p\(xk\)𝟏\[xk∈Aρ\]\)=∏k=1K\(Zρpρ\(xk\)\)=ZρK∏k=1Kpρ\(xk\)\.\\prod\_\{k=1\}^\{K\}\\left\(p\(x^\{k\}\)\\mathbf\{1\}\[x^\{k\}\\in A\_\{\\rho\}\]\\right\)=\\prod\_\{k=1\}^\{K\}\\left\(Z\_\{\\rho\}p\_\{\\rho\}\(x^\{k\}\)\\right\)=Z\_\{\\rho\}^\{K\}\\prod\_\{k=1\}^\{K\}p\_\{\\rho\}\(x^\{k\}\)\.Multiplying both sides by the batch diversity indicator in Eq\. \([4](https://arxiv.org/html/2610.02372#S3.E4)\) preserves this equality\. SinceZρK\>0Z\_\{\\rho\}^\{K\}\>0is constant with respect tox1:Kx^\{1:K\}, it is absorbed into the batch normalizer\. The diversity indicator remains in the target, exactly as in Eq\. \([10](https://arxiv.org/html/2610.02372#S3.E10)\)\.

The following lemma records a standard conditioning property of KL divergence; see[Csiszár and Shields \(2004\)](https://arxiv.org/html/2610.02372#bib.bib4)for background on information projections\.

###### Lemma 5\(Among distributions overAρA\_\{\\rho\},pρp\_\{\\rho\}is uniquely closest to the base model\)\.

For a feasible floorρ\\rho, the distributionpρp\_\{\\rho\}uniquely minimizesKL\(⋅∥p\)\\mathrm\{KL\}\(\\cdot\\\|p\)among distributions that assign probability one toAρA\_\{\\rho\}\. Moreover, every such distributionν\\nuwith finiteKL\(ν∥p\)\\mathrm\{KL\}\(\\nu\\\|p\)has support contained insupp⁡\(pρ\)\\operatorname\{supp\}\(p\_\{\\rho\}\)\.

###### Proof\.

We compute the KL divergence ofpρp\_\{\\rho\}, identify where a competing distribution can assign probability, and then compare their KL divergences directly\.

Step 1: Compute the reference KL and establish support containment\.Whereverpρp\_\{\\rho\}is positive,pρ/p=1/Zρp\_\{\\rho\}/p=1/Z\_\{\\rho\}\. Substituting this ratio and using∫Aρpρ​\(x\)​𝑑x=1\\int\_\{A\_\{\\rho\}\}p\_\{\\rho\}\(x\)dx=1gives

KL\(pρ∥p\)=∫Aρpρ\(x\)log\(1/Zρ\)dx=−logZρ∫Aρpρ\(x\)dx=−logZρ<∞\.\\mathrm\{KL\}\(p\_\{\\rho\}\\\|p\)=\\int\_\{A\_\{\\rho\}\}p\_\{\\rho\}\(x\)\\log\(1/Z\_\{\\rho\}\)dx=\-\\log Z\_\{\\rho\}\\int\_\{A\_\{\\rho\}\}p\_\{\\rho\}\(x\)dx=\-\\log Z\_\{\\rho\}<\\infty\.\(14\)An infinite\-KL distribution cannot improve this finite value\. Consider anyν\\nuwith finite KL toppandν⁡\(Aρ\)=1\\nu\(A\_\{\\rho\}\)=1\. Finite KL requires its support to lie within that ofpp\. The reward constraint also confines it toAρA\_\{\\rho\}\. Its support therefore lies within their intersection, which is exactly the support ofpρp\_\{\\rho\}\.

Step 2: Every competitor has at least as much KL divergence\.The support containment makes the ratios below defined almost everywhere underν\\nu\. We insertpρ/p=1/Zρp\_\{\\rho\}/p=1/Z\_\{\\rho\}into the density ratio and split the logarithm\. The constantlog⁡Zρ\\log Z\_\{\\rho\}can then be taken outside the integral\. Finally,∫Aρν⁡\(x\)​𝑑x=1\\int\_\{A\_\{\\rho\}\}\\nu\(x\)dx=1gives

KL\(ν∥p\)\\displaystyle\\mathrm\{KL\}\(\\nu\\\|p\)=∫Aρν⁡\(x\)​log⁡ν⁡\(x\)p⁡\(x\)​𝑑x\\displaystyle=\\int\_\{A\_\{\\rho\}\}\\nu\(x\)\\log\\frac\{\\nu\(x\)\}\{p\(x\)\}dx=∫Aρν⁡\(x\)​log⁡\(ν⁡\(x\)pρ​\(x\)​1Zρ\)​𝑑x\\displaystyle=\\int\_\{A\_\{\\rho\}\}\\nu\(x\)\\log\\left\(\\frac\{\\nu\(x\)\}\{p\_\{\\rho\}\(x\)\}\\frac\{1\}\{Z\_\{\\rho\}\}\\right\)dx=∫Aρν⁡\(x\)​log⁡ν⁡\(x\)pρ​\(x\)​𝑑x−log⁡Zρ​∫Aρν⁡\(x\)​𝑑x\\displaystyle=\\int\_\{A\_\{\\rho\}\}\\nu\(x\)\\log\\frac\{\\nu\(x\)\}\{p\_\{\\rho\}\(x\)\}dx\-\\log Z\_\{\\rho\}\\int\_\{A\_\{\\rho\}\}\\nu\(x\)dx=KL\(ν∥pρ\)−logZρ\.\\displaystyle=\\mathrm\{KL\}\(\\nu\\\|p\_\{\\rho\}\)\-\\log Z\_\{\\rho\}\.\(15\)KL divergence is nonnegative, with equality only for identical distributions\. ThusKL\(ν∥p\)≥−logZρ=KL\(pρ∥p\)\\mathrm\{KL\}\(\\nu\\\|p\)\\geq\-\\log Z\_\{\\rho\}=\\mathrm\{KL\}\(p\_\{\\rho\}\\\|p\), with equality only forν=pρ\\nu=p\_\{\\rho\}\. This proves the unique minimum; support containment was established above\. ∎

###### Proof of Theorem[3](https://arxiv.org/html/2610.02372#Thmtheorem3)\.

The proof has four steps\. First,pρp\_\{\\rho\}retains maximal diversity at each floor\. Second, we identify its highest guaranteed reward, which need not equal the specified floorρ\\rho\. Third, we show that improving this guarantee requires losing diversity\. Finally, we show that the family represents every frontier point in the stated comparison class\.

Step 1:pρp\_\{\\rho\}has maximal diversity at each floor\.By Eq\. \([12](https://arxiv.org/html/2610.02372#A2.E12)\),pρp\_\{\\rho\}assigns probability one toAρA\_\{\\rho\}and is positive exactly whereppis positive withinAρA\_\{\\rho\}\. Letρ′≥ρ\\rho^\{\\prime\}\\geq\\rhobe another feasible floor\. SinceR⁡\(x\)≥ρ′R\(x\)\\geq\\rho^\{\\prime\}impliesR⁡\(x\)≥ρR\(x\)\\geq\\rho, we haveAρ′⊆AρA\_\{\\rho^\{\\prime\}\}\\subseteq A\_\{\\rho\}\. Intersecting these sets with the support ofppgives

supp⁡\(pρ′\)=supp⁡\(p\)∩Aρ′⊆supp⁡\(p\)∩Aρ=supp⁡\(pρ\)\.\\operatorname\{supp\}\(p\_\{\\rho^\{\\prime\}\}\)=\\operatorname\{supp\}\(p\)\\cap A\_\{\\rho^\{\\prime\}\}\\subseteq\\operatorname\{supp\}\(p\)\\cap A\_\{\\rho\}=\\operatorname\{supp\}\(p\_\{\\rho\}\)\.\(16\)By assumption, restricting support cannot increase diversity, and a proper restriction strictly decreases it\. Equation \([16](https://arxiv.org/html/2610.02372#A2.E16)\) therefore implies thatpρ′p\_\{\\rho^\{\\prime\}\}is no more diverse thanpρp\_\{\\rho\}\. The inequality is strict whenp⁡\(Aρ∖Aρ′\)\>0p\(A\_\{\\rho\}\\setminus A\_\{\\rho^\{\\prime\}\}\)\>0\. Otherwise, the two conditioning sets differ only on a set of zero base\-model probability, sopρ′=pρp\_\{\\rho^\{\\prime\}\}=p\_\{\\rho\}almost everywhere\.

