Discount Model Search for Quality Diversity Optimization in High-Dimensional Measure Spaces
Authors: Bryon Tjanaka
Henry Chen
Matthew Christopher Fontaine
Stefanos Nikolaidis
Venue: ICLR 2026 Oral (2026)
Keywords: quality diversity optimization
black-box optimization
derivative-free optimization
latent space exploration
Primary area: optimization
TL;DR: We present a method that enhances exploration in quality diversity (QD) optimization and show how this method enables new applications for QD. Project page: https://discount-models.github.io/

Abstract: Quality diversity (QD) optimization searches for a collection of solutions that optimize an objective while attaining diverse outputs of a user-specified, vector-valued measure function. Contemporary QD algorithms are typically limited to low-dimensional measures because high-dimensional measures are prone to distortion, where many solutions found by the QD algorithm map to similar measures. For example, the state-of-the-art CMA-MAE algorithm guides measure space exploration with a histogram in measure space that records so-called discount values. However, CMA-MAE stagnates in domains with high-dimensional measure spaces because solutions with similar measures fall into the same histogram cell and hence receive the same discount value. To address these limitations, we propose Discount Model Search (DMS), which guides exploration with a model that provides a smooth, continuous representation of discount values. In high-dimensional measure spaces, this model enables DMS to distinguish between solutions with similar measures and thus continue exploration. We show that DMS facilitates new capabilities for QD algorithms by introducing two new domains where the measure space is the high-dimensional space of images, which enables users to specify their desired measures by providing a dataset of images rather than hand-designing the measure function. Results in these domains and on high-dimensional benchmarks show that DMS outperforms CMA-MAE and other existing black-box QD algorithms.

Published as a conference paper at ICLR 2026
DISCOUNT MODEL SEARCH FOR QUALITY DIVERSITY OPTIMIZATION IN HIGH-DIMENSIONAL MEASURE SPACES
Bryon Tjanaka, Henry Chen, Matthew C. Fontaine, Stefanos Nikolaidis
'Thomas Lord Department of Computer Science, University of Southern California 2 Archimedes AI ft janaka, hchen 65, mfontain, aoklaldecee
ABSTRACT
Quality diversity (QD) optimization searches for collection of solutions that optimize an objective while attaining diverse outputs of user-specified, vector- valued measure function. Contemporary QD algorithms are typically limited low-dimensional measures because high-ding gigh-dimensional measures are prone to dis- tortion, where many solutions found by the QD algorithm map to similar mea- sures. For example, the state-of-the-art CMA-MAE algorithm guides measure space exploration with histogram measure space that records so-called dis- count values. However, CMA-MAE stagnates in domains with high-dimensional measure spaces because solutions with similar measures fall into the same his- togram cell and hence receive the same discount value. To address these lim- itations, we propose Discount Model Search (DMS), which guides exploration with model that provides smooth, continuous representation of discount val- ues. In high-dimensional measure spaces, this model enables DMS to distinguish between solutions with similar measures and thus continue exploration. We show that DMS facilitates new capabilities for QD algorithms by introducing two new domains where the measure space is the nigh-dimensional space of images, which enables users to specify their desired measures by providing dataset of images rather than designing the measure function. Results in these domains and on high-dimensiona, benchmarks show that DMS outperforms CMA-MA and other existing black-box QD algorithms.
INTRODUCTION
We present method that enhances exploration in quality diversity (QD) optimization and show how this method enables new applications for QD (Pugh et al., 2016) is a branch of stochas- tic optimization that seeks to find diverse, gigh-performing solutions
problem. with applica- tions like robotics (Mouret & Clune, 2015)), generative modeling (Ding et al.) 2024). and LLM red-teaming (Samvelyan al. (025). To illustrate, consider searching for "a photo of hiker." By itself, this problem is ambiguous since hikers vary widely in appearance, depending on, for instance, where they hiking and the time of year. If we single-objective optimization algorithm like Adam (Kingma & Ba 2015) or CMA-ES (Hansen, 2016), the image we find would optimize the objective I of "a photo of a hiker," but the hiker could take on one of many different appearances. In contrast, QD (Pugh et al. 2016) can manage the ambiguity by searching for an archive (set) of images that both optimize the objective f and diversify along the outputs of measure function m.
Prior work (Pugh al.) 2016) typically hand-designs m to output low www-dimensional (< 1OD) vectors. To illustrate, our measure function could output two measures: the hiker's age and the temperature for which they are dressed. Then, our archive would contain images like younger hiker dressed for cold weather and an older hiker dressed for warm weather, as well as all images in between.
One reason prior works focus on low-dimensional measures is that high-dimensional measure spaces are prone to distortion, where many solutions (images) map to a small region of measure space
Project page available at httphttpp////m. //d / www.iticontttttt siiodels.githu..111

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2D L. (Sphere)
0g
02
0
0
10D (Sphere)
f
400-
A
Unique Cells
A
A2
fA
A2
fA
200
Cell 1
Cell 2 3ell
Cell
Cell 2
Cell 3
0 5000
10000
(0)
m() (b)
Iterations
(a)
(c)
Figure I: (a): One failure mode of CMA-MAE. On a flat objective f. solutions @ and 02 fall in the same archive cell based on their measures, resulting in identical discount values from the discount function fA- (b): In our proposed DMS, the discount model provides a smooth discount function that assigns distinct discount values 0g and 02, showing that 0g has greater archive improvement than 02 (A2 > A) and thus providing a stronger signal to guide search. (c): Number of unique cells where solutions sampled by CMA-MAE land in two benchmarks (mean over 20 trials: Sec.
(i.e.. the solutions have similar measures). For example, it may be easy to search for images of hikers in warm weather, while hikers in cold weather are hard find. Although distortion exists in low-dimensional measure spaces (Fontaine & Nikolaidis, 2021; Fontaine et al.) (020), it is more prominent in high-dimensional measure spaces because there are exponentially larger volumes to which solutions can map SSec. 4).
We propose to scale to high-dimensional measure spaces by addressing the effects of distortion in Covariance Matrix Adaptation MAP-Annealing (CMA-MAE) (Fontaine & Nikolaidis, 2023), state-of-the-art black-box QD algorithm. To fill the archive of solutions, CMA-MAE searches for solutions 0 that maximize archive improvement, defined as A()) = f() fm)) where f(0) is the solution's 's objective value and fA is a discount function that returns scalar discount values based the solution's 's measures m m0). CMA-MAE represents fA as histogram by tessellating the measure space into cells and storing a scalar value in each cell. In domaing with high distortion, particularly domains with high-dimensional measures, solutions with similar measures fall into the same cell. As a result, CMA-MAE incorrectly assigns these solutions the same discount value, which creates inaccurate improvement values that cause the search to stagnate Fig. lap.
Our key insight is that a smooth, continuous representation of the discount function will enhance exploration in nigh-dimensional measure spaces. As such, we propose Discount Model Search (DMS), where smooth discount model accurately assigns distinct discount values to solutions even when distortion causes them to have similar measures (Fig. 1b). The discount values guide DMS to explore the measure space and discover solutions long after CMA-MAE would stagnate.
We show that by scaling to high-dimensional measure spaces, DMS facilitates new capabilities for QD. For example, it can be challenging to design low-dimensional measure function to describe "where the hiker is located," as locations vary widely from beaches to mountains to forests. In general, creating mcasure functions can be tedious and unintuitive
similar to objective func- tions (Skalse et al. 2022, Lehman et al. 2020). the design of the measure function vastly affects the quality of solutions produced (Bossens et al., 2020 Pugh et al. 2015). In contrast, consider that by defining the measures as age and temperature in our earlier example, we essentially speci- fied a 2D grid of (age, temperature) points for which we sought images of hikers. If we treat the high-dimensional space of images (e.g., 256 x 256 x 3 RGB vectors) as the measure space, we can replace the 2D points with images, i.e., we can easily specify "where the hiker is located" by pro- viding a dataset of landscape images (Fig. 2}. Thus, by scaling to high- nigh-dimensional measure spaces, we believe DMS makes QD more accessible: it enables specifying measures via datasets, which can alleviate the need for manual measure function design and enable specifying new types of measures. We refer to this setup as Quality Diversity with Datasets of Measures (QDDM).
In our work, the term "domain" is synonymous with "problem."
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Figure 2: In the LSI (Hiker) domain, the objective is "A photo of the face of a hiker," and the measure space the space of images. We specify desired measures with landscape images from LHQ (Skorokhodov et al. 2021). Thus, DMS finds images depicting what a hiker might look like in each landscape: hikers in thick jackets for the mountains or lighter clothing for the beach, and even a baby bundled for the snow. Each hiker is shown to the left of their corresponding landscape.
Our contributions are follows: (1) We propose Discount Model Search (DMS), which improves exploration in nigh-dimensional measure spaces by searching over a smooth representation of the discount function (Sec. 5). (2) We benchmark DMS standard QD domains (Sec. 6.1). showing outperforms CMA-MAE and other black-box QD algorithms (Sec. 7). (3) We propose the QDDM setting and introduce two QDDM domains where images form the measure space (Sec. 6.2), showing
that DMS also outperforms existing algorithms in these domains (Sec. 7p. Overall, given the ubiq- uity of datasets in machine learning, we excited for applications that can be framed as QDDM problems and tackled with DMS.
PROBLEM FORMULATION
As formulated in Fontaine & Nikolaidis (02TT. black-box quality diversity (QD) optimization considers scalar-valued objective function f R - IR and vector-valued measure function m RR -> R Both functions take as input a solution 0 E IR", and m outputs measures. The image of m forms the measure space S. The QD objective is to find, for every E S, solution O such that m(0) = s and [() is maximized. As stated, this QD objective requires infinite memory since S is continuous space. Hence, algorithms based on MAP-Elites (Mouret & Clune 2015) dis- cretize S into ressellation of M cells, leading a relaxed QD objective maxe, Σ f(O)). Each solution 0; has measures located in the region of measure space indicated by cell i in T, and the set of solutions 01.,,s as is referred to an archive A.
3 BACKGROUND
Our work builds on several black-box QD algorithms. We include further related work in QD and image generation in Appendix
(Mouret & Clune 2015) produces "grid archive" where the tessellation T divides the measure space into grid axis-aligned sreptr-)rcttnnles. Each cell the archive stores one solution. Each iteration, MAP-Elites selects archive solution 0, mutates it to create new solution 0', and adds 0' to the archive. As 0' added, assigned cell e based on measure values. 0' replaces the solution in cell if has higher objective value. In this manner, MAP-Elites retains elites, i.c., the best solution found in each cell. We consider a version of MAP- Elites that mutates solutions by adding isotropic Gaussian noise, i.e., solutions created 0' 0 + (0, I)-
MAP-Elites
Since grid archives require exponentially more memory in digh-dimessional measure spaces, prior work (Vassiliades et al. 2018) proposes defining the tessellation T centroidal Voronoi tessella- tion (CVT), where a number centroids (e.g., 10,000) divide the measure space into equally Voronoi cells. We use these CVT archives in experiments in high-dimensional measure spaces.
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MAP-Elites (line) \Vassiliades & Mouret 2018) augments MAP-Elites with the Iso+LineDD oper- ator, which leverages correlations between solutions in the archive by generating new solutions as 0' 0 + I)) + ()) 01). 0 and 02 are sampled from the archive.
Covariance Matrix Adaptation MAP-Annealing (CMA-MAE) (Fontaine & Nikolaidis 2023) is state-of-the-art black-box QD algorithm that integrates the CMA-ES (Hansen 2016) optimizer into MAP-Elites to directly optimize the QD objective (Sec. 2). CMA-MAE maintains a CMA- ES instance that samples solutions O, from a Gaussian distribution NN E). Each solution O, is evaluated and added to the archive. For each O,, CMA-MAE computes an improvement value A; that represents how much 0, improves the archive. The ranking of the improvement values enables the CMA-ES instance to update the distribution parameters 0 and Σ in the direction of greatest archive improvement; thus, CMA-ES continues to sample solutions that improve the archive in future iterations. This update causes CMA-ES to optimize the QD objective. Multiple CMA-ES instances may operate in parallel, with each instance referred to as an emitter.