Now consider any distribution that assigns probability one toAρA\_\{\\rho\}and has finite divergence topp\. Lemma[5](https://arxiv.org/html/2610.02372#Thmtheorem5)places its support withinsupp⁡\(pρ\)\\operatorname\{supp\}\(p\_\{\\rho\}\)\. Its diversity therefore cannot exceed the diversity ofpρp\_\{\\rho\}\. Thus,pρp\_\{\\rho\}has maximal diversity at each floor\.

Step 2: Find the highest guaranteed reward\.To compare satisfaction, we need the highest floor actually guaranteed bypρp\_\{\\rho\}\. This can exceed the specifiedρ\\rhowhen there is a gap in the reward values\. Writeg=sup\{a∈ℝ:pρ​\(R≥a\)=1\}g=\\sup\\\{a\\in\\mathbb\{R\}:p\_\{\\rho\}\(R\\geq a\)=1\\\}for the supremum of the guaranteed floors\. Sinceρ\\rhoitself is guaranteed,g≥ρg\\geq\\rho\. We now check thatggis finite and is itself guaranteed, so it can be used in the Pareto comparison\.

First,ggis finite\. If it were infinite, every event\{R≥n\}\\\{R\\geq n\\\}, for positive integersnn, would have probability one\. These events decrease to the empty set because rewards are real\-valued\. Continuity of probability would then give probability one to the empty set, a contradiction\.

Next, the floorggis itself guaranteed\. For each positive integernn, the supremum definition supplies a guaranteed floor greater thang−1/ng\-1/n\. This impliespρ​\(R≥g−1/n\)=1p\_\{\\rho\}\(R\\geq g\-1/n\)=1\. These events decrease to\{R≥g\}\\\{R\\geq g\\\}, so continuity of probability gives

pρ​\(R≥g\)=limn→∞pρ​\(R≥g−1/n\)=1\.p\_\{\\rho\}\(R\\geq g\)=\\lim\_\{n\\to\\infty\}p\_\{\\rho\}\(R\\geq g\-1/n\)=1\.Thusggis the highest floor attained with probability one, not just a supremum of smaller guaranteed floors\. The same decreasing\-event argument applies to any distribution with finite guaranteed reward\.

Step 3: No competitor dominatespρp\_\{\\rho\}\.A distribution dominates another if neither satisfaction nor diversity is lower and at least one is higher\. We rule out both ways a competitor could improve onpρp\_\{\\rho\}\.

*Greater diversity without lower satisfaction\.*Consider a distributionν\\nuwithKL\(ν∥p\)<∞\\mathrm\{KL\}\(\\nu\\\|p\)<\\inftythat guarantees reward at leastgg\. Sinceg≥ρg\\geq\\rho, it satisfiesν⁡\(Aρ\)=1\\nu\(A\_\{\\rho\}\)=1\. Lemma[5](https://arxiv.org/html/2610.02372#Thmtheorem5)therefore places its support within that ofpρp\_\{\\rho\}, soν\\nucannot have greater diversity\.

*Higher satisfaction without lower diversity\.*Supposeν\\nuguarantees a strictly higher floorh\>gh\>g\. Thenpρ​\(R<h\)\>0p\_\{\\rho\}\(R<h\)\>0: otherwisehhwould also be guaranteed underpρp\_\{\\rho\}, contradicting the definition ofgg\. In contrast,ν⁡\(R<h\)=0\\nu\(R<h\)=0\. Sincepρp\_\{\\rho\}assigns probability one toAρA\_\{\\rho\}, substituting its density from Eq\. \([12](https://arxiv.org/html/2610.02372#A2.E12)\) gives

p⁡\(Aρ∖Ah\)=Zρ​pρ​\(R<h\)\>0,ν⁡\(Aρ∖Ah\)≤ν⁡\(R<h\)=0\.p\(A\_\{\\rho\}\\setminus A\_\{h\}\)=Z\_\{\\rho\}p\_\{\\rho\}\(R<h\)\>0,\\qquad\\nu\(A\_\{\\rho\}\\setminus A\_\{h\}\)\\leq\\nu\(R<h\)=0\.The competitor therefore excludes a set of positive base\-model probability from the support ofpρp\_\{\\rho\}\. By strict support monotonicity,ν\\nuhas lower diversity\. No competitor can improve either guaranteed reward or diversity without reducing the other, sopρp\_\{\\rho\}is Pareto\-optimal\. Floors producing the same distribution give the same frontier point\.

Step 4: Every frontier point is represented by the family\.Step 3 establishes that eachpρp\_\{\\rho\}is Pareto\-optimal\. We must also show that no undominated satisfaction–diversity pair is missing from this family\.

Take anyν\\nuwith finite divergence toppand finite guaranteed rewardh=sup\{a∈ℝ:ν⁡\(R≥a\)=1\}h=\\sup\\\{a\\in\\mathbb\{R\}:\\nu\(R\\geq a\)=1\\\}\. The decreasing\-event argument in Step 2 givesν⁡\(Ah\)=1\\nu\(A\_\{h\}\)=1\. We must haveZh=p⁡\(Ah\)\>0Z\_\{h\}=p\(A\_\{h\}\)\>0: otherwiseν\\nuwould assign probability one to a set of zeroppprobability, contradicting finite KL\. Thusphp\_\{h\}exists\. Lemma[5](https://arxiv.org/html/2610.02372#Thmtheorem5)givessupp⁡\(ν\)⊆supp⁡\(ph\)\\operatorname\{supp\}\(\\nu\)\\subseteq\\operatorname\{supp\}\(p\_\{h\}\), sophp\_\{h\}has at least as much diversity asν\\nu\.

We also need to check thatphp\_\{h\}has the same satisfaction asν\\nu\. It guarantees reward at leasthhby construction\. Suppose it guaranteed some floorb\>hb\>h\. Its conditional density would givep⁡\(Ah∖Ab\)=Zh​ph​\(R<b\)=0p\(A\_\{h\}\\setminus A\_\{b\}\)=Z\_\{h\}p\_\{h\}\(R<b\)=0\. Finite KL would then implyν⁡\(Ah∖Ab\)=0\\nu\(A\_\{h\}\\setminus A\_\{b\}\)=0\. Together withν⁡\(Ah\)=1\\nu\(A\_\{h\}\)=1, this givesν⁡\(Ab\)=1\\nu\(A\_\{b\}\)=1, contradicting the definition ofhh\. Therefore, the highest guaranteed reward ofphp\_\{h\}is exactlyhh\.

We have shown thatphp\_\{h\}has the same satisfaction asν\\nuand at least as much diversity\. A proper support inclusion gives strictly greater diversity, so such aν\\nuis dominated byphp\_\{h\}\. Equal supports give the same diversity and hence the same satisfaction–diversity pair\. Every undominated pair of finite satisfaction and diversity is therefore represented by the family, although more than one distribution can represent the same pair\.

∎

### B\.3Zero\-Temperature Limit of SatisDive’s Objective

###### Proof of Proposition[2](https://arxiv.org/html/2610.02372#Thmtheorem2)\.

We first show that the objective’s weight approaches the indicator of a feasible batch\. We then show that its normalizing constant converges to the probability of that event\. Finally, we show that the normalized distribution converges toπ⋆\\pi^\{\\star\}\.

Setup\.Fix a current candidate batch and letMMbe its maximum reward\. Fix the toleranceΔ\>0\\Delta\>0and setτ=ρ=M−Δ\\tau=\\rho=M\-\\Delta\. Also fix the diversity cutoffδ∈ℝ\\delta\\in\\mathbb\{R\}for this objective evaluation\. These values remain fixed as we integrate over batches and take temperature limits\. In particular,MMis not recomputed for each integration variable\.

For a generic batchx1:Kx^\{1:K\}, writerk=R⁡\(xk\)r\_\{k\}=R\(x^\{k\}\)for candidatekk’s reward\. The density of a batch of independent base\-model samples isPc\(x1:K\)=∏k=1Kp\(xk\)P\_\{c\}\(x^\{1:K\}\)=\\prod\_\{k=1\}^\{K\}p\(x^\{k\}\)\. Integrals over batches use the product reference measuredx1⋯dxKdx^\{1\}\\cdots dx^\{K\}\.

Letdi​j=di​j\(x1:K\)d\_\{ij\}=d\_\{ij\}\(x^\{1:K\}\)be the measurable cosine distances between candidates’ features\. We hold the feature map fixed and assume these distances are well defined\. They lie in\[0,2\]\[0,2\]and do not depend on the temperatures\. There are\(K2\)=K⁡\(K−1\)/2\\binom\{K\}\{2\}=K\(K\-1\)/2candidate pairs\. For a reward temperatureβr\>0\\beta\_\{r\}\>0, write the reward gate and weighted diversity as

qk=11\+e\(ρ−rk\)/βr,Dβr=∑i<jqi​qj​di​j∑i<jqi​qj\.q\_\{k\}=\\frac\{1\}\{1\+e^\{\(\\rho\-r\_\{k\}\)/\\beta\_\{r\}\}\},\\qquad D\_\{\\beta\_\{r\}\}=\\frac\{\\sum\_\{i<j\}q\_\{i\}q\_\{j\}d\_\{ij\}\}\{\\sum\_\{i<j\}q\_\{i\}q\_\{j\}\}\.HereDβrD\_\{\\beta\_\{r\}\}is theDDin Eq\. \([8](https://arxiv.org/html/2610.02372#S3.E8)\)\. The subscript makes its dependence on the reward temperature explicit\. The hard target instead uses the unweighted meanD0=\(K2\)−1​∑i<jdi​jD\_\{0\}=\\binom\{K\}\{2\}^\{\-1\}\\sum\_\{i<j\}d\_\{ij\}\. We will show below thatDβr→D0D\_\{\\beta\_\{r\}\}\\to D\_\{0\}whenever all rewards exceed the floor\.

Define the feasible batch setC=\{x1:K:rk≥ρfor everyk,D0≥δ\}C=\\\{x^\{1:K\}:r\_\{k\}\\geq\\rho\\ \\text\{for every \}k,\\ D\_\{0\}\\geq\\delta\\\}\. The proposition assumesPc​\(C\)\>0P\_\{c\}\(C\)\>0, so conditioning on this set is possible\. It also assumesPc​\(rk=ρ\)=0P\_\{c\}\(r\_\{k\}=\\rho\)=0for everykkandPc​\(D0=δ\)=0P\_\{c\}\(D\_\{0\}=\\delta\)=0\. Since there are finitely many candidates, the union of these boundary events also has probability zero\. We may therefore establish the limit outside these events\.

For temperaturesβr,βd\>0\\beta\_\{r\},\\beta\_\{d\}\>0, define the weightwβr,βd=exp⁡\(−𝒥\)w\_\{\\beta\_\{r\},\\beta\_\{d\}\}=\\exp\(\-\\mathcal\{J\}\)using Eq\. \([9](https://arxiv.org/html/2610.02372#S3.E9)\)\. Its normalizing constant isZβr,βd=𝔼Pc​\[wβr,βd\]Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}=\\mathbb\{E\}\_\{P\_\{c\}\}\[w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\]\.

These quantities are well defined at every positive temperature\. Everyqkq\_\{k\}lies in\(0,1\)\(0,1\), andK≥2K\\geq 2ensures that there is at least one pair\. Thus the denominator ofDβrD\_\{\\beta\_\{r\}\}is positive, and its weighted average is finite\. Both penalties are consequently finite and nonnegative, giving0<wβr,βd≤10<w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\\leq 1\. Taking expectations gives0<Zβr,βd≤10<Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\\leq 1\. The induced distribution is therefore

π~βr,βd\(x1:K\)=Pc\(x1:K\)wβr,βd\(x1:K\)Zβr,βd\.\\widetilde\{\\pi\}\_\{\\beta\_\{r\},\\beta\_\{d\}\}\(x^\{1:K\}\)=\\frac\{P\_\{c\}\(x^\{1:K\}\)w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\(x^\{1:K\}\)\}\{Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\.\(17\)
Step 1: The weights converge to the feasible\-batch indicator\.Equation \([7](https://arxiv.org/html/2610.02372#S3.E7)\) gives𝒥r=−∑klogqk\\mathcal\{J\}\_\{r\}=\-\\sum\_\{k\}\\log q\_\{k\}\. We substitute this andsoftplus⁡\(z\)=log⁡\(1\+ez\)\\operatorname\{softplus\}\(z\)=\\log\(1\+e^\{z\}\)intoexp⁡\(−𝒥r−𝒥d\)\\exp\(\-\\mathcal\{J\}\_\{r\}\-\\mathcal\{J\}\_\{d\}\)\. Exponentiating the logarithms gives the quotient below\. Rewriting its denominator usingsigmoid⁡\(z\)=1/\(1\+e−z\)\\operatorname\{sigmoid\}\(z\)=1/\(1\+e^\{\-z\}\)gives a product of reward gates and a diversity gate:

wβr,βd\\displaystyle w\_\{\\beta\_\{r\},\\beta\_\{d\}\}=e∑klog⁡qk​e−log⁡\(1\+e\(δ−Dβr\)/βd\)\\displaystyle=e^\{\\sum\_\{k\}\\log q\_\{k\}\}\\,e^\{\-\\log\(1\+e^\{\(\\delta\-D\_\{\\beta\_\{r\}\}\)/\\beta\_\{d\}\}\)\}\(18\)=∏k=1Kqk1\+e\(δ−Dβr\)/βd=\(∏k=1Kqk\)​sigmoid⁡\(Dβr−δβd\)\.\\displaystyle=\\frac\{\\prod\_\{k=1\}^\{K\}q\_\{k\}\}\{1\+e^\{\(\\delta\-D\_\{\\beta\_\{r\}\}\)/\\beta\_\{d\}\}\}=\\left\(\\prod\_\{k=1\}^\{K\}q\_\{k\}\\right\)\\operatorname\{sigmoid\}\\left\(\\frac\{D\_\{\\beta\_\{r\}\}\-\\delta\}\{\\beta\_\{d\}\}\\right\)\.Takeβr,βd→0\\beta\_\{r\},\\beta\_\{d\}\\to 0through positive values\. Outside the boundary events, there are two reward cases\.

*Case 1: At least one candidate is below the reward floor\.*Ifrk<ρr\_\{k\}<\\rho, thene\(ρ−rk\)/βr→∞e^\{\(\\rho\-r\_\{k\}\)/\\beta\_\{r\}\}\\to\\infty, soqk→0q\_\{k\}\\to 0\. All factors in Eq\. \([18](https://arxiv.org/html/2610.02372#A2.E18)\) lie between zero and one\. Hence0≤wβr,βd≤qk→00\\leq w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\\leq q\_\{k\}\\to 0: one failed reward requirement makes the batch weight vanish, regardless of diversity\.