The improvement value in CMA-MAE defined as A()) - f(0)=f a discount function. CMA-MAE represents LA as a histogram in measure space by associating a discount value with each cell e. In CMA-MAE's predecessor, CMA-ME (Fontaine et al. 2020). the discount value is the objective value of the solution 6g currently in the cell, i.e., f())). W When a new solution is added to cell e, the discount value is updated to the objective of the solution. As in MAP-Elites, a new solution can only replace the cell solution 9. if it has a higher objective value.
where /A R + I
CMA-MAE builds the insight that CMA-ME's discount values can cause the search quickly leave areas of the archive that require further optimization of the objective (Tjanaka et al.) 2022). For example, consider if the maximum objective attainable for cell e is 100, and CMA-ME finds a solution with objective 90. Future solutions that land in e garner little improvement since the discount value associated with cell is now f(O.), ). which is 90. Even a solution O with objective f() = 100 only receives an improvement of 100 - 90 = 10. Thus, CMA-ME immediately searches for solutions in other areas of the archive that offer higher improvement values. In contrast, to continue optimizing for solutions that land in cell e, CMA-MAE sets the discount value to an acceptance threshold fe, rather than the objective value of the solution in the cell. te is initialized to minimum value fmin. As new solution O' enters cell te is updated as te < (1- atte + (O'), where S a S 1 is an archive learning rate that controls how quickly le updates. Consider cell with f fmin O. Given a = 0.1, solution with objective f(0') 90 updates te as = 9. A new solution in e with objective 100 would receive improvement 100 9 = 81, CMA-MAE still receives high improvement for discovering solutions in cell e.
te + (1-
Notably, when a = 1.0, CMA-MAE focuses on exploration and is equivalent to CMA-ME, since te will always be set to the objective value of new solutions. When a 0.0, CMA-MAE performs single-objective optimization because te will be constant, so the improvement only considers the objective, i.e., A0) = s() - C. Thus, adjusting the learning rate enables smoothly balancing between optimization of the objective and exploration of the measure space.
Density Descent Search (DDS) (Lee et al. 2024) removes the discount function from CMA-MAE and introduces a kernel density estimator (KDE) that models the density of previously discovered solutions in measure space. Instead of archive improvement, DDS ranks solutions by the KDE's density estimates, prioritizing solutions in areas of low density. The KDE provides smoother signal than CMA-MAE's discrete histogram, enabling DDS to excel at exploring measure spaces. However, since the KDE does not consider the objective, DDS does not optimize the objective, mak- ing a diversity optimization algorithm, i.e., it only searches for solutions with diverse measures. Nevertheless, we draw inspiration from how the smooth signal in DDS enhances exploration.
4 UNDERSTANDING DISTORTION IN HIGH-DIMENSIONAL MEASURE SPACES
Distortion in QD refers to when large areas of solution space map to a small region of measure space (Fontaine & Nikolaidis 2023). When CMA-MAE encounters distortion, the solutions it sam- ples land in fewer archive cells since they have similar measures. Fig. la shows one scenario where solutions that land in the same cell interfere with CMA-MAE's improvement mechanism. On flat (constant) objective like the one in the figure, CMA-MAE's s histogram represents how often it has visited each area of measure space, and the improvement ranking guides it towards areas that it has
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not visited before. For instance, Cell 3 has the lowest discount value among the three cells since it has not been explored yet, so the direction of greatest archive improvement is to generate solutions that land in Cell 3. However, since 0 and 0₂ both land in Cell 2, they have the same discount value. Since they also have the same constant objective, they receive the same improvement value
(A, = A2). Hence, CMA-MAR cannot identify the direction of greatest archive improvement.
In Fig. lc. we present an experiment that illustrates how distortion causes solutions to land in the same cell, which activates failures like the one above. We run CMA-MAE in the 2D and 10D LP (Sphere) benchmarks, which are designed to exhibit distortion (See. 6.1 2D and 10D indicate the measure space dimensionality. We plot the number of unique archive cells where solutions sampled by CMA-MAE land according to their measures; CMA MMAE samples 540 solutions per iteration. The plot shows that in both benchmarks, CMA-MAF, begins searching in areas of low distortion, as many solutions land in unique cells. Over time, solutions more often land in the same cell (i.e., the number of unique cells goes down), indicating CMA-MAE has reached areas with higher distortion.
While low-dimessional measure spaces can exhibit distortion (Lee et al. 2024), our experiment shows how higher dimensions can amplify its effects. To elaborate, in Fig. lc the number of unique cells falls to only 30 in the higher-dimensional 10D LP (Sphere). In part, this occurs because al- though both the 2D and 10D benchmarks have archives with 10,000 cells, the cells in the 10D domain are exponentially larger by nature of being digeer-dimensional. As such, there is larger area of measure space where solutions sampled by CMA-MAE can fall and still be assigned the same discount value, leading inaccurate improvement values that stall the search.
Since large cells can amplify the effects of distortion, Fontaine & Nikolaidis (2023) suggest increas- ing the archive resolution (i.e., adding more cells), albeit only in 2D measure space. With higher resolution, cells are smaller, so solutions with similar measures can still fall in different cells and receive different discount values. However, this approach entails large amounts of memory, and this amount grows exponentially with measure space dimensionality. Since increasing the archive resolution effectively makes the histogram closer to a continuous function, we propose eliminate the histogram entirely and instead search with a continuous representation of the discount function.
DISCOUNT MODEL SEARCH
To improve exploration in domains with distorted, high-dimensional measure spaces, we propose Discount Model Search (DMS). DMS trains a discount model to provide smooth, continuous rep- resentation of the discount function. The key insight DMS that such a representation provides distinct discount values and hence improvement values, even when solutions have similar measures, making it easier to guide search towards solutions that improve the archive. For example, in Fig. lb. the higher improvement 2₂ correctly indicates that generating solutions in the direction of 0₂ would create greater archive improvement, as such solutions would land in Cell 3, which currently has a low discount value. Below we describe DMS's s components, with pseudocode shown in Algorithm 1.
Archive and Emitters. DMS maintains a MAP-Elites-style archive that retains the best solutions found (line 14 16). and CMA-ES-based (Hansen 2016) emitters that optimize for archive improve- ment. Unlike CMA-MAE, DMS does not store discount or threshold values in the archive.
Discount Model. The primary component of DMS is its discount model, which approximates the true discount function fA : R + R. The discount model is a neural network (a(+;v) param- eterized by zb. It takes measures as input and outputs scalar discount values. While alternative models like kernel-based methods (Chen) 2017) are feasible, we select neural networks because the inductive biases of their various architectures make them suitable for many types of measures. For example, if the measures are low-dimensional vectors as is common QD, an MLP (Multi-Laayr Perceptron) would suffice. If the measure space includes high-dimensional data like images or text, a convolutional network (He et al. 2016) or transformer (Vaswani et al. 2017) would be suitable.
DMS trains the discount model as follows. First, to reflect that the archive is initially empty, DMS regresses the discount model to output the minimum objective fmm at the centers of ninit cells sampled from the archive (line B) In the main loop (line 5 24), DMS regresses the discount model to match a dataset D, of input measure values s and their corresponding discount value targets t-- The dataset entries (s, LA) come from two sources. The first source is solutions sampled by the
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Algorithm 1: Discount Model Search (DMS)
Discount Model Search (eval, 0b N, W, A, a, Rinits a, Nemptyy
Input: eval function that computes objective and measures initial solution Oo. iterations N, num. emitters W, batch size X, initial step size o, initial training points Minits archive learning rate num, empty points Nempty- minimum objective m Result: Generates NWX solutions, storing elites in an archive A
Initialize empty archive A; discount model fa(::)) with parameters /
Sample Pinut cells from A and regress fA output fmin at the centers of these cells Initialize W emitters, each with mean 6 <- 0b. covariance E t oI, internal parameters p for iter <- 1..N do
DA [
// Dataset of measures and discount value targets.
for Emilter 1., Emilter W do for i + 1..A do N00 ((0,))) Σ)
f(0.), m(0)) < eval(0;)
// Compute improvement based on discount model. Compute LA,; with Eq. where m(0;)
A, f(O)) fA (.))
ax.append( (m,)) (A..)
e <- calculate
Sishivve_ell(AA. (emee(,
15
f(,)) > f(O,) then Replace Og (the solution in cell e) with O,
Rank 0, by A;
Adapt 0" E. based on improvement ranking A; if CMA-ES converges then
20
Restart emitter with 0 f- a random solution from A, E + aI, new internal parameters p
Sample Rempty unoccupied cells El..mempty from archive A without replacement
22
Compute centers $1..ne of cells Cl.mnemptty
D ( Memmmly fmin).
23
Regress FA on DA
24
emitters (line 13 For each emitter solution 0, DMS creates entry with the solution's measure values 8 m0 and a target LA that reproduces CMA-MAE's threshold update rule (Sec. 3):
if f(0) S fA(s) t = (1- +a) if f(0)
fa(s)
(1)
SA(8)
Here, if the objective value f() is worse than its discount value (s),,, tA set to the current discount value. If the objective value exceeds the discount value, f is linear combination between the objective value and the current discount value, weighted by archive learning rate a. Similar CMA-MAE's threshold update rule, this target aims slowly increase the discount values (and hence decrease the improvement values) in areas of the measure space that DMS has explored, so that the emitters are required find solutions in new areas of measure space.
The second source of data for D is "empty points," i.c., the centers of unoccupied archive cells. During preliminary experiments, we noticed that updating the discount model caused change its outputs in areas of the measure space that were not represented in the dataset D.A- In particular, in areas that had not been explored yet, the discount model should have output the minimum objective fmin. but it instead output high arbitrary values. To prevent this issue by "clamping down" the discount model in unexplored areas of the archive, we sample Nempty unoccupied cells from the archive (line 21). We then add the center of each cell to the dataset DA, with associated target of t = fmi (line 22 23). If there are fewer than nempty unoccupied archive cells, we select all such cells. Note that in the CVT archive, the "center" of the cell is that cell's centroid.

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Summary. DMS performs two phases. First, it searches for solutions that improve the archive A by sampling solutions with the emitters. Since the emitters contain CMA-ES instances, solutions are sampled from Gaussian (line 9. As DMS progresses, each emitter updates its Gaussian based
on archive improvement rankings (line (8) computed via the discount model (line [T. If CMA-ES converges, the emitters reset (line 20). Second, DMS trains the discount model fA. ensuring
improvement values remain accurate. It collects the dataset DA (line 12 13 21 23 and regresses
fA to match these values (line 24 Thus, fa guides the emitters to fill a highhhigghooeee high-ppeforming archive.
DOMAINS
We evaluate DMS on standard QD benchmarks and in a setting we refer to as Quality Diversity with Datasets of Measures (QDDM). We provide further details of all domains in inppenddix E.
6.1 BENCIMARKS
Linear Projection (LP) (Fontaine et al. 2020) benchmarks distortion by creating a measure func- tion that projects the majority of an n--dimensional solution space into the center of a k-dimensional measure space. We set n = 100 and instantiate LP with the Sphere. Rastrigin, and Flat objectives. With the Flat objective (Lee al. 2024), which only outputs 1.0, LP becomes a diversity optimiza- tion domain where solutions differ only by their measures. To provide a range of objective functions and measure space dimensionalities, we use eight instantiations of LP, named by the measure space dimensionality k and the objective function: 2D LP (Sphere), 10D LP (Sphere), 20D LP (Sphere), 50D LP (Sphere), 2D LP (Rastrigin), 10D LP (Rastrigin), 2D LP (Flat), 10D LP (Flat).
Arm Repertoire (Vassiliades & Mouret 2018 is an inverse kinematics domain where solutions are joint angle configurm configurations of a 2D planar arm with 100 joints. The measure function outputs the 2D position of the arm's 's end effector. The objective indicates the variance of the n joint angles.
6.2 QUALITY DIVERSITY WITH DATASETS OF MEASURES
We propose the QDDM setting, where instead of designing measure functions, user provides a dataset indicating their desired measure values. The defining feature of QDDM that the user provides nigh-dimensional data, e.g., images, audio, or text, and the measure space S is the space of such data, e.g. C 252=250-256+3-1166 if the data are 256 x 256 3 RGB images.