*Case 2: Every candidate is above the reward floor\.*Now everyrk\>ρr\_\{k\}\>\\rho, soe\(ρ−rk\)/βr→0e^\{\(\\rho\-r\_\{k\}\)/\\beta\_\{r\}\}\\to 0and everyqk→1q\_\{k\}\\to 1\. Each pair weightqi​qjq\_\{i\}q\_\{j\}then tends to one\. Taking limits in the finite sums, whose denominator tends to\(K2\)\>0\\binom\{K\}\{2\}\>0, gives

limβr→0Dβr=∑i<j1⋅1⋅di​j∑i<j1⋅1=∑i<jdi​j\(K2\)=D0\.\\lim\_\{\\beta\_\{r\}\\to 0\}D\_\{\\beta\_\{r\}\}=\\frac\{\\sum\_\{i<j\}1\\cdot 1\\cdot d\_\{ij\}\}\{\\sum\_\{i<j\}1\\cdot 1\}=\\frac\{\\sum\_\{i<j\}d\_\{ij\}\}\{\\binom\{K\}\{2\}\}=D\_\{0\}\.\(19\)We must now check the diversity gate, whose numerator and temperature both change\. SinceD0≠δD\_\{0\}\\neq\\delta, the limiting diversity has a nonzero gap from the cutoff\. Convergence toD0D\_\{0\}ensures that, for sufficiently smallβr\\beta\_\{r\},\|Dβr−D0\|<\|D0−δ\|/2\|D\_\{\\beta\_\{r\}\}\-D\_\{0\}\|<\|D\_\{0\}\-\\delta\|/2\. ThusDβrD\_\{\\beta\_\{r\}\}stays on the same side ofδ\\deltaasD0D\_\{0\}\. The triangle inequality also bounds its distance from the cutoff:

\|Dβr−δ\|≥\|D0−δ\|−\|Dβr−D0\|\>\|D0−δ\|2\>0\.\|D\_\{\\beta\_\{r\}\}\-\\delta\|\\geq\|D\_\{0\}\-\\delta\|\-\|D\_\{\\beta\_\{r\}\}\-D\_\{0\}\|\>\\frac\{\|D\_\{0\}\-\\delta\|\}\{2\}\>0\.The positive lower bound does not depend on either temperature\. Dividing byβd→0\\beta\_\{d\}\\to 0therefore makes the magnitude of the sigmoid argument diverge\. WhenD0\>δD\_\{0\}\>\\delta, its sign is positive and the diversity gate tends to one\. WhenD0<δD\_\{0\}<\\delta, its sign is negative and the gate tends to zero\. This holds regardless of the relative temperature rates\.

Combining these diversity cases with the reward cases gives

limβr,βd→0wβr,βd=\(∏k𝟏\[rk≥ρ\]\)𝟏\[D0≥δ\]=𝟏Calmost everywhere underPc\.\\lim\_\{\\beta\_\{r\},\\beta\_\{d\}\\to 0\}w\_\{\\beta\_\{r\},\\beta\_\{d\}\}=\\left\(\\prod\_\{k\}\\mathbf\{1\}\[r\_\{k\}\\geq\\rho\]\\right\)\\mathbf\{1\}\[D\_\{0\}\\geq\\delta\]=\\mathbf\{1\}\_\{C\}\\quad\\text\{almost everywhere under \}P\_\{c\}\.\(20\)
Step 2: The normalizer converges to the feasible\-batch probability\.To normalize the limiting weights, we also need the limit of their expectation\. Since0≤wβr,βd≤10\\leq w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\\leq 1andPcP\_\{c\}is a probability distribution, the constant one is an integrable bound\. Dominated convergence therefore allows us to move the limit inside the expectation in Eq\. \([20](https://arxiv.org/html/2610.02372#A2.E20)\):

limβr,βd→0Zβr,βd=limβr,βd→0𝔼Pc​\[wβr,βd\]=𝔼Pc​\[𝟏C\]=Pc​\(C\)\>0\.\\lim\_\{\\beta\_\{r\},\\beta\_\{d\}\\to 0\}Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}=\\lim\_\{\\beta\_\{r\},\\beta\_\{d\}\\to 0\}\\mathbb\{E\}\_\{P\_\{c\}\}\[w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\]=\\mathbb\{E\}\_\{P\_\{c\}\}\[\\mathbf\{1\}\_\{C\}\]=P\_\{c\}\(C\)\>0\.\(21\)
Step 3: The normalized distribution converges toπ⋆\\pi^\{\\star\}\.The target in Eq\. \([4](https://arxiv.org/html/2610.02372#S3.E4)\) is the base\-batch distribution conditioned onCC:

π⋆\(x1:K\)=Pc\(x1:K\)𝟏C\(x1:K\)Pc​\(C\)\.\\pi^\{\\star\}\(x^\{1:K\}\)=\\frac\{P\_\{c\}\(x^\{1:K\}\)\\mathbf\{1\}\_\{C\}\(x^\{1:K\}\)\}\{P\_\{c\}\(C\)\}\.\(22\)Equations \([20](https://arxiv.org/html/2610.02372#A2.E20)\) and \([21](https://arxiv.org/html/2610.02372#A2.E21)\) givewβr,βd/Zβr,βd→𝟏C/Pc​\(C\)w\_\{\\beta\_\{r\},\\beta\_\{d\}\}/Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\\to\\mathbf\{1\}\_\{C\}/P\_\{c\}\(C\)almost everywhere underPcP\_\{c\}\. Division is valid because the limiting normalizerPc​\(C\)P\_\{c\}\(C\)is positive\.

To conclude convergence in total variation, we need to integrate the absolute difference of these ratios\. For sufficiently small temperatures, convergence of the normalizer ensuresZβr,βd≥Pc​\(C\)/2Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\\geq P\_\{c\}\(C\)/2\. The triangle inequality and0≤wβr,βd,𝟏C≤10\\leq w\_\{\\beta\_\{r\},\\beta\_\{d\}\},\\mathbf\{1\}\_\{C\}\\leq 1then give

\|wβr,βdZβr,βd−𝟏CPc​\(C\)\|≤wβr,βdZβr,βd\+𝟏CPc​\(C\)≤2Pc​\(C\)\+1Pc​\(C\)=3Pc​\(C\)\.\\left\|\\frac\{w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\{Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\-\\frac\{\\mathbf\{1\}\_\{C\}\}\{P\_\{c\}\(C\)\}\\right\|\\leq\\frac\{w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\{Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\+\\frac\{\\mathbf\{1\}\_\{C\}\}\{P\_\{c\}\(C\)\}\\leq\\frac\{2\}\{P\_\{c\}\(C\)\}\+\\frac\{1\}\{P\_\{c\}\(C\)\}=\\frac\{3\}\{P\_\{c\}\(C\)\}\.This finite constant is integrable under the probability distributionPcP\_\{c\}, so dominated convergence applies again\. Total variation is half the integral of the absolute density difference\. We substitute the densities from Eqs\. \([17](https://arxiv.org/html/2610.02372#A2.E17)\) and \([22](https://arxiv.org/html/2610.02372#A2.E22)\) into this integral\. Factoring out the common densityPcP\_\{c\}turns the integral into an expectation:

2​‖π~βr,βd−π⋆‖TV\\displaystyle 2\\left\\\|\\widetilde\{\\pi\}\_\{\\beta\_\{r\},\\beta\_\{d\}\}\-\\pi^\{\\star\}\\right\\\|\_\{\\mathrm\{TV\}\}=∫\|π~βr,βd\(x1:K\)−π⋆\(x1:K\)\|dx1⋯dxK\\displaystyle=\\int\\left\|\\widetilde\{\\pi\}\_\{\\beta\_\{r\},\\beta\_\{d\}\}\(x^\{1:K\}\)\-\\pi^\{\\star\}\(x^\{1:K\}\)\\right\|dx^\{1\}\\cdots dx^\{K\}=𝔼Pc​\[\|wβr,βdZβr,βd−𝟏CPc​\(C\)\|\]⟶0\.\\displaystyle=\\mathbb\{E\}\_\{P\_\{c\}\}\\left\[\\left\|\\frac\{w\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\{Z\_\{\\beta\_\{r\},\\beta\_\{d\}\}\}\-\\frac\{\\mathbf\{1\}\_\{C\}\}\{P\_\{c\}\(C\)\}\\right\|\\right\]\\longrightarrow 0\.\(23\)The integrand tends to zero by the ratio limit above\. Dividing by two proves convergence toπ⋆\\pi^\{\\star\}in total variation asβr,βd→0\\beta\_\{r\},\\beta\_\{d\}\\to 0\. ∎

#### Pareto optimality of the batch targets\.

The preceding proof connects the objective toπ⋆\\pi^\{\\star\}\. We now show that this batch target is Pareto\-optimal under the support\-based diversity assumption\. The argument follows the same two comparisons as the single\-image theorem: a competitor cannot increase diversity while preserving the reward guarantee, and a strictly higher guarantee requires lower diversity\.

Setup\.FixPcP\_\{c\},D0D\_\{0\}, andδ\\deltaas above\. WriteS\(x1:K\)=minkR\(xk\)S\(x^\{1:K\}\)=\\min\_\{k\}R\(x^\{k\}\)for worst\-candidate reward\. The feasible set is thenC=\{S≥ρ,D0≥δ\}C=\\\{S\\geq\\rho,\\ D\_\{0\}\\geq\\delta\\\}, andπ⋆=Pc​𝟏C/Pc​\(C\)\\pi^\{\\star\}=P\_\{c\}\\mathbf\{1\}\_\{C\}/P\_\{c\}\(C\), withPc​\(C\)\>0P\_\{c\}\(C\)\>0\.

Assume distribution\-level diversity is nonincreasing under support restriction\. Assume also that it strictly decreases when a set of positivePcP\_\{c\}probability is removed\. This diversity compares distributions over batches; it is distinct from the per\-batch statisticD0D\_\{0\}\.

Step 1: Apply the conditioning lemma on batch space\.The conditioning calculation in Lemma[5](https://arxiv.org/html/2610.02372#Thmtheorem5)uses only the positive probability of the conditioning event\. We can therefore substitutePcP\_\{c\},CC, andπ⋆\\pi^\{\\star\}forpp,AρA\_\{\\rho\}, andpρp\_\{\\rho\}\. This givesKL\(π⋆∥Pc\)=−logPc\(C\)<∞\\mathrm\{KL\}\(\\pi^\{\\star\}\\\|P\_\{c\}\)=\-\\log P\_\{c\}\(C\)<\\infty\. It also shows that every finite\-KL distribution assigning probability one toCChas support contained in that ofπ⋆\\pi^\{\\star\}\.

Step 2: Preserving the reward guarantee cannot increase diversity\.Letggbe the highest guaranteed value ofSSunderπ⋆\\pi^\{\\star\}\. HereSSis real\-valued and is at leastρ\\rhowith probability one underπ⋆\\pi^\{\\star\}\. Step 2 of Theorem[3](https://arxiv.org/html/2610.02372#Thmtheorem3)’s proof therefore applies:ggis finite,g≥ρg\\geq\\rho, andπ⋆​\(S≥g\)=1\\pi^\{\\star\}\(S\\geq g\)=1\.

Consider a competitorν\\nuwith finite KL toPcP\_\{c\}that satisfiesD0≥δD\_\{0\}\\geq\\deltaalmost surely and guarantees reward at leastgg\. Becauseg≥ρg\\geq\\rho, this competitor also satisfies the reward requirement definingCC\. The union bound makes both requirements explicit:

ν⁡\(Cc\)≤ν⁡\(S<ρ\)\+ν⁡\(D0<δ\)=0\+0=0,ν⁡\(C\)=1\.\\nu\(C^\{c\}\)\\leq\\nu\(S<\\rho\)\+\\nu\(D\_\{0\}<\\delta\)=0\+0=0,\\qquad\\nu\(C\)=1\.The lemma therefore placesν\\nu’s support within that ofπ⋆\\pi^\{\\star\}, soν\\nucannot have greater diversity\.

Step 3: A higher reward guarantee requires lower diversity\.Ifν\\nuguarantees a strictly higher floorh\>gh\>g, thenπ⋆​\(S<h\)\>0\\pi^\{\\star\}\(S<h\)\>0\. Otherwisehhwould also be guaranteed underπ⋆\\pi^\{\\star\}, contradicting the definition ofgg\. In contrast,ν⁡\(S<h\)=0\\nu\(S<h\)=0\. Substituting the target density gives

Pc\(C∩\{S<h\}\)=Pc\(C\)π⋆\(S<h\)\>0,ν\(C∩\{S<h\}\)=0\.P\_\{c\}\(C\\cap\\\{S<h\\\}\)=P\_\{c\}\(C\)\\pi^\{\\star\}\(S<h\)\>0,\\qquad\\nu\(C\\cap\\\{S<h\\\}\)=0\.This is a proper support restriction, soν\\nuhas strictly lower diversity\. Thusπ⋆\\pi^\{\\star\}is Pareto\-optimal between guaranteed worst\-candidate reward and support\-based diversity within the stated class of batch distributions\.

At a fixed objective evaluation,ρ=M−Δ\\rho=M\-\\Delta\. VaryingΔ\>0\\Delta\>0therefore traverses this target family over the feasible floors belowMM\.

## Appendix CExperiment Settings

#### Sampling and update schedules\.

The main comparisons use499499Pick\-a\-Pic prompts and32003200HPDv2 prompts; the200200DrawBench prompts are evaluated separately in Appendix[D](https://arxiv.org/html/2610.02372#A4)\. We compute95%95\\%percentile bootstrap confidence intervals by resampling prompts with replacement, keeping each prompt’s four candidates together and holdingρ\\rhofixed\. The displayed±\\pmvalues are the interval half\-widths\. For the reward gaps at matched DreamSim in Figure[3](https://arxiv.org/html/2610.02372#S4.F3), we linearly interpolate FK steering’s worst\-candidate reward between adjacent evaluated points, without extrapolating beyond its sampled DreamSim range\.

We sample FLUX at1024×10241024\\times 1024with 28 Euler steps and guidance3\.53\.5, SANA at1024×10241024\\times 1024with 20 Euler steps and guidance4\.54\.5, and SD1\.5 at512×512512\\times 512with 100 DDIM\([Song et al\., 2021](https://arxiv.org/html/2610.02372#bib.bib42)\)steps,η=1\\eta=1, and guidance7\.57\.5\. For FLUX and SANA, every method uses the marginal\-preserving SDE of[Mark et al\. \(2025\)](https://arxiv.org/html/2610.02372#bib.bib33)with noise amplitudea=0\.3a=0\.3\. At steps where reward is evaluated, we estimatex^0\\hat\{x\}\_\{0\}by averaging the velocity predictions from the current and previous steps\. Because the SDE correction is applied through a first\-order Euler update, we replace SANA’s shipped second\-order solver with Euler for each method\. For methods with scheduled scoring or resampling, the common steps are4,8,…,244,8,\\dots,24on FLUX and3,6,…,183,6,\\dots,18on SANA\. On SD1\.5, the baselines use their published configurations, and SatisDive follows FK steering’s schedule, updating latents at steps20,40,60,8020,40,60,80and9999\. We used an NVIDIA A100 GPU for the original experiments and NVIDIA H100 GPUs for runtime and memory profiling\. Unless stated otherwise, each method returns four candidates per prompt using matched starting seeds\.

#### Baseline settings\.

DAS uses Srinivasan Sampling Process \(SSP\) resampling, while FK steering uses multinomial resampling; both resample when the effective sample size is below0\.5​K0\.5K\. For DAS, the annealing rateγ\\gammais determined by the number of sampling steps, and we setα\\alphausing its recommended range for the ratio of the reward\-guidance norm to the model\-drift norm\. FK steering applieseλ​re^\{\\lambda r\}to the unstandardized reward, so we selectλ\\lambdaaccording to the reward scale\. We use the published NegToMe settings\.

#### SatisDive settings\.

We use a fixed step\-size multiplier of11for the latent updates\. We setδ\\deltato the median of the\(K2\)\\binom\{K\}\{2\}pairwise distances at the first update step and setβd=12​δ\\beta\_\{d\}=\\tfrac\{1\}\{2\}\\delta\. At each update, the diversityDDin Eq\.[8](https://arxiv.org/html/2610.02372#S3.E8)is the mean of the current pairwise distances weighted byqi​qjq\_\{i\}q\_\{j\}\. Eachdi​jd\_\{ij\}is the cosine distance between pooled features from FLUX joint block 12, SANA transformer block 7, or the SD1\.5 UNet bottleneck\. We apply late\-stage latent replacement only during the second half of denoising \(s\>T/2s\>T/2\) for every reward and base\-model setting\. We recomputeτ\\tauat each update step but stop gradients throughmaxk⁡rk\\max\_\{k\}r\_\{k\}; otherwise, increasing the current maximum reward would raiseτ\\tauand increase the penalties on the other candidates, producing a gradient that reduces the maximum reward\. The only SatisDive setting swept in the main experiments is the toleranceΔ\\Delta\. For each reward and base model, the selectedΔ\\Deltais approximately three\-quarters of the median per\-prompt difference between the largest and smallest candidate rewards\. We also report sweeps overΔ\\Delta\.