Initially, it seems problematic to construct an archive for QDDM due to the high dimensionality of the measure space. However, we adopt the manifold hypothesis (Fefferman et al.] (0166). i.e., the assumption that most high-dimensional data lie on a low-dimensional manifold embedded within the high-dimensional space. We recognize that the distribution of measures relevant to a user occupies only small region of the overall measure space, and this distribution is reflected in the user's dataset. Hence, we propose to construct a CVT archive where the centroids are the points in the dataset. Here, the CVT no longer uniformly partitions the measure space. Rather, it only partitions the small region of measure space desired by the user and indicated in the dataset. However, the CVT archive introduces new consideration for QDDM, viz., the choice of distance function. To locate the cell where a solution belongs, CVT archives find the centroid closest to the solution's measures. While Euclidean distance is common here, is not always ideal (Vassiliades et al. 2018). which may be especially true when the measures are as nigh-dimensional as images or text.
Triangle Arrangement (TA). We introduce two QDDM domains. First, TA builds on computational creativity domains (Tian & Ha 2022) and involves arranging prespecified number of triangles to create images. A solution consists of the vertices, brightness (we consider grayscale images), and alpha (transparency) of each triangle. A solution's measure is created by rendering the triangles as a raster image. We specify desired images (measures) by sampling 1000 images from either MNIST (Lecun et al. 1998) or Fashion MNIST (iiao et al. (017). leading to two versions of this domain: TA (MNIST) and TA (F-MNIST). Since the images in these datasets are 28 x 28, the measure space is dem-sinnl Drawing from the loss function in Tian & Ha 2022), , we define the distance function as Euclidean distance, so each solution is placed in the archive cell of the digit it most resembles. To make the triangle images resemble the MNIST images, we define the objective
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Table
Mean QD Score ("QD") and Coverage ("Cov") for each algorithm in each domain.
2D LP (Sphere)
10D LP (Sphere)
20D LP (Sphere)
50D LP (Sphere)
QD
Cov
QD
Cov
QD
Cov
QD
Cov
DMS
6,978.20
95.89%
6,409.50
89.21%
7,406.01
95.97%
6,991.00
87.00%
CMA-MAE
6.327.90
80.95%
608.53
6.95%
881.76
9.13%
2.327.11
24.21%
DDS
3,156.24
70.75%
4.237.72
60.07%
6,990.51
84.31%
6,373.24
74.30%
MAP-Elites (line)
4,908.81
60.42%
2,570.74
29.20%
5,936.66
65.86%
5.204.80
55.77%
MAP-Elites
4,163.41
50.76%
228.65
2.35%
3,280.34
35.62%
3,870.48
40.89%
2DI (Rastrigin)
10D I.P (Rastrigin)
2D LP (Flat)
10D I.P (Flat)
QD
Cov
QD
Cov
QD
Cov
QD
Cov
DMS
5,738.90
91.67%
5,138.81
88.19%
7,902.05
79.02%
7.982.15
79.82% %
CMA-MAE
5.258.59
80.14%
246.55
2.98%
5,675.90
56.76%
1.554.90
15.55%
DDS
2,495.11
71.68%
3,331.70
59.54%
6,967.75
69.68%
6,004.95
60.05%
MAP-Elites (line)
3,841,05
56.63%
2,001.76
28.04%
4.510.65
45.11%
757.75
7.58%
MAP-Elites
3,172.59
48.21%
499.66
7.09%
4,327.00
43.27%
125.65
1.26%
Ann Repertoire
TA (MNIST)
TA (F-MNIST)
LSI (lliker)
QD
Cov
QD
Cov
QD
Cov
QD
Cov
DMS
7.963.44
80.15%
951.56
99.84%
701.14
72,28%
214.91
3.77%
CMA-MAE
7.902.43
79.22%
954.27
99.48%
625.65
63.92%
14.61
1.56%
DDS
5.568.23
80.24%
MAP-Elites (line)
7.458.67
75.60%
945.60
98. 88.86%
551.13
56.68%
-51827.44
7.49%
MAP-Elites
7,411.10
75.42%
941.94
98.42%
513.13
52.68%
-18,917.87
5.06%
as the negative (to facilitate maximization) mean squared error between the triangle image and the archive centroid (MNIST image) to which is assigned.
Latent Space Illumination (LSI). LSI entails exploring the latent space of a generative model to create images with diverse measures. While prior work (Fontaine et al., 2021b) considers LSI with low-dimensional measures, we consider LSI where the measures are images. As described in /Sec. 1 we search for images of hikers in different landscapes. The solutions are latent vectors (w-space, but not w -spacc) of StyleGAN3 (Karras et al. 2021a),, with images of faces output by StyleGAN3 serving as measures. Thus, the measure space is the space of 256 x 256 x 3 images. The desired mea- sures are specified with a dataset of 10,000 landscape images sampled from LHQ256 (Skorokhodov jet al. 2021). To associate hikers with landscapes in the CVT archive, the distance function is the CLIP score (Radford et al. 2021) between the face and landscape; CLIP score is more semantically meaningful than Huclidean distance. The objective is the CLJP score between the face image and the prompt "A photo of the face of hiker." We also add regularizer loss (Fontaine & Nikolaidis) 2023) to penalize latent vectors that fall outside the StyleGAN3 training distribution, and similar to TA, we reward higher alignment between faces and landscapes by adding the CLIP score between the face and the landscape to which it is assigned. We refer to this domain as LSI (Hiker).
7 EXPERIMENTS
We evaluate DMS in low- and high-dimensional measure spaces through experiments in 9 bench- marks [2D LP (Sphere). 10D LP (Sphere), 20D LP (Sphere). 50D LP (Sphere), 2D LP (Rastrigin), 10D LP (Rastrigin), 2D LP (Flat), 10D LP (Flat), Arm Repertoire] and 3 QDDM domains [TA (MNIST), TA (F-MNIST), LSI (Hiker)). In each domain, we conduct a between-groupe study with the algorithm as the independent variable: besides DMS, we consider the black-box QD algorithms described in Sec. 3 CMA-MAF, DDS, MAP-Elites (line), and MAP-Elites. We consider two de- pendent variables. QD Score (Pugh et al. 2016) represents overall performance by summing the objectives all solutions in the archive, as is done in the QD objective (Sec. 2). Coverage indicates how much of the measure space has been explored by computing the percentage of archive cells that have a solution in them. Note that the objective is normalized to have a maximum value of 1 in all domains. Our hypothesis is that DMS will outperform all other algorithms in both QD Score and Coverage. We implement all algorithms with pyribs (Tjanaka et al.) 2023); hyperparameters and further details are in Appendix Cand F
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7.1 ANALYSIS
Table shows the results from 20 trials in the benchmark domains and 5 trials in the QDDM do- mains. Fig. 2jshows sample images from DMS in LSI (Hiker). Refer topApeeddix for performance plots, error bars, archive heatmaps, and discount function visualizations. We could not run DDS in the QDDM domains due to the KDE's runtime, which grows linearly with measure space dimen- sionality. Visual inspection showed the results were normally distributed, but Levene's test showed most settings were non-homoscedastic. Thus, to analyze the results, we conducted Welch's 's one-way ANOVA each domain for each dependent variable. All ANOVAS were significant (p < 0.001; F values in Appendix B), so we performed pairwise comparisons with the Games-Howell test.
Benchmark Domains. In the benchmarks, DMS significantly outperformed all baselines in QD Score and Coverage, except in Arm Repertoire, where DDS had significantly better coverage. The high performance on LP shows that DMS better overcomes distortion than previous algorithms, as these domains are designed to benchmark distortion. Since DDS is a diversity optimization algorithm, we expect it to achieve the best coverage in all domains, so it is surprising that DMS outperforms it in almost all domains, even 2D and 10D LP (Flat), which are diversity optimization domains where the objective is always 1.0.
TA Domains. In TA (MNIST), for both metrics, DMS significantly outperformed the two MAP- Elites algorithms, but there was no significant difference with CMA-MAE. In TA (F-MNIST), DMS significantly outperformed all baselines in both metries. The coverage results in TA (MNIST) il- lustrate that not all QDDM domains exhibit high distortion - low distortion is reflected in how all algorithms achieve nearly perfect coverage in TA (MNIST), We believe the high coverage stems from how MNIST images are fairly similar in appearance. As such, an algorithm can fill the archive by generating triangle images that differ only slightly from one another. TA (F-MNIST) seems more challenging, as no algorithm achieves perfect coverage there.
On the other hand, the difficulty of TA (MNIST) seems lie in optimization of the objective, the difference in QD Score between DMS/CMA-MMAE and the two MAP-Elites algorithms is small yet statistically significant. This property suggests a potential limitation of DMS that may be suitable for exploration in future work. We speculate that by nature of being a model, the discount model in DMS exhibits errors, which we can imagine as adding noise to discount values. In domains where objective optimization is less important, we hypothesize that the noise is small enough that improvement rankings remain unaffected. However, the objective like TA (MNIST), this noise interferes with improvement rankings, hindering DMS. CMA-MAE maintains exact values in its histogram and would not have such noise, potentially explaining why DMS does not outperform CMA-MAE's QD Score here.
domain that requires fine optimization of
LSI (Hiker). The results in LSI (Hiker) highlight the difficulty of complex QDDM domains. Here, DMS significantly outperformed CMA-MAE in both metrics, but there was no significant difference with the two MAP-Elites algorithms. While DMS outperforms CMA-MAE, it covers only 3.77% of the archive, although this still represents 377 hiker images. We note that the two MAP-Elites algorithms receive large negative QD Scores and high coverages by generating latent vectors far outside the training distribution of StyleGAN3 and incurring large regularization losses. Similarly, they exhibit high performance variance, so they have no significant difference with DMS.
Ablation Study. We ablate the hyperparameters of DMS in Appendixx D We find that the archive learning rate a behaves similarly as in CMA-MAE: intermediate values enable DMS to balance optimization of the objective and exploration of the measure space, while a causes DMS to over-emphasize the objective. Meanwhile, the "empty points" are necessary for training the discount model: removing them by setting Nempty
causes performance to drop since the discount model takes on arbitrary values in areas of the measure space that have not been explored (Fig. 11). In contrast, setting Teempty = 10, 100, or 1000 resolves this issue by "clamping down" the discount model (Fig. 8). Overall, our experimental results show how the discount model successfully guides optimization in DMS, leading to high performance across domains with different measure spaces.
Computation Time. Since DMS trains a discount model, it may be more computationally expensive than other methods like CMA-MAE. To quantify this tradeoff, we recorded the wallelock time when running all the QD algorithms in our experiments. These results summarized in Table 2 We focus on comparing DMS and CMA-MAE, as their primary difference is in training the discount model. On the benchmark domains (L.P and Arm Repertoire), DMS takes noticeably longer than

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Published as a conference paper at ICLR 2026
Table Computation (wallclock) time (in seconds) of each algorithm in each domain. We show the mean and standard error of the mean over 20 trials for the benchmark domains and 5 trials for the QDDM domains.
2D LP (Sphcre)
10D LP (Sphere)
20D LP (Sphere)
50D LP (Sphere)
DMS
397.83 +0.37
876.33 #44.22
2,797.61 #6.57
7,843.98
+228,28
CMA-MAE
121.13 =0.