#### Numerical latent updates\.

For each candidate, we normalize the reward and diversity gradients separately\. We weight the normalized reward gradient by\(1−qk\)/maxj⁡\(1−qj\)\(1\-q\_\{k\}\)/\\max\_\{j\}\(1\-q\_\{j\}\), then clip each gradient separately to a maximum norm of0\.10\.1times the mean candidate latent norm at the first update\. After clipping, we multiply the reward gradient by0\.5\+1\.5​\(s−1\)/\(T−1\)0\.5\+1\.5\(s\-1\)/\(T\-1\)and subtract the sum of the two gradients from the latent\. We hold the pair weightsqi​qjq\_\{i\}q\_\{j\}fixed when differentiating the diversity penalty\.

#### Settings across rewards and base models\.

For each reward and base model,ρ\\rhois the median reward of the base model\. With HPSv3, FLUX usesρ=9\.66\\rho=9\.66,Δ=1\.5\\Delta=1\.5, DASα=0\.1\\alpha=0\.1, and FKλ=1\.5\\lambda=1\.5\. With ImageReward, FLUX usesρ=1\.23\\rho=1\.23,Δ=0\.5\\Delta=0\.5, DASα=0\.005\\alpha=0\.005, and FKλ=10\\lambda=10; SANA usesρ=1\.19\\rho=1\.19,Δ=0\.5\\Delta=0\.5, DASα=0\.005\\alpha=0\.005, and FKλ=10\\lambda=10; and SD1\.5 usesρ=0\.18\\rho=0\.18,Δ=0\.75\\Delta=0\.75, DASα=0\.001\\alpha=0\.001, and FKλ=10\\lambda=10\. DAS usesγ=0\.024\\gamma=0\.024on FLUX,γ=0\.035\\gamma=0\.035on SANA, andγ=0\.008\\gamma=0\.008on SD1\.5, as determined by their respective sampling\-step counts\.

The SANA and SD1\.5 qualitative examples with ImageReward use the baseline parameters listed above andΔ=0\.5\\Delta=0\.5for SatisDive\. Each method returns four candidates from matched starting seeds\.

### C\.12D Toy Setup

The base model is a rectified\-flow velocity network trained on a mixture of4949Gaussians on a7×77\\times 7grid with spacing55, using the same interpolant as FLUX, and is frozen for all runs\. The reward is the maximum of4949Gaussian bumps, one per mode\. The1616bumps on every other row and column have height11, and the other3333have heights drawn in\[0\.15,0\.45\]\[0\.15,0\.45\]\. We use the floorρ=0\.5\\rho=0\.5, so only the 16 high\-reward modes contain points that satisfy the floor\. The centers of the 16 high\-reward bumps are displaced from the grid by Gaussian noise with standard deviation1\.51\.5\. For the displayed SatisDive operating point, we useΔ=0\.325\\Delta=0\.325,βr=0\.2\\beta\_\{r\}=0\.2,βd=0\.5\\beta\_\{d\}=0\.5, diversity cutoffδ=20\\delta=20, and latent\-update multiplier0\.70\.7, with scoring and updates at steps3,6,…,243,6,\\ldots,24of the2828sampling steps\. These toy\-specific settings were selected from a hyperparameter sweep;βr\\beta\_\{r\},βd\\beta\_\{d\}, andδ\\deltaare fixed rather than computed from each batch\. Hereβd\\beta\_\{d\}is independent ofδ\\delta, unlike the12​δ\\tfrac\{1\}\{2\}\\deltaused for the image models\. The plotted SatisDive metrics average all200200runs with seeds4242–241241\. The candidate illustration uses seed8888, selected to show four candidates above the floor in distinct high\-reward regions; it is not an average batch\. The base model, reward landscape, and baseline configurations are unchanged\.

## Appendix DAdditional Experiments

Unless stated otherwise, the experiments below use Pick\-a\-Pic prompts, withρ\\rhoset to the base model’s median reward and the metrics defined in Section[4](https://arxiv.org/html/2610.02372#S4)\.

### D\.1Effect of Increasing the Number of Candidates

We test whether SatisDive remains more diverse than reward\-steering baselines when the number of candidates increases fromK=4K\{=\}4toK=16K\{=\}16\. As Table[5](https://arxiv.org/html/2610.02372#A4.T5)shows with the SD1\.5 base model and ImageReward, SatisDive achieves a much higher DreamSim of0\.770\.77than DAS \(0\.040\.04\) or VASR \(0\.470\.47\)\.

Table 5:AtK=16K\{=\}16, SatisDive maintains a competitive satisfaction rate and higher diversity than the reward\-steering baselines\.We use SD1\.5 with ImageReward,Δ=0\.75\\Delta\{=\}0\.75, andρ=0\.16\\rho\{=\}0\.16, the median reward of the base model’sK=16K\{=\}16outputs\.
### D\.2Additional Results on DrawBench

We use DrawBench\([Saharia et al\., 2022](https://arxiv.org/html/2610.02372#bib.bib37)\)to evaluate whether our improvements extend beyond the Pick\-a\-Pic and HPDv2 prompts used in the main text\. Its prompts cover counting, spatial relations, rare combinations, and long descriptions\. As Table[6](https://arxiv.org/html/2610.02372#A4.T6)shows, SatisDive attains a satisfaction rate of0\.630\.63and DreamSim of0\.480\.48, compared with0\.600\.60and0\.050\.05for DAS and0\.570\.57and0\.180\.18for FK steering\. Although NegToMe produces greater diversity, it has a substantially lower satisfaction rate of0\.340\.34\.

Table 6:SatisDive has the highest observed satisfaction rate on DrawBench and higher observed diversity than every reward\-steering method\.We use FLUX\.1\-dev with HPSv3 on the200200DrawBench prompts and setρ=11\.60\\rho\{=\}11\.60\.
### D\.3Effect of a Higher Reward Floor

We repeat the SANA comparison withρ\\rhoset to the7575th percentile rather than the median of the base model’s ImageReward distribution\. As Figure[5](https://arxiv.org/html/2610.02372#A4.F5)shows, SatisDive has a higher observed satisfaction rate and DreamSim than every reward\-steering baseline atΔ∈\{0\.1,0\.3,0\.5,0\.75\}\\Delta\\in\\\{0\.1,0\.3,0\.5,0\.75\\\}\.

Figure 5:AtΔ∈\{0\.1,0\.3,0\.5,0\.75\}\\Delta\\in\\\{0\.1,0\.3,0\.5,0\.75\\\}, SatisDive has a higher observed satisfaction rate and DreamSim than every reward\-steering baseline under the stricter floor\.We use SANA\-1\.6B with ImageReward and setρ=1\.69\\rho\{=\}1\.69, the7575th percentile of the base model’s candidate rewards\. Error bars are95%95\\%bootstrap confidence intervals over prompts\. For legibility, we show five of the sevenΔ\\Deltasettings\.
### D\.4CMMD on MS\-COCO

CMMD\([Jayasumana et al\., 2024](https://arxiv.org/html/2610.02372#bib.bib17)\)measures the distance between distributions of generated and real images\. We compute CMMD between each method’s generated images and the MS\-COCO\([Lin et al\., 2014](https://arxiv.org/html/2610.02372#bib.bib29)\)val2017 images to test whether SatisDive’s diversity advantage over reward\-steering methods coincides with greater distance from the real\-image distribution\. SatisDive obtains a CMMD of0\.810\.81, compared with0\.830\.83–0\.880\.88for the baselines\.

Table 7:SatisDive has the lowest observed CMMD among the evaluated methods\.We report CMMD \(×103\\times 10^\{3\}; lower is better\) using20002000images generated from500500MS\-COCO captions per method and all50005000MS\-COCO val2017 images as the reference distribution\. We use FLUX\.1\-dev with HPSv3,Δ=1\.5\\Delta\{=\}1\.5for SatisDive,λ=1\.5\\lambda\{=\}1\.5for FK steering,α=0\.1\\alpha\{=\}0\.1andγ=0\.024\\gamma\{=\}0\.024for DAS, and the published NegToMe settings\.
### D\.5Component Ablations

Table[8](https://arxiv.org/html/2610.02372#A4.T8)removes late\-stage latent replacement and the diversity penalty separately from SatisDive to measure how each component changes satisfaction and diversity\. Removing late\-stage latent replacement reduces worst\-candidate HPSv3 from9\.399\.39to8\.588\.58and satisfaction from0\.550\.55to0\.500\.50, while removing the diversity penalty reduces DreamSim from0\.560\.56to0\.540\.54\.

Table 8:Removing late\-stage latent replacement lowers worst\-candidate reward, while removing the diversity penalty lowers DreamSim\.We remove one component at a time from SatisDive using FLUX\.1\-dev with HPSv3,Δ=1\.5\\Delta\{=\}1\.5, andρ=9\.66\\rho\{=\}9\.66\. The95%95\\%bootstrap confidence intervals computed from per\-prompt differences exclude zero for the changes in worst\-candidate and mean HPSv3 after removing late\-stage latent replacement and for the changes in DreamSim and mean HPSv3 after removing the diversity penalty\.We additionally ablate late\-stage latent replacement on Stable Diffusion 1\.5 with ImageReward\. As Table[9](https://arxiv.org/html/2610.02372#A4.T9)shows, removing latent replacement reduces worst\-candidate reward from0\.540\.54to−0\.07\-0\.07and satisfaction from0\.770\.77to0\.640\.64\. DreamSim increases from0\.720\.72to0\.880\.88because the batch retains a low\-reward candidate that the full method would replace\.

Table 9:late\-stage latent replacement also improves worst\-candidate reward and satisfaction on Stable Diffusion 1\.5\.We compare the full method with a variant without latent replacement using Stable Diffusion 1\.5 with ImageReward on Pick\-a\-Pic,Δ=0\.75\\Delta\{=\}0\.75, andρ=0\.18\\rho\{=\}0\.18\. The±\\pmvalue is the half\-width of a95%95\\%prompt\-bootstrap confidence interval, where available\.We also test whether the replaced candidate and the selected higher\-reward candidate, whose latents are identical immediately after replacement, remain identical after subsequent denoising\. The full FLUX\.1\-dev run in Table[8](https://arxiv.org/html/2610.02372#A4.T8)records521521replacement events on314314Pick\-a\-Pic prompts, all during the final two update steps\. Immediately before replacement, the replaced candidate has a mean reward of8\.498\.49, compared with10\.8710\.87for the selected candidate\. Across503503distinct pairs of replaced and selected candidates from these events, the corresponding final images have a mean DreamSim of0\.1950\.195\. Thus, identical latents do not necessarily remain identical after subsequent denoising\.

### D\.6FLUX\.1\-dev with ImageReward

Table[10](https://arxiv.org/html/2610.02372#A4.T10)repeats the FLUX\.1\-dev comparison with ImageReward instead of HPSv3 to test whether SatisDive remains more diverse than reward\-steering methods at comparable satisfaction\. SatisDive and FK steering both attain a satisfaction rate of0\.610\.61, while their DreamSim values are0\.570\.57and0\.230\.23, respectively\.

Table 10:With ImageReward, SatisDive matches FK steering’s satisfaction while maintaining higher diversity\.We repeat the FLUX\.1\-dev comparison using ImageReward,Δ=0\.5\\Delta\{=\}0\.5, andρ=1\.23\\rho\{=\}1\.23\. The±\\pmvalues are half\-widths of95%95\\%percentile bootstrap confidence intervals over prompts\.
### D\.7Stable Diffusion 1\.5 with ImageReward

Table[11](https://arxiv.org/html/2610.02372#A4.T11)tests whether the satisfaction–diversity comparison holds on SD1\.5 with ImageReward\. AtΔ=0\.5\\Delta\{=\}0\.5, SatisDive attains a satisfaction rate of0\.800\.80and DreamSim of0\.650\.65, compared with0\.720\.72and0\.490\.49for FK steering and0\.770\.77and0\.360\.36for VASR\. DAS attains a higher satisfaction rate of0\.840\.84but a DreamSim of0\.040\.04\. VASR \(soft\) uses reward\-weighted resampling without redirecting duplicate selections to the highest\-reward candidate\.

Table 11:SatisDive achieves higher satisfaction and diversity than FK steering on Stable Diffusion 1\.5\.We use ImageReward withρ=0\.18\\rho\{=\}0\.18and report SatisDive at three values ofΔ\\Delta\. The±\\pmvalues are half\-widths of95%95\\%percentile bootstrap confidence intervals over prompts\.We use DAS’s published SD1\.5 settings\. DAS attains the highest ImageReward satisfaction rate,0\.840\.84, but its mean held\-out HPSv3 is−4\.01\-4\.01, compared with5\.775\.77for the base model and6\.246\.24–6\.576\.57for SatisDive\.

### D\.8Comparison with CREPE

Table[12](https://arxiv.org/html/2610.02372#A4.T12)compares SatisDive with CREPE\([He et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib9)\)at reward scales1010and100100on the same Pick\-a\-Pic prompts using SD1\.5 and ImageReward\. AtΔ=1\.0\\Delta\{=\}1\.0, SatisDive achieves higher worst\-candidate reward at similar DreamSim than both CREPE settings; atΔ=1\.25\\Delta\{=\}1\.25, it has higher observed means for both metrics\.

Table 12:SatisDive improves worst\-candidate reward at comparable diversity to both evaluated CREPE settings\.Both methods returnK=4K\{=\}4images per prompt using SD1\.5 with ImageReward\. We useρ=0\.18\\rho\{=\}0\.18and report means on Pick\-a\-Pic;±\\pmvalues are half\-widths of95%95\\%percentile confidence intervals from200,000200\{,\}000prompt\-bootstrap resamples, keeping each prompt’s four candidates together andρ\\rhofixed\.We adapt the released CREPE sampler to SD1\.5, generating512×512512\\times 512images with guidance7\.57\.5,6464noise levels, and200200replica\-exchange iterations\. We evaluate effective terminal reward scales1010and100100; scale100100corresponds to the released reward maximum, with all other settings unchanged between runs\. Four outputs are retained at fixed late iterations and denoised to completion, without reward\-based selection\. SatisDive uses the SD1\.5 configuration in Appendix[C](https://arxiv.org/html/2610.02372#A3)\. We evaluate independently generated batches on the same prompts using common ImageReward and DreamSim implementations\.

### D\.9Comparison with SGI

We compare with SGI\([Parmar et al\., 2026](https://arxiv.org/html/2610.02372#bib.bib34)\), which begins withM=128M\{=\}128candidates and progressively prunes them to returnK=4K\{=\}4\. Its generation budget is therefore not matched to methods that maintain four candidates throughout sampling\. We use HPSv3 to score individual candidates and sweep SGI’s diversity weight, denotedλSGI\\lambda\_\{\\mathrm\{SGI\}\}to distinguish it from FK steering’sλ\\lambda\. AsλSGI\\lambda\_\{\\mathrm\{SGI\}\}increases, DreamSim increases from0\.6470\.647to0\.8230\.823, while satisfaction rate decreases from0\.5920\.592to0\.5070\.507\.