333.92 +1.70
1.316.87
+22.32
4,707.9
=86,53
DDS
1,918.68 +22.42
3,535.78
+1.53
1,481188 +30.98
2,600.24
+78,20
MAP-Elites (linc)
31.10 #0.10
231.50 +0.73
883.23 #1.04
4,147.29
100..89
MAP-Elite
39.01 #0.21
190.25 #2.46
873.06 20.35
4,523.16 #51,12
2D LP (Rastrigin)
1OD LP (Rastrigin)
2D LP (Flat)
10D LP (Flat)
DMS
707 221.33
711.08 #30.76
683.53
#27.73
723.34 #32.31
CMA-MAE
182.07 #0.36
465 +3.81
165.33 =0.25
800.12 #0.92
DDS
1,957.29 2957.29-20 +0.87
3,598 86 +2.56
1.937.4
+1.51
3,581.79
+2.05
MAP-Elites (line)
39.74 +0.14
274.86 +0.62
40.09 30.84
339.73
#0.42
MAP-Elites
51.27 #0.17
367.56 +1.57
38.41 20.14
451.46 #0.41
Arm Repertoire
TA (MNIST)
TA (F-MNIST)
LSI (Hiker)
DMS
687.85 +27.18
489.90
546.60 #21.46
4,740.24
#482.74
CMA-MAE
130.83 #0.26
495.95 =13.12
525.42 #19.22
4.690.12 #654.47
DDS
1,577.54 +1.70
MAP-Elites (line)
39.68 #0.10
247.21
#4.07
231.91 =0,38
3,470.08 12288
MAP-Elites
20.80
291.98 #1.46
309.01 #3.92
3,460.71 #5.86
CMA-MAE due to this training. However, on the QDDM domains (TA and LSD,, this difference much less noticeable, because the evaluation of solutions becomes the bottleneck, rather than the QD algorithm itself. Meanwhile, we note that MAP-Elites and MAP-Elites (line) are the fastest algorithms because their search operations are the least computationally expensive, since they are rooted in fixed Gaussian noise. In contrast, DMS and CMA-MAE manipulate multivariate Gaussian distributions their CMA-ES emitters.
CONCLUSION
By searching in distorted and high-dimensional measure spaces, DMS offers two benefits for QD practitioners. First, DMS can improve performance in current QD applications. As the results in the benchmark domains show (Sec. 7.1), DMS outperforms current algorithms in various domains with low-dimensional measure spaces. Since most current applications involve low-dimensional measure spaces, we believe DMS will also outperform current QD algorithms in current applications. Second, DMS enables new applications by addressing the proposed QDDM setup (Sec. 6.2), where diversity in a high-dimensional measure space like images is specified by providing dataset. We believe framing measures in terms of datasets makes QD more accessible by not only alleviating the need to hand-design measure functions, but also making it possible to specify measures that cannot easily be represented by low-dimensional values. For example, the TA and LSI (Hiker) domains showed, we can now specify the measure space vision and art domains with image datasets. Overall, given that datasets are central to machine learning, we believe it will prove fruitful to frame problems across machine learning as QDDM problems and solve them with algorithms like DMS.
Our paper has several limitations. First, while searching over the discount model garners high per- formance, training it induces computational overhead (Sec. 7.1). Second, while DMS trains the discount model with targets that reproduce CMA-MAE's threshold update, alternative targets may improve properties like smoothness. Finally, we primarily consider small MLPs for the discount model. We are excited for future work in domains that require more advanced discount model chitectures.
ETHICS STATEMENT
DMS has the potential for some negative societal impacts. For example, its abilities in QDDM set- tings may exacerbate biases in the dataset of measures, image generator (Karras et al. 2021a), and even distance metric (Radford et al. 202T). These biases can be mitigated by carefully managing all datasets, including that used to train the generator (Perera & Patel 2023 Maluleke al. 2022
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In general, QD also debias models by generating balanced datasets (Chagg et al. 2024). In addition, using QDDM methods to drive generative search towards specific artistic styles defined in the measure space can discount unique styles created by human artists (Shan al. 2023). Altern-- tive objective functions (Tseng et al. (2021) may enable finding solutions that reflect the dataset of measures without mimicking artists' creativity.
10 REPRODUCIBILITY STATEMENT
We have included details of DMS Sec. 5 including pseudocode in Algorithm | We include de- tails of our domains and experimental setup in Sec. (Sec 7| along with hyperparameters in An-A pendix C extended domain details in Appendii EJ and further implementation details in/Appendix F We have released
code at https:: //gi:////tttbttttt com/ ca ras susessuuc//eee scount- Inside the code repository, a README file lists instructions for setting the environment and reproducing all experiments this paper.
ACKNOWLEDGMENTS
We thank Varun Bhatt, Saeed Hedayatian, and Aaquib Tabrez for their insightful feedback. This work was partially supported by the NSF CAREER (#2145077), NSF NRI (#2024949), NSF GRFP (#DGE- 1842487,, and the DARPA EMHAT project.
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A RELATED WORK
Quality Diversity Optimization. Applications of QD include robotics (Cully et al. 2015 Hu- ber et al. 2025 Morrison et al., 2020 drug discovery (Boige et al., 2023 Verhellen Van den Abeele| 2020), urban planning (Galanos et al. 2021). and finance (Gasperov et al. (025). p. In com-
puter Vision, QD can create diverse images by exploring the latent space of generative models, e.g., StyleGAN (Karras et al. (02 lb) in our work and in (Fontaine & Nikolaidis 2021), and Stable Dif- fusion (Rombach et al., 2022) (Ding et al. 2024). Such images can form synthetic dataset for debiasing downstream models CChang et al. 2024). In reinforcement learning. QD can generate di-
verse locomotion policies (Nilsson& & Cully] 2021 Pierrot et al. 2022; Batra et al. 20224 Chalumeau QD can probe a large language model (L.LM) to
et al. 2023 Xue et al. 2024), and in red-teaming,
produce harmful outputs (Samvelyan et al.) 2025).
Our work fits a growing trend of integrating models into QD. To accelerate evaluations, multiple
works (Bhatt et al.] 2022 2023 Kent et al.) 2025; Zhang et al. 2022 Gaier et al., [2018]
Hagg
al. 2020) train surrogate models to approximate expensive objectives and/or measures. Oth-
ers (Gravina et al. 2016 2019 Salehi et al.]
202T) build models that guide the creation of new
solutions, especially reinforcement learning Tjanaka et al. 2022 Nilsson & Cully] 2021 Pierrot
iet al. 2022 Batra et al.) [2024] Chalumeau et al. [2023] Grillotti et al. [2024). Furthermore, whereas
DMS searches directly the space of high-dimensional measures, several approaches (Cully) 2019,
Paolo et al., 2020) Grillotti & Cully, 2022, McCormack & Cruz Gambardella] (0022) build models
that compress high-dimensional measures into low-dimensional measures. Finally, as discussed in
Sec. 6.2
we apply the manifold hypothesis (Fefferman et al., 2016) in measure space when creating the archive in QDDM domains. Prior work (Vassiliades & Mouret 2018 Gaier et al. [2200] Rakice-
vic et al.) 2021 Hegde et al., (023) applies the manifold hypothesis in solution space by searching over the elite hypervolume, a low-dimensional manifold where the solutions to each QD problem
hypothesized to exist.
Related to our proposed QDDM setting, developing intuitive ways to specify measures of diversity is an active research area in QD. For example, QD through AI Feedback (QDAIF) (Bradley al. (024) evaluates measures by querying LLMs for feedback, while the LSI domain in prior work (Fontaine & Nikolaidis 2021) computes the CLIP score between text prompts and generated images. QD through Human Feedback (QDHF) (Ding et al. 2024) learns measures from human preferences via contrastive loss. Each method's suitability depends on which user effort casiest. For example, QDAIF excels when measures can be conveniently specified to an L.M evaluator that outputs vector of measures. Conversely, QDAIF can be limited by the challenges of prompt engineering and the stochasticity of L.M outputs. Moreover, it may be difficult to create vector measures that elicit the desired diversity, like in LSI (Hiker), where QDDM makes it easy to specify "where a hiker is located" with landscape images. A key distinction is that QDDM specifies desired measure values, while the above methods all define measure functions. an appropriate dataset does not exist, designing a measure function may be more appropriate, since datasets require significant effort to curate. On the other hand, we believe QDDM will be applicable in many problems since datasets abundant in machine learning.
Computational Creativity. Our TA domains draw from prior work Tian & Ha 2022) that arranges triangles represent images and text prompts with evolution strategies. Similarly ly, various works ar- range basic shapes to create artistic images (Paauw & Van den Berg. 2019 Berg et al. 2019 Cason 2016 Fogleman| 2016). To our knowledge, QD algorithms have not been applied in this setting, but they have generated other forms of art, such as line drawings (McCormack & Cruz Gambardella 2022 and images (Zammit al. 2022).
Latent Space Exploration. LSI (Hiker) is an example of latent space illumination, where a QD algorithm searches for latent vectors that elicit diverse, high-performing outputs of a generative
model. LSI was first introduced to generate video game levels (Fontaine et al.) 2021b a Sarkar &
Cooper 2021 Schrum et al. 2020 Steckel & Schrum 202 and 2D shapes (Hagg et al. 2021).
Later work (Fontaine & Nikolaidis 2021: 2023) explored the latent space of StyleGAN Karras
et al. 2021b) to generate celebrity images with dww-iimensional (2D) measures based on the CLIP
score (Radford et al. 2021 Herrera- Berg 202I). Beyond LSI, various methods aim to navigate
latent spaces. Several works discover interpretable directions in GAN latent spaces, where these
directions manipulate pose or facial features (Shen & Zhou 2021
Shen et al. 2020 Voynov &l
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Babenko, 2020:
Shen et al.
2022) transform position and scale (Plumerault et al. 2020) In
single-objective optimization. prior works search the latent spaces of generative models to create video game levels (Volz et al.
2018 Merino et al. 2023 Tanabe et al. 2021 or synthetic finger- prints (Bontrager et al. 2018 that satisfy desired characteristics.
B RESULTS
We present further results obtained in our main experiments in Sec. 7 Fig. and Fig. 4 show the mean and standard error of the mean of both dependent variables (QD Score and Coverage), for all algorithms in all domains. The plots shows the values over 10,000 iterations, and the table shows the final values. Note that since the objective is always 1.0 in the LP (Flat) domains, the QD Score and Coverage differ by a factor of the number of cells in the archive, i.c., the QD Score is 10,000 times the Coverage. In the plot for LSI (Hiker), the QD Score initially negative since the algorithms receive regularization penalty due to generating images outside the training distribution of StyleGAN3. Neither MAP-Elites variant's QD Score visible due to being large negative values.
Fig. 5 and Fig. 6 show sample images from DMS in the TA (MNIST) and TA (F-MNIST) domains. Note that by default, these domains render the triangles into 28 28 image. However, since the triangles in each solution form vector graphic. they can be rendered any resolution. Thus, for visualization purposes, we rendered them at 280 x 280 resolution in these figures.
Fig. 7 shows heatmaps of a randomly selected archive of each algorithm, in domains with 2D mea- sure spaces. Fig. 8) shows how the archive and discount model in DMS progresses over iterations in the 2D LP (Sphere) benchmark. We include further descriptions and analyses of both of these figures in their captions.
Table 3
shows the results of Welch's one-way ANOVA for both dependent variables in all domains.
Note that large between-groue variances led to many large F statistics. [Talle 4 and Table 3 show the results of pairwise comparisons between each pair of algorithms in each domain. Each entry compares the method in the row to the method in the column, with > indicating significantly greater, significant difference, and indicating an invalid
< indicating significantly less, indicating comparison. For example, DMS significantly outperforms DDS in QD Score in 2D LP (Sphere). Note that since we were unable to run DDS in the QDDM domains [TA (MNIST), TA (F-MNIST), and LSI (Hiker)), the degrees of freedom are smaller for those domains' ANOVAs, and there is no comparison between DDSS and other algorithms in the pairwise comparison tables.
B.1 COMPARABILITY OF RESULTS TO PRIOR WORK
Variations of the linear projection and arm repertoire domains have appeared in a number of prior works, each with slightly different setups than the one in our work. Here, to facilitate better com- parability with the results in prior work, we clarify the distinctions between setup and previous ones. The first difference is in the order of magnitude of the scores reported in our work. Prior work like Fontaine 2021 (Fontaine & Nikolaidis (021) and Fontaine 2023 (Fontaine & Nikolaidis, 2023) reported normalized QD scores, which are QD scores divided by the number of cells in the archive (10,000). Those works also normalized objective values be 0-100, whereas we normal- ize to 0-1 to make the objective values more amenable to the discount model. These two changes explain the difference in orders of magnitude between our results and those works: when divided by 10,000 and multiplied by 100, our results for CMA-MAE, MAP-Elites, and MAP-Elite (line) on 2D LP (Sphere), 2D LP (Rastrigin), and Arm Repertoire are nearly identical to those for LP (sphere), LP (Rastrigin), and Arm Repertoire in Fontaine 2023 (our remaining benchmark domains were not included in Fontaine 2023).
The second difference is in the number of iterations run. Like Fontaine 2023, run all algorithms for 10,000 iterations. However, Lee 2024 (Lee et al. (024)) ran DDS and other algorithms for 5,000 iterations. As result, Coverage for DDS, CMA-MAE, and MAP-Elites (line) in 2D LP (Flat) and 10D LP (Flat) is higher than that in the LP and Multi-feature LP domains of Lee 2024.
The third difference is in the solution space dimensionality in the benchmarks. We follow Fontaine 2023 in considering a 100D solution space in each benchmark domain. In contrast, Fontaine 2021 considered 1000D solution spaces to emphasize the scalability of differentiable quality diversity in
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high-dimensional solution spaces. Due the difference in setting, results not comparable to those from Fontaine 2021.
Finally, Fontaine 2020 (Fontaine et al. 2020) considers fundamentally different hyperparameters than those in any of these works, c.g., it runs for 4500 iterations and has archives with 250,000 cells, whereas our work and most of these works run for 10,000 iterations and have archives with 10,000 cells.
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Figure 3: Mean and standard error the mean for QD Score and Coverage of each algorithm in each domain. Standard error may not be visible in some plots.
DMS
DIS
MAP-FiBeS
CMA-MAE
(line)
(Sghere)
(Sphase)
(Sphere)
(Sphere)
8000
8000
7500
6000
Spore
4000
4000
4000
2500
2000
0-
100%
1.00%
0.75
Contrage
30%
50%
0.5
0.50
0.25
0%
0.0
0.00
5000
5000
5000
Dicrations
Iteniiioss
Iterations
Iterations
2D LP (Sphece)
10D LP (Sphere)
20D LP (Sphere)
50D LP (Sphere)
QD Score
Covenge
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
DMS
6,978.20 #17.94
95.89
=0.43%
6,409.50
#13.50
89,21 10.18%
7,406.01 +23.37
95.97 20.07%
6,991.00 14.48
87.00 20.04%
CMA-MAE
6.327.90 +3.60
80.95 10.06%
6018.53 +24.78
6.95 20.31%
881.76
9.13
+0.80%
2,327.11 118.75 78
24.21 20 20.20%
3,156.24
+0.65%
4,237.72
+21.18
=0.28%
6,990.51
6,373.24
MAP-Elites (linc)
4,908 01008.8122 +2.48
60.42
+0.04%
2,570,74
#39.16
29.20
20.45%
5,936.66
+6.90
65.86
20. 10%
5,204.80
#11.47
55.77
20.14%
MAP-Elites
4,163.44 24.10
50.76
20.07%
228.65
48.79
2.35
20.09%
3,280.34 294.26
35.62
11.10%
3,870.48
#12.47
40.89 20.14%
DMS CMA-MAE
DDS MAP-Blies (line)
MAR-Elius
2D LP (Rastrignn)
LOD LP (Rastrigin)
(Plat)
10D (Plata))
8000
8000
5000
9000
QU Store
100%
1.00%
100%
1005
Covergo
50%
50%
50%
5000I Bicratioons
10000
5000
10000
5000
10000
5000
10000
Dieniions
Themaions
Iurationss
2D LP (Rastrigin)
10D LP (Rastrigin)
2D LP (Flat)
10D LP (Flat
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
DMS
5,738.90 +19.78
91.67
=0.44%
5,138.81
+10.31
88.19
10.12%
7,902.05
+35.68
79.02 =0.36%
7,982.15 =24.43
79,82
+0.24%
CMA-MAE
5,258. 59 +5 97.97
80. 11%
246.55
436.93
2.98
+0.49%
5.675.90 90 +46.04
56.76 +0.46%
1,554.90
#26 04
15.55 20 20.26%
DDS
2.495.11 193.14
71.68
10.96%
3.331.70 218.20
59.54
20.27%
6,967.75
298.80
69.68
10.99%
6,004.95
#21.52
60.05
20 22%
Elitees (line)
3,841.05
#4.67
+0.07%
2,001.76
#23.09
20.33%
4,510.65
#41.43
757.75
=20.
MAP-Elites
3,172.59
+8.43
48.21
14%
499.66
+14.06
7.09
+0.22%
4,327.00
+17.29
43.27
+0. 17%
125.65
+15.88
1,26 +0. 16%
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Figure 4: Mean and standard error of the mean for QD Score and Coverage of each algorithm in each domain. Standard error may not be visible in some plots.
DMS
DDS
MAP Elites
MAP-Eline (line)
Repertoire
LSI (lliker)
8000
1000
1000
100-
4000
500
500
Q13
100%
100%
100%
10%
Covergg
30%
50%
50%
555-
10000
5000
10000
5000
10000
5000
10000
Bienations
Denniios
Dieiiions
Derations
Arm Repertoire
TA (MNIST)
TA (F MNIST)
LSI (liker)
QD Score
Covemge
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
DMS
7,963.44 +2.47
80.15 +0.01%
951.56 +0.95
99.84 10.
701.14 +5.65
72.28 +0.60% 63.92 =0 88%
214.91 +15.23
3.77 #0.13%
CMA-MAE
7,902.43 +1.93
79.22
+0.02%
954.27 +0.69
99.48 +0.09%
625.65 +8. 60
14.61 126.18
=0.05%
5,568.23
80.24
20.00%
MAP Elites (line)
7,458.67
+2.56
75.60
+0.03%
945.60
+1.38
98 86 +0. 15%
551.13 16.
56.68 20.64%
-11,82744 11753.11
7.49 #1.03%
MAP-Elites
7,411.10
#2.56
75.42
+0.03%
941.94 +2.01
98.42 +0 22%
513.13 +6.80
52.68 #0.70%
-18,917.87 +5705.28
5.06 11.1%
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g
5
3
9
5900
6
6
567
4
305
6
0
5
4
3
3
Figure 5: A random subset of images generated by DMS in the TA (MNIST) domain, where desired measures are sampled from the MNIST dataset. The goal in this domain is to arrange triangles to look like the given MNIST digits. Each rendered triangle image is shown to the left of its corre- sponding MNIST digit.
Figure 6: A random subset of images generated by DMS in the TA (F-MNIST) domain, where desired measures are sampled from the Fashion MNIST dataset. The goal in this domain to arrange triangles to look like the images of fashion items. Each rendered triangle image is shown to the left of its corresponding fashion item.
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DM5
CHA-MAE
DD5
MAP-lia [Bac)
MAR-EIK
0.8
20 LPP.
225
256
-26
226
216
-226
LP (oariign
0.1
-256
236
1.0
2D (P
04
220
DMS
nOS
Fllme (Brx)
MAPP
100
1.5
Arm Repernine
100
03
101
100
100
Figure 7: Heatmaps of randomly selected archive produced by each algorithm in domains with 2D measure spaces. Each row contains heatmaps for single domain. The axes of the heatmaps are the measures, while the color of each cell indicates the objective value. Notably, the heatmaps show how DMS achieves high coverage of the measure space. They also show how DDS achieves good coverage but cannot achieve high objective values since it is diversity optimization algorithm.
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Ittration 0
Archive
Discoun Model
Samples
Empry Points
2.56
Iteration 250
Archive
Discoun Model
256
256
1.00
0.75
0.50
0.25
Emitter Samples
Empty
-256
-256
0.00
-256
255
-256
256
Iteration 1000
Archive
Discorat Model
Samplex
Empty Points
Iteration 5000
Archive
Discomm Medel
256
256
1.00
0.5
0.50
0.25
Emitter Samples
Emety Points
2.56
256
0.00
-256
255
-256
256
Iteration 10000
Archive
Discount Model
2.56
Emikier Samplex Empty Points
-256
Figure 8: Progression of the archive and discount model in DMS in the 2D LP (Sphere) benchmark. The left heatmap shows the archive, while the right heatmap shows the discount model. To plot the discount model, we computed its output at points discount model heatmap also shows the dataset D of points on a given iteration - blue circles indicates points created with solutions from the emitters, and yellow triangles indicate empty points. On Iteration 0, the discount model initializes to output fmin everywhere. On Iteration 250, as the emitters begin to populate the archive, discount model begins to output higher values areas that have been explored. However, unexplored areas still maintain low values (shown dark colors) due to the empty points the dataset. further iterations, the discount model outputs higher and higher values the emitters populate the archive further, until it outputs high values nearly everywhere Iteration 10000.
200 200 grid in measure space. The
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Table 3: Welch's one-way ANOVA results in each domain. All p-values are less than 0.001.
QD Score
Coverage
2D LP (Sphere)
Welch's F(4, 44.8) = 44362.7
Welch's 44.2) = 32645.6
10D LP (Sphere)
Welch's 44.8) = 38532.9
Welch's 434,43.7) = 51051.2
20D L.P (Sphere)
Welch's 444,44.0)
4887.9
Welch's F(4,45.3) 18271.4
50D (Sphere)
Wolch's F(4,44.1)
27718.5
Welch's F(4,44.4)
55904.6
2D LP (Rastrigin)
Welch's 454,45.0)
= 14008.9
Welch's 4(4,44.3) = 11766.9
10D LP (Rastrigin)
Welch's F(4,44.8)
= 19493.4
Welch's F(4,444 = 32616.3
2D LP (Flat)
Welch's F(1,41.2)
2121.44
Welch'ss[4,441
21214
1OD LP (Flat)
Welch's F(4,17.1)
25859.6
Welch's F(4,47.
25859.6
Arm Repertoire
Welch's 464,
10820.9
Welch's 1(4.88.1)
15340.2
TA (MNIST)
Welch's F(3,8.4)
16.5
Welch's F(3,8.6)
-5.00
TA (F-MNIST)
Welch's #(3,8.8)
160.6
Welch's F(3,888)
157.4
LSI (Hiker)
Welch's s F.6,
18.0
Welch's F(3.7.2) 86.0
Table 4: Pairwise comparisons (Games-Howell test) of the QD Score of each algorithm.
(Sphere)
(Sphere)
(Sphere)
(Sphere)
MAP-Elies (Ime)
MAP-Ellies (lie)
MAP-Elies (linc)
Mliee (line)
CMA-MAE
MAP-Ellies
CMA-MAT
MAP-Elies
CMA-MAE
MAP Ellies
CMA-MAE
MAP-Elies
DDS
DMS
DDS
DMS
DDS
DMS
DDS
DMS CMA-MAB
AAAV
VV 22 VVV
ADAV
TVVVV
A AAAAA 0
V &
AA VAAB
VVVV
AA TAAAS &
VV V G
MAV
A VSVAV
VVVAV
KAAAS S
S V
R VVVV
S WAAV
AAAS
DDS
VV V
MAP MAP-Elites (line)
MAP-Elites
2D LP (Rastrigia)
10D LP (Rastrigin)
2D LP (Hla)
LDD UP (Flat)
MAP Biies (lme)
MAP-Elies (line)
MAP-Elies (line)
MAP BBies (lime)
CMA MAB
MAP-Elies
CMA-MAE
MAP-Elies
CMA-MAE
MAP Eiiles
CMA-MAB
MAP-Elees
DMS
DDS
DMS
DDS
DMS
DDS
DMS
DDS
CMA-MAE
AAAA S
AA AAA S V
V
8 V A V V
V V & V
AA VSSA
V
SV A V
AA & V
AA D AAAV
V
S SVVV VVV V
AA A
V S V
AA 00 A V
8 GVVVA VV
MAP-Elia (line)
DDS
AAR
AAAA
1
MAP-Elites
Arm Repertoize
TA (MNIST)
IA (F-MNIST)
LSI (Hiker)
MAP-Eines (lme)
(ime)
MAP-ELies (lnn)
MAP-Fliee (linc)
CMA-MAE
MAP-Fllies
CMA-MAF
MAP-Eliee
CMA-MAE
MAP. Iliies
MAR-Elies
DDS
DMS
DDS
DMS
DDS
DMS
DDS
DMS
CMA-MAB
AAAAS
AAASS
V.V VVV
ABAVV
S AVV
AA 1 &
AA S
S 19996
VVV
GVV
AA MA &
AA 2
A 1SSVV
VVV
8 S
DDS
& 30
S 10
MAP MAP-Elite (line)
MAP-Elites
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Table 5: Pairwise comparisons (Games-Howdr test) of the Coverage of cach algorithm.
(Sphere)
(Sphere)
(Sphere)
(Sphere)
MAP (Ine)
MAP (line)
MAP-Ellos Eilres (line)
MAP Flites (line)
CMA-MAE
MAP- Elires
CMA-MAF
MAP-EIG Elikes
CMA-MAE
MAP Elites
CMA-MAE
MAP-Elies
AAAA DMS
DDS
DMS
DDS
DMS
DDS
DMS
DDS
DMS
CMA-MAR -AA
AAAS
AA AAVV V
ASVVY
VVVVV
A NAAAC Ø
BAV A
A SSVV V
VVVV
A AAAA R
V VVVV S V
A ASVAV V
AAAA G
V VVVA & V
S VAVV
I SVVAY
MAP-Elites (line)
MAP-Elites
2D LP (Rastrigin)
10D LP (Rastrigin)
2D (Flla)
10D LP (Flat)
MAP Eiles (line)
MAP-Elies (line)
MAP-Elies (lime)
MAP-Elles (Iiee)
CMA MAB
MAP-Elles
CMA-MAE
MAP-Elies
CMA-MAE
MAP-Eiies Eiiee
CMA-MAH
MAP-Elies
DMS
DDS
DMS
DDS
DMS
DDS
DMS
DDS
DMS
IS DVVE A V
AA SAAA V
1 VVV V V
AA 0 A V
8 VVVVV V
MAAMMAE
6AAAA
AAA 1
V
V S
AA PASAY
AA 5
DDS
AA 1
0 DVV.
AAA
VAVE
MAP-Elites (line)
MAP Elites
Anm Repertoire
(MNIST)
TA (F-MNIST)
(Hiker)
MAP-Elees (Ime)
MAP- (lim))
MAP-Elies (line)
MAP-Elies (ime)
CMA-MAE
MAP-Ellies
CMA-MAF
MAP-Elies
CMA-MAE
MAP- Elltes
CMA-MAE
MAP-Elies
DDS
DMS
DDS
DMS
DDS
DMS
DDS
DMS
CMA-MAE MAR
AAVAS &
AAVS
AA SAA
VVVVA
VVVV
AAS S 8
S S 199960 0
VVV
VVVV
A MAA &
AA G
A VVVV
& SVVV
SA &
02 SA
S 19
MAP-Elites (line)
MAP-Elites
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C
HYPERPARAMETERS
Table 6: Hyperparameters.
10D (Sphere)
2D LP (Rastrigin)
10D LP (Rastrigin)
2D LP (Sphere)
Arm Repertoire
TA (NISSS
2D LP (Flat)
10D LP (Flat)
TA (MNIST)
LSI (Hiker)
DMS
Number emitters W
15
15
5
Emitter batch size À
36
36
36
36
36
36
36
36
36
36
Initial step size da
0.5
0,5
0.5
0.5
0.5
0,5
0.2
0.1
0.1
0.02
Archive learning rate
0.1
0.1
0.1
0.1
0.1
0.1
0.001
0.001
0.1
1.0
Restart
Basic
100
Basic
100
Basic
100
Basic
50
50
Basic
Selection rule
"
/
/
/A
/
Empty points Rempty
100
100
100
100
100
100
100
100
100
100
Initial points
1000
1000
1000
1000
1000
1000
1000
1 1000
1000
1000
CMA-MAE
Number of emitters W
15
15
15
15
15
5
5
Emitter batch size À
36
36
36
36
36
36
36
36
36
36
Initial step size do
0.5
0.5
0.5
0.5
0.5
0.5
0.2
0.1
0.1
0.02
Archive learning
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.001
0.1
0.01
Restart rule
Basic
Basic
Basic
Basic
Basic
Basic
Basic
50
50
Basic
Selection rule
DDS
Number of emitters W
1,5
15
155
15
15
Emitter batch size
36
36
36
36
Initial step size do
1.5
1.5
1.5
1.5
1.5
1.5
0.5
Bandwidth Buffer size
25.6
5.12
25.6
5.12
25.6
5.12
10.0
10,000
10,000
10,000
10,000
10,000
10,000
10,000
Restart
Imp.
No Imp. Filter
No Imp. Filter
Imp. Filter
Imp. Filter
No Imp. Filter
No Imp.
Selection rule
Filter
Filter
MAP-Elites (line)
À (batch size)
540
540
540
540
540
540
540
180
180
36
0.5
0.5
0.5
0.5
0.1
d2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
MAP-Elites
À (batch size)
540
540
540
540
540
540
540
180
180
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.1
0.1
0.1
Table lists the hyperparameters of DMS and all baseline algorithms. All algorithms run for 10,000 terattions. The number of solutions generated and evaluated on each iteration is equal across all algorithms. In DMS, CMA-MAE, and DDS, this number is equivalent to the number of emitters W times the emitter batch size X. In the two MAP-Elites algorithms, this number is equivalent to the batch size X. For DDS, we use the KDE version TIDDS-KDET) (Lee et al. 2024). In all domains, the objective is normalized be between and so the minimum objective Smin set to 0. For the benchmark domains, parameters for the basclines adapted from prior work (Fontainc & Nikolaidis 2023 Lee et al. 2024).
Archive. In each domain, all algorithms use the same archive configuration. In benchmark domains, the archive has 10,000 cells, arranged as a 100 x 100 grid for domains with 2D measure spaces: 2D LP (Sphere), 2D LP (Rastrigin). 2D LP (Flat), Arm Repertoire. The cells arranged as 10,000- cell CVT archive for domains with 10D measure spaces: 10D LP (Sphere), 10D LP (Rastrigin), 10D I.P (Flat). In QDDM domains, the archive is CVT archive consisting of centroids sampled from
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the dataset. There are 1,000 cells in the archive for TA (MNIST) and TA (F-MNIST), and 10,000 cells in the archive for LSI (Hiker). The same CVT is used across all trials of all algorithms per domain (as opposed to randomly regenerating the CVT every trial).
Restart Rule. The restart rule refers to the conditions upon which emitters are restarted from solu- tions in the archive. "No Imp." refers to restarting when the emitter no longer discovers solutions that are added to the archive (i.e., solutions that improve the archive) (Fontaine et al.) (020). "Basic" refers to restarting only when default CMA-ES (Hansen 2016) termination rules are met. An integer value A refers to restarting every I iterations (Fontaine & Nikolaidis) 2023). We study the role of the restart rule in DMS in further detail in Appendix D
Discount Model Architecture and Training, In all domains except for LSI (Hiker), the discount model is three-layer MLP with layer sizes [As, 128, 128, 1]. where is the dimensionality of the measure space. In LSI (Hiker), the architecture differs slightly in that measures (images) are embed- ded with the vision transformer of CLIP (Radford et (00T)) before being passed into a three-layer MLP with layer sizes [512, 128, 12 128, 1]. Beyond that, the details of all MLPs are identical. There is ReLU activation after every layer except the output layer. Inputs to the network are normalized to the range [-1, 1] based on the bounds of the measure space; in the case of images, these bounds are assumed to be [0. 1]. Networks are instantiated with the default Py initialization. The discount models are trained with Adam optimizer with settings of learning rate 0.001 and 31 0.9, 32 = 0.999. The loss function is mean squared error (MSE), and we train with a batch size of 32. Each iteration (including during the initial training of the model output fmin), the model trains until an average cutoff' loss of at most 0.05 is reached over the whole dataset DA. with a maximum of five epochs allowed. In practice, we found that training almost always required only one epoch to reach a cutoff loss of 0.05. The optimizer is maintained throughout the entire run.
C.I ON THE CHOICE OF MLP DISCOUNT MODELS
Our selection of an MLP as the discount model (as opposed to kernel-based methods or more com- plex architectures) is motivated by two reasons. First, MLPs are relatively straightforward to im- plement and train. In designing DMS, we focused on making the discount model output accurate discount values. For a trainable model like an MLP, this process breaks down into two components: providing the correct training data for the discount values, and ensuring the model can learn to output those values. Prior work on compositional pattern-producin producing networks (CPPNs) (Ha) 2016) shows that MLPs with similar setups as our discount models can be trained output complex training data like images, giving us confidence that the MLPs can accurately represent the discount function. Thus, we are able to focus on providing the correct training data (i.e., the targets described in Sec. 5). Second, choosing a neural network architecture like the MLP opens the door to scale to more com- plex neural network- based models in the future. Namely, now that we know how to train the MI.P, we believe it will be feasible to scale to more complex domains by inserting larger architectures like transformers.
C.2 ON NEAREST NEIGHBOR SEARCH IN ARCHIVES
A key consideration in using CVT archives is how solutions can be assigned to centroids. This process entails a nearest neighbor search in the measure space to identify the closest centroid each new solution. In lower-dimensional measure spaces, this search can be efficiently performed with a k-D tree (Bentley 1975) in logarithmic time. However, as the dimensionality of the measure space grows, the performance of k-D trees degrades to that of brute force. For our experiments, we were able to use k-D trees for all the benchmark domains and the TA domains. The benchmark domains had sufficiently low-dimensional measure spaces that the k-D tree was able to operate efficiently. Meanwhile, although the TA domains had 784-dimensional measure space, the archives only had 1000 centroids, making brute foree performance acceptable. We note that for higher dimensions (10D or more), we further improved efficiency by allowing the k-D tree to use multiple workers (8 workers, specifically).
For LSI (Hiker), nearest neighbor search was slightly more involved since the measures were larger RGB images. However, we note that the CLIP score is computed by first embedding images (or text) into a 512-dimensional latent space, and then computing cosine similarity between the latent vectors. Thus, to make archives efficient, we precomputed the embeddings for all 10,000 archive
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centroids. Then, associating solutions with centroids in the archive could be accomplished by first embedding the new solutions, and then performing nearest neighbor search with the embeddings of the archive centroids. From this perspective, the nearest neighbor search was only over 10,000 centroids, each of 512 dimensions, rather than each being a large image. We performed this search using the Weaarette ghbors implementation from scikit-learn (Pedregosa al. 2011).
We anticipate that for QDDM domains, the computational efficiency of nearest neighbor searches will become major bottleneck, especially since machine learning datasets often consist of millions of images. In such cases, it may suffice to continue to use brute force search, perhaps implemented on a hardware accelerator like GPU. However, it may also become necessary turn approximate nearest neighbor methods like those implemented in the FAISS library (Douze et al. 2024)
D ABLATION STUDY
D.1 ARCHIVE LEARNING RATE
In DMS, the discount model trained on dataset DA consisting entries derived from points sampled by the emitters, and entries created from empty cells in the archive (Sec. 5). The sampled
points create targets that reproduce the threshold update rule from CMA-MAE (Eq. Ij. As such, this target contains archive learning rate that controls how quickly the discount model adapts its values. Here, we empirically analyze whether this archive learning rate a induces similar effects as the one in CMA-MAE by varying from 0.0 to 1.0 in the benchmark domains with 2D and 10D measure spaces.
Fig. displays the results of this ablation. We observe that similar to CMA-MAF, low values of a essentially turn DMS into a single-objective optimization algorithm. This can seen in the low 0.0, which indicate that DMS only optimizing the objective and not
coverage values for exploring the measure space. We also observe that QD Score and Coverage typically peak around 0.1 in the LP benchmarks. Due low distortion, it easy to explore the measure space Arm Repertoire, so there is greater focus on optimization, and thus lower learning rate of = 0.001 is more helpful.
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Figure 9: Mean and standard error the mean of QD Score and Coverage when varying the archive learning rate in DMS in the benchmark domains. Highlighted lines indicate results from the main paper Table
Mean over 20 trials.
8000
8000
8000
8000
4000
41010
41000
40000
100%
Contrnge
509S
50%
016
0%/
055
10
10
10
10
10
10
10
10"
10"
10'
(Sphere)
10D LP (Splacre)
2D (Rastrigin)
10D LP (Rastrigia)
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
QD Soore
Coverage
5.77 40%4
5.17
=0.04%
0.001
6,317.68
+141
75.69 +2.1 77.17%
5,058. 38 +39.31
59.00 +0.48%
1,666.56 +42 82
19.72
+0.51%
2,179.26 =18.72
3. +0.27%
0.01
7,057.29
29 +28.30
91.83 +0.61%
6,649.11
#21.16
85.38 +0.31%
5,114,61
#42.14
71.00
50.80%
4,791.17 #37.63
74.67 =0.64%
0.1
117.94 69788.00
95.89
30.43%
6,409
#13.5
89.21
10.18
5.78.90
219.78
91.67
20.44%
5,13881
#10.31
88.19
10.12%
1.0
7,107.31
#15.99
97.19 =0..4%%
6,449.26
#27.17
82.47 30.66%
3.644.14:18.99
91.20 20.30%
4,802.90 #37.32
76.70 =0.85
2D (Fla))
10D (Plal)))
Arm Repertoize
100%6
100%
100%
Coverage
5056
10
01
LP (Flat)
LP (Hlat)
Repertoize
Score
Coverage
Score
Coverage
Score
Coverage
0.0
2,614.85 +303.
26. +3.03%
483,10 #57.52
4.83 =0.58%
5,523.36 +277 61
55.94 #2.77%
0.001
6.777.
+187.7 78
67.77 #1 88%
7,112.80 #61
71.13 20.618
7,963.44 #3.47
80.15 =0.01
0.01
7.770.30
#66
71.70 +0.67%
#42.68 7,772.00
77.72 =0.43%
+3.41 1,874.43
80.23
+0.00%
7,902.05
7,982.15
79.82
7,784.08
1.0
7,405.05 157.10
74.05
a0..7%
6,483.45
268.55
64.83
=0.69%
7,861.87 +6.34
80.15
=0.01%
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Figure 10: Mean and standard error of the mean of QD Score and Coverage when varying the number of empty points Pempty in DMS in the benchmark domains. Highlighted lines indicate results from the main paper in Table Mean over trials.
(Sphere)
(Sphee)
(Rasingin)
8000
8000
8000
8000
4000
41000
40000
4000
0-
0-
100%
Corenge
50%
50%
076
07%
07
10'
30
10
10'
10"
10
10"
Nomas
LP (Sphere)
10D (Sphere)
2D LP (Rastrigin)
10D LP (Rastrigin)
QD Score
Coverago
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
Hemply
#12.29
192.25
2.55
nempty 10
7,301.27 +8.53
97.41 =0.23%
6,658. 16 +11.48
91.10 +0.13%
5.629.27 #25.66
87.22 40.45%
5,325.70 #12.45
89.49 +0.
Bemply 6,978.20
+0.43%
6,409.50
89.21 18.18%
5.738.0 +19,78
40.4%%
5,13881
88. 20.12%
nempty 1000
6.726.24 128.83
92.32 10.68%
6,131.24 113.54
87.84 20.22%
5,415.26 125.07
85.80 50.57%
4922819913..1
87.80 20.22%
2D (Flla)
10D (Fluu)
Arta Repettoie
Score
4000
10006
1007%
100%
Coverige
5056
50%
50%
056
0%
10'
10'
10'
2D LP (Flat)
10D LP (Flat)
Arm Repertoire
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
Pempty U
1.573.05 =97.23
15.73 20.97%
2,357.20 1:49 99
23.57
20.50%
5.672.67 =102.41
57.24
21.0%
Remply 10
8,049.20 #63.60
80.49 40.64%
7,975.20 #25.60
79.75 40.26%
7,953.08 +3.00
60.12 a0.01%.
Rempty 100
+3.668 7,902.05
79.02 40.3
7,982.
+24.43
79.82 #0.24%
7,963.44 +2.47
80.15
=0.01
Nempty 1000
1.73875550.22
71.39 20.50%
7,972.80 #32.19
79.73 20.32%
(924.8338..2
79.58 20.08%
Table 7: Mean and standard error the mean of QD Score and Coverage with a basic restart rule and restart rule of 100 in the benchmark domains. Highlighted lines indicate results from the main paper in Table
Mean over 20 trials.
2D I.P (Sphere)
10D I.P (Sphere)
2D I.P (Rastrigin)
10D I.P (Rastrigin)
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
restart = basic
6,978.20 117.94
95.89 20.43%
5.636.33 +17.84
82.58 a11.23%
5,738.90 #19.78
91.67 40.44%
4.29%. 35 #22.15
76.94 20.39%
restart 100
6,472.23
6,409.50
5,304.59
5,138.81
2D LP (Flat)
10D LP (Flat)
Arm Repertoire
QD Score
Coverage
QD Score
Coverage
QD Score
Coverage
bostc
7,902.05 #35 688
79.02 36%
2,064, #22.56
20.64 +0.23%
7,963.44 #2.47
80.15 =0.01
restart 100
8,390.00 #21.51
83.90 a0.22%
7,982.15 a24.43
79.82 20.24%
7,659.79 24.68
77.38 10.03%
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Iteration 0
Discoan Milel
256
256
1.00
0.75
0.50
0.25
Emitter Samples
Emppy
-256
0.00
256
255
256
256
Iteration 250
Archive
Discount Model
Samples
Engty Peints
-2.5
-256
0.00
Iteration 10000
Archäve
Discount Model
256
236
1,00
0.75
0.50
0.25
Emkkee Samples
Empoy
-2.5
0,00
255
235
-256
256
Figure 1I: Similar to Fig. 8 this figure shows how the archive and discount model in DMS progress across iterations. However, this time, DMS does not train the discount model with any empty points, i.., Nempty = 0. As result, the discount model takes on arbitrary values in areas of the measure space that have not been explored yet, as evinced by the high values across the discount model heatmap on Iteration 250 and 10000. Because the discount values are high everywhere, the emitters in DMS mistakenly believe they have explored all areas of the measure space, even though the archive is essentially empty.
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D.2 EMPTY POINTS
The second set of entries in the dataset created by DMS are "empty points," which originate as the centers of unoccupied cells in the archive (Sec. 5). To understand the necessity of these points, vary the number such points nempty from 0 (no empty points) to 1000 in the benchmark domains with 2D and 10D measure spaces.
Fig. 10 shows the results of this ablation. When there are no empty points, both QD Score and Coverage drop because the discount model takes arbitrary values in areas of the measure space that have not been explored yet, as shown in Fig. 11] Arbitrary values make it appear as those areas have already been explored. In contrast, when Hempty O, performance increases because the discount function now outputs the minimum objective smin in those areas, which reflects that those areas have not been explored yet. Surprisingly. across all domains, performance remains relatively even with respect to the number of empty points - DMS with Rempty = 10, nempty = 100, and Dempty 1000 all achieve fairly similar scores. However, we note that increasing the number of empty points also increases runtime since the discount model must be trained with these points. In short, these results show that the empty points are a necessary addition to the training set DMS.
D.3 RESTART RULE
The "restart rule" in DMS and CMA-MAE refers to the conditions upon which the emitters restart search from new emitter in the archive AAgoriihm I] line 19 20). By default, emitters in DMS and CMA-MAE use a "basic" restart rule, which makes the emitter restart when the default termination conditions for CMA-ES (Hansen 2016) are met, e.g., the area of the search distribution becomes too small. While tuning DMS, we found it helpful in some domains to make the emitters restart on fixed schedule as introduced in prior work (Fontaine & Nikolaidis, 2023), e.g., restarting every 100 iterations. To better understand the effect of the restart rule in DMS, we thus present an ablation where we run DMS in the benchmark domains with both a "basic" restart rule and restart rule of 100.
Table 7 shows the results of this ablation. We observe that in the 2D LP benchmarks and Arm Reper- toire, the "basic" restart rule usually achieves better performance, while in the 10D LP benchmarks, the restart rule of 100 always achieves better performance. The difference particularly prominent in 10D LP (Flat), where the "basic" restart rule receives mean QD Score of 2,064. 10 while the restart rule of 100 receives mean QD Score of 7,982.15. As such, it seems the restart rule of 100 is particularly helpful in the benchmarks with higher-dimensione measure spaces.
We speculate that this result occurs due to "hacking" of the discount model. In short, it may be possible for emitter to generate solutions that achieve similar measures. The discount model then updates to reflect that that specific area of measure space now has a high discount value. However, the emitter now slightly modify its distribution such that it generates solutions in nearby that still has a low discount value. is especially easy to do this in a nigh-dimensiooal measure space because there are exponentially more "nearby areas" with low discount values. Thus, with basic restart rule, the emitter can continue this process and receive high discount values. On the other hand, fixed restart rule forces the emitter restart before it can converge to a small distribution that hacks the discount model.
E DOMAIN DETAILS
E.1 LINEAR PROJECTION
Introduced in prior work (Fontaine et al., 2020) and extended later work (Lee et al. 2024 LP considers measure function that maps n-dimensional solution space to k-dimensional measure space. Given f. the measure values are bounded by [-5.12 r, 5.12 r]* and defined
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(+1Dr Σ clip(0:) :j {0,..., -0,,,.. k -1)
m(0)
i=j++1
0 5.12/00;
if 00:1 - 5.12
clip(0;)
otherwise
where 0; is the ith component of 0 (one-indexed). The measure function, m(0), partitions O into contiguous, non-overlapping blocks of size r, applies clip(.) clement-wise, and then sums each block, thus producing k-dimensional measure vector. The clip(-) function bounds 0g to the interval [-5.12, 5.1 so m(0) bound by [-5.12 5.12 r|*.
r-1
y=
Density
m0))
Figure 12: Irwin-Hall distribution when summing r uniformly random variables with range 5-1.12
To understand how the measure function creates distortion, consider if we randomly sample each solution value 0g from the range [-5.12, 5. 5.12]. Since each measure value made by summing block o values from the solution vector, the distribution of each measure value forms an Irwin-Hall distribution (Fig. 12). Importantly, observe that for higher values of r, the probability of attaining extreme measure values becomes less and less likely. For example, with uniformly at random, it is quite likely that we obtain a solution with measures close to 0 or 10. In comparison, for 10, there is virtually
2, if we sample all 0,
probability of obtaining the extreme values in the range [0, 10) or [40, 50). In other words, there is higher distortion because a larger portion of solution space maps the relatively small region of measure space in the center of the distribution. However, it is possible to overcome this distortion and reach the edges of the measure space by intelligently sampling 0,,
Our experiments instantiate the LP domain with the Sphere and Rastrigin objectives from the black- box optimization benchmark (Hansen et al. 2010):
JSpbere ())
87
fRastrigia (0)
100 +
Σ
10 10 (2]
While these objectives require minimization by default, convert them for maximization by taking their negative values. Furthermore, following prior work (Fontaine et al. 2020), we shift the global optimum to 6; 5.12 2.048. We also normalize the objective the range [0, 1] to do this,
we consider the minimum possible value of the objective to be at 0; = -5.1 5. 5.122 1.:---77.10 7.168.
In addition Sphere and Rastrigin, consider the Flat objective from prior work (Lee al. 2024]
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(0)
=1
The Flat objective turns the domain into diversity optimization domain since all solutions have the same objective value, an algorithm only needs to find solutions with diverse measures.
For each objective, considered solution dimensionality
100 and measure space dimension- ality & = {2. 10}. Thus, we obtained six LP benchmarks, named according to their measure space dimensionality and objective function: 2D LP (Sphere), 10D LP (Sphere), 2D LP (Rastrigin), 10D LP (Rastrigin), 2D LP (Flat), 10D LP (Flat).
E.2
ARM REPERTOIRE
Arm Repertoire Cully & Demiris 2017: Vassiliades & Mouret, 2018) considers solutions 0 E II"" that represent the n joint angles of planar robotic with joints and links of length 1. The measure function, m0), uses forward kinematics to compute the (z, y) position of the end effector thus, the measure space two-dimensional. The objective function seeks to minimize the variance between each joint angle:
f(0) var
Our experiments consider n 100. We also add the objective normalize to maximum value of I.
E.3
TRIANGLE ARRANGEMENT (TA)
This QDDM domain considers arranging triangles to create images, as inspired by prior work (Tian
Ha 2022). We consider grayscale images our work, with the triangles all drawn on black background. Each triangle is parameterized by the following eight values:
(wo, Vo. F1, 31 22, y2, brightness,
(Io, 9o). (I1,31) and (I2, y2) are the vertices of the triangle, and the triangle is shaded/filled ac- cording to the given brightness and alpha (transparency) value a. A single solution vector 0 con- sists of the parameters for 30 triangles concatenated together; thus, each solution has 30 8 240 parameters. The triangles are drawn the order specified in the solution vector, i.c., the last triangle would end up top of the first triangle if they intersect. To render the triangles raster image, we adapt the JAX (Bradbury et al. 2018) implementation from EvoJAX (Tang et al.) 2022 EvolAX 2022).
We define the measure space as the space of 28 x 28 grayscale images, making it 784-dimensional.
Each dimension has bounds [0, 1]. i.e., the possible brightness values of a grayscale pixel. For each solution O, we compute the measures by rendering a 28 x 28 raster image of its triangles. We specify desired measures by sampling 1000 images from either MNIST (Lecun al., 1998) or Fashion MNIST (Xiao et al.) 2017). leading to the two versions of this domain considered in this work, TA (MNIST) and TA (F-NNIST). These 1000 images form the centroids of the CVT archive. To determine the closest centroid when inserting solutions into the CVT archive, we compute the Euclidean distance between the solution's 's measures (i.c., its rendered image) and each centroid.
For each solution, the objective is the negative mean squared error between the solution's measures and the centroid to which closest. normalize this objective to range of [0, 1], we add to it. This objective enables solutions that resemble an MNIST image to replace solutions that do not.
E.4 LATENT SPACE ILLUMINATION (LSI)
We construct this QDDM domain by adapting prior work that introduced LSI domains with 2D measures (Fontaine Nikoladds 2021 2023), and prior work that shows how to guide StyleGAN3 to generate images that match text prompts (Herrera-Berg. 202T). In domain, which refer to
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as LSI (Hiker), the goal is generate face images of hikers in different landscapes, e.g., hiker who is ready for the mountains.
To that end, consider StyleGAN3 Karras al. 2021a the pretrained st stylegan3 1e05st-ffhuue wfuuuuuuuu-----------uu-uuuu-uuu 2562256.2... pki that generates 256 x 256 RGB images of faces. Each solution 0 is latent vector in the w-space of StyleGAN3, making it 512-dimensional. Note that w-space differs from w -space, which assigns distinct style vector to each layer of the GAN and is much higher-dimensional (e.g., 8,192- or 7, (68-dimensional)
generative model, specifically
We consider the space of 256 x 256 x 3 RGB images as the measure space. The measure function m0) outputs the face image generated by passing the solution 0 through StyleGAN3. To specify desired measures, we sample 10,000 landscape images from the LHQ256 (Skorokhodov et al. 2021) dataset. These images form the centroids of CVT archive, and when inserting solutions. we com- pute the closest centroid with CLIP score (Radford et al. 2021), specifically with the ViT-B/33 32 model of CLIP. The intuition is that the CLIP score will cause hiker be associated with the centroid/landscape that is semantically closest to them. For example, a hiker wearing a thick jacket most suited for a cold landscape like the mountains, while a hiker wearing thin clothes is most suited for the beach (Fg.. 2D.
The objective in this domain considers several factors:
1. f frrompe: CLIP score between the generated image and the prompt "A photo of the face of hiker."
2. feentroid: CLIP score between the generated image and the centroid to which the solution is assigned. This results in images that more closely match the landscape specified in the centroid.
3. Treg: We sample 10,000 points in w-space and compute their mean and standard deviation. If the latent vector strays too far outside this distribution, we apply an L2 penalty.
The final objective is computed as:
Sprompt + feentroid
freg
2
an algorithm stays within the training distribution StyleGAN3, it should not incur any regular- ization penalty, and the objective should stay between 0 and 1. However, the objective can become negative if an algorithm goes outside the training distribution and thus receives high values for freg-
F IMPLEMENTATION
Compute Resources. We run our experiments on workstation with 64-core (128-thread) AMD Ryzen Threadripper, NVIDIA RTX A6000 GPU, and 64GB of RAM.
Compute Usage. In benchmark domains, we run all 20 trials of each algorithm in parallel. In QDDM domains, we run algorithms serially. For wallclock times, refer to Table 2 In all domains, we train the discount model of DMS on the GPU. The discount model typically requires 300-400MB of GPU memory, In TA (MNIST) and TA (F-MNIST), we accelerate rendering of the triangles on the GPU. LSI (Hiker), we accelerate StyleGAN3 and CLIP with the GPU, The final experimental results required 100GB of storage.
Preliminary Experiments. We estimate that developing DMS and tuning the baselines required used in the final experiments and ablations, viz., all our
about the same amount of compute preliminary results occupied another 100GB of storage.
Software. We implement DMS and all baselines with the pyribs (Tjanaka et al. 2023 library, which is available under the MIT License.
Datasets.
We the MNIST (eecun et al., 1998), Fashion MNIST (Xiao et al. 2017). and LHQ (Skorokhodov et al. 2021) datasets in our work. MNIST in the public domain, Fashion MNIST is under an MIT License, and LHQ is available under the CC BY 2.0 license.
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G DIFFERENTIABLE QUALITY DIVERSITY IN QDDM DOMAINS
While our work considers a black-box QD setting as defined in Sec. prior work (Fontaine & Nikolaidis 2021) introduced differentiable quality diversity (DQD), a QD setting where the objec- tive and measure functions are first-order differentiable. Since DQD methods make assumptions that DMS and our other baselines do not, viz., that the objective and measures are differentiable, we have not included DQD methods in our experiments. Nevertheless, here we discuss considerations in applying DQD methods QDDM domains.
The primary consideration is that DQD methods, such CMA-MEGA (Fontaine & Nikolaidis, 202T), were designed with the assumption of high-dimensional solution space that could be re- duced to a low-dimensional objective-measure space, under the assumption that the measure space low-dimensional.
elaborate, during operation, CMA-MEGA and other DQD methods com- pute gradient for the objective and for each measure. In low-dimensional measure spaces, e.g., 2D measure space, this works well because only few gradients. for the objective and 2 for the measures in this case, need to be computed. In contrast, in QDDM domains, the measure spaces much higher-dimensional than even the solution space. In LSI (Hiker), the solution space is 512D while the measure space is 196,608D (256 256 3 RGB image). In the TA domains, the solution space is 240D while the measure space is 784D (it is also worth noting that the TA domains are non- differentiable due the rendering process). As such, running DQD in these domains would mean computing to 196,608 measure gradients on every iteration, which is computationally intractable. Given this limitation, we believe developing DQD method that operates in QDDM domains is an exciting avenue for future work.
39


Reviews:
[{"review_id": "leDmoSmQz0", "reviewer": "Reviewer_Rx9F", "summary": "This paper introduces Discount Model Search (DMS), a new algorithm for quality diversity (QD) optimization that addresses the challenges of exploring high-dimensional measure spaces. Traditional algorithms like CMA-MAE rely on discrete histograms to represent discount values in measure space, which causes exploration to stagnate when multiple solutions map to the same histogram cell. DMS replaces this histogram with a continuous, learned discount model—typically a neural network, that can assign smoother and more fine-grained discount values, even for similar measures.\nThe paper also introduces Quality Diversity with Datasets of Measures (QDDM), where the measure space is defined by a dataset (e.g., images), enabling new applications in domains like generative modeling. Extensive experiments across benchmark tasks and new QDDM domains (e.g., Triangle Arrangement and Latent Space Illumination) show that DMS outperforms state-of-the-art QD algorithms such as CMA-MAE and DDS in both QD Score and coverage.", "questions": "- In Figure 2, a lot of the characteristics of the person (e.g., skin tone, facial features) remain the same even though the input image looks different. And although the image looks more similar sometimes with lakes in different images, the person looks more different. Do the authors have any insight or analysis on why this happens?\n- How do the authors foresee the algorithm to be applicable to text domains? Could we reuse pretrained LLMs as the DMS?\n- Is there a possibility of transferring the trained DM across tasks?\n- Could the same mechanism be applied to the fitness function?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper tackles a well-motivated problem of QD algorithms in high-dimensional measure spaces.\n- The paper is well written.\n- there is extensive experimentation with diverse domains and baselines.", "weaknesses": "- If discount values of two data points are the same, then the DMS would treat them the same way. As with every quantitative measure, by Goodhart's law, pathologies may appear when we optimize for quantitative measures. Did the authors observe any of such phenomenon? How might the authors foresee mitigating such a limitation in future?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0KAeRlfaGo", "reviewer": "Reviewer_uYMX", "summary": "The paper addresses the challenge of \"distortion\" in Quality Diversity (QD) optimization, where high-dimensional measure spaces cause many solutions to map to the same regions, leading to stagnation in state-of-the-art algorithms like CMA-MAE that rely on discrete archive histograms for guiding search. The authors propose Discount Model Search (DMS), which replaces the discrete discount histogram with a continuous, learned \"discount model\" (a neural network) to provide smoother, more accurate improvement signals in high-dimensional spaces. Additionally, the paper introduces a new QD setting called Quality Diversity with Datasets of Measures (QDDM), allowing users to specify desired diversity via datasets (e.g., images) rather than hand-crafted functions. Experiments on standard high-dimensional benchmarks and new QDDM domains (rendering triangles to match MNIST, latent space illumination with StyleGAN3) show DMS significantly outperforms current black-box QD algorithms in terms of QD score and coverage.", "questions": "1. In the QDDM setting, how does the approach scale with the size of the user-provided dataset? If a user provides a dataset of 1 million images, does constructing the CVT archive and finding nearest centroids become a bottleneck, and does DMS still effectively navigate such a large, sparsely populated space?\n\n2. Regarding the \"noise\" limitation in TA (MNIST): Have you considered a hybrid approach where the discrete histogram is used for highly local, fine-grained optimization (if cell capacity allows), while the discount model guides broad exploration?\n\n3. Could you provide more concrete details on the wall-clock time difference between DMS and CMA-MAE on the 10D benchmarks to better illustrate the computational trade-off mentioned in the conclusion?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- Originality & Significance in addressing high-dimensional measures: The core insight—replacing the discrete, low-resolution histogram of CMA-MAE with a continuous learned model—is a novel and effective solution to the well-documented problem of distortion in high-dimensional QD. This allows QD specifically to scale to measure spaces previously considered intractable for standard MAP-Elites-based approaches.\n\n- Introduction of QDDM: The proposal of Quality Diversity with Datasets of Measures (QDDM) is a highly practical and significant contribution. By leveraging the \"manifold hypothesis\" and using datasets to define centroids for CVT archives, it opens QD to a vast array of new applications where hand-crafting low-dimensional measure functions is difficult, but data is abundant (e.g., using raw images as measures).\n\n- Strong Empirical Results: DMS demonstrates superior performance across specifically designed high-distortion benchmarks (e.g., 10D Linear Projection) compared to strong baselines like CMA-MAE and DDS. The inclusion of complex, creative domains like LSI (Hiker) demonstrates practical utility beyond simple synthetic benchmarks.\n\n- Clarity of Exposition: The paper is exceptionally clear. Figure 1 effectively illustrates the failure mode of discrete histograms in distorted spaces. The explanation of distortion and how it is amplified by dimensionality is well-reasoned.\n\n- Thorough Ablations: The appendices contain valuable ablations, specifically demonstrating the critical necessity of \"empty points\" in training the discount model to prevent it from hallucinating explored regions.", "weaknesses": "- Computational Overhead: As noted by the authors, training the neural network discount model introduces computational overhead compared to the simple histogram updates of CMA-MAE. While likely acceptable given the performance gains in high dimensions, a more explicit quantification of this wall-clock time trade-off in the main text would be beneficial.\n\n- Potential Precision Limitations: The results in the TA (MNIST) domain show DMS performing slightly (though not significantly) worse than CMA-MAE in QD Score, despite similar coverage. The authors speculate this is due to noise in the discount model interfering with fine-grained objective optimization. This suggests a potential limitation of DMS in domains where extremely precise objective optimization in already-explored regions is paramount.\n\n- Dependence on \"Empty Points\": The method heavily relies on explicitly sampling \"empty points\" to clamp the discount model. While effective in the tested domains, one might question how this scales if the measure space is so vast that almost all points are empty. Is a fixed number $n\\_{empty}$ sufficient, or does it need to scale with the measure