Table 13:Increasing SGI’s diversity weight increases DreamSim but decreases satisfaction rate\.We evaluate SGI on Pick\-a\-Pic using FLUX\.1\-dev with HPSv3\. SGI prunesM=128M\{=\}128candidates toK=4K\{=\}4, andλSGI\\lambda\_\{\\mathrm\{SGI\}\}weights its DINO diversity score\. We useρ=9\.66\\rho\{=\}9\.66; the±\\pmvalues are half\-widths of95%95\\%bootstrap confidence intervals over prompts\.
### D\.10Inference\-Time Computation

Figure[6](https://arxiv.org/html/2610.02372#A4.F6)reports diffusion\-model passes separately from forward evaluations of auxiliary scoring models\. Under the FLUX\.1\-dev configuration, SatisDive uses more diffusion\-model passes than Base, NegToMe, FK steering, FK \+ NegToMe, and VASR, but fewer diffusion\-model passes and reward\-model forward evaluations than DAS even when every partial SatisDive backward pass is counted as a full pass\. SGI uses more diffusion\-model passes and repeatedly applies HPSv3 and DINO while pruningM=128M\{=\}128candidates toK=4K\{=\}4\. These are operation counts rather than wall\-clock measurements\.

For SatisDive, the152152diffusion\-model forward passes comprise112112denoising passes,2424additional gradient\-evaluation passes,1212replacement\-check passes, and44final feature\-extraction passes\. Adding2424full and2424partial backward passes gives the reported upper bound of200200\. The4040reward evaluations comprise2424update evaluations,1212replacement checks, and44final evaluations\. Panel \(b\) reports auxiliary\-model forward evaluations\.

Figure 6:Inference\-time operation counts for the evaluated FLUX\.1\-dev methods\.One image\-equivalent pass processes one candidate once through the indicated model; processing a batch ofbbcandidates therefore counts asbbpasses\. We count operations per prompt forK=4K\{=\}4returned candidates and2828denoising steps\.\(a\)SatisDive uses at most200200diffusion\-model passes, compared with112112for Base, NegToMe, FK steering, FK \+ NegToMe, and VASR,224224for DAS, and340340for SGI\. The SatisDive count is an upper bound because2424of its4848backward passes stop at the intermediate feature capture\. DAS computes a reward gradient at every denoising step; its SSP resampling condition is checked at steps4,8,…,244,8,\\ldots,24\.\(b\)We count auxiliary\-model forward evaluations\. SGI uses252252HPSv3 evaluations and252252DINO evaluations while pruningM=128M\{=\}128candidates toK=4K\{=\}4\. The counts exclude VAE decoding and SGI’s candidate\-selection optimization\.Table[14](https://arxiv.org/html/2610.02372#A4.T14)separately reports measured generation time and GPU memory for the FLUX\.1\-dev implementations\. We use the same three prompts and starting seeds for all methods, following one excluded warmup prompt\.

Table 14:Measured runtime and GPU memory for FLUX\.1\-dev\.Each prompt returnsK=4K\{=\}4images at1024×10241024\\times 1024resolution using2828denoising steps and HPSv3 for reward\-guided methods\. Time is the mean over three prompts, excluding model loading and image\-file writes\. Memory is the largest sampled resident GPU memory, summed across GPUs over generation and reward\-model processes, rather than an exact peak\. All runs use H100 80 GB GPUs; two\-GPU methods place the generator and reward model on separate devices\. Figure[3](https://arxiv.org/html/2610.02372#S4.F3)reports the corresponding worst\-candidate reward improvements at matched diversity\.
### D\.11Reward\-Temperature Robustness

Becauseβr\\beta\_\{r\}is computed from the standard deviation of the candidate rewards, it depends on the initial\-noise seed\. We repeat the Stable Diffusion 1\.5 experiment with five seeds\. Across these seeds, worst\-candidate reward ranges from0\.540\.54to0\.590\.59, satisfaction from0\.750\.75to0\.780\.78, and DreamSim from0\.710\.71to0\.730\.73\. None of the pairwise differences between seeds is statistically significant for any reported metric at the5%5\\%level\.

Table 15:SatisDive’s satisfaction and diversity remain stable across five initial\-noise seeds\.We use Stable Diffusion 1\.5 with ImageReward on the same Pick\-a\-Pic prompts, setΔ=0\.75\\Delta\{=\}0\.75, and computeβr\\beta\_\{r\}separately for each prompt and seed\. The±\\pmvalues are half\-widths of95%95\\%bootstrap confidence intervals over prompts\.
### D\.12Reward Capping without the Diversity Penalty

To test whether stopping further reward optimization atρ\\rhois sufficient to preserve diversity, we modify DAS by replacingrkr\_\{k\}withmin⁡\(rk,ρ\)\\min\(r\_\{k\},\\rho\)in its guidance gradient and resampling weights\. This makes rewards at or aboveρ\\rhoequivalent in both computations without adding SatisDive’s diversity penalty\. All other DAS settings remain unchanged\. Reward capping increases DreamSim from0\.0160\.016to0\.0660\.066, while SatisDive reaches0\.6580\.658on the same prompts and reward floor\. The change in DAS’s satisfaction rate is not statistically significant\.

Table 16:Reward\-capped DAS remains less diverse than SatisDive\.We use SANA\-1\.6B with ImageReward on Pick\-a\-Pic and setρ=1\.19\\rho\{=\}1\.19\. The reward\-capped variant replacesrkr\_\{k\}withmin⁡\(rk,ρ\)\\min\(r\_\{k\},\\rho\)in the guidance gradient and resampling weights\. The±\\pmvalues are half\-widths of95%95\\%bootstrap confidence intervals over prompts\.

## Appendix EAdditional Qualitative Comparisons

Figures[7](https://arxiv.org/html/2610.02372#A5.F7)–[9](https://arxiv.org/html/2610.02372#A5.F9)extend the SANA\-1\.6B comparison to all seven methods\. Each row contains four candidates generated from matched starting seeds and reports their minimum ImageReward and mean pairwise DreamSim; the pink outline marks the candidate with the lowest ImageReward in that row\. The reward floor isρIR=1\.19\\rho\_\{\\mathrm\{IR\}\}\{=\}1\.19\. Figure[10](https://arxiv.org/html/2610.02372#A5.F10)provides a corresponding comparison with Stable Diffusion 1\.5 at its reward floorρIR=0\.18\\rho\_\{\\mathrm\{IR\}\}\{=\}0\.18\.

![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sana_qualitative_pickapic_appendix_00420.png)Figure 7:On the Japanese\-highway car prompt, SatisDive returns four visually distinct candidates above the reward floor\.SatisDive attains a minimum ImageReward of1\.641\.64and a DreamSim of0\.730\.73\. FK steering, VASR, DAS, and FK \+ NegToMe also satisfy the reward floor, but their DreamSim values are0\.110\.11,0\.100\.10,0\.020\.02, and0\.220\.22\. Base and NegToMe remain diverse, but their minimum rewards of1\.131\.13and−0\.03\-0\.03fall below the reward floor\.![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sana_qualitative_pickapic_appendix_00036.png)Figure 8:On the fairy\-and\-owl prompt, SatisDive has the highest minimum ImageReward,1\.731\.73, and a DreamSim of0\.680\.68\.FK steering, VASR, DAS, and FK \+ NegToMe have DreamSim values between0\.020\.02and0\.230\.23\. Base and NegToMe remain diverse, but their minimum rewards of0\.690\.69and−1\.44\-1\.44are below the reward floor\.![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sana_qualitative_pickapic_appendix_00339.png)Figure 9:On the dog\-driving\-a\-bus prompt, SatisDive attains a minimum ImageReward of1\.461\.46and a DreamSim of0\.840\.84\.VASR and DAS satisfy the reward floor but return visually similar candidates\. Base and NegToMe have comparable diversity to SatisDive, but each includes candidates below the reward floor\.![Refer to caption](https://arxiv.org/html/2610.02372v1/figures/sd15_qualitative_pickapic_appendix_00065.png)Figure 10:For Stable Diffusion 1\.5, SatisDive returns four visually distinct images with a minimum ImageReward of1\.721\.72\.The prompt describes a man on a rocky precipice above clouds\. Base and NegToMe have minimum rewards of0\.490\.49and0\.030\.03, while the candidates from FK steering, VASR, and DAS are visually similar within each row\. All methods use matched starting seeds\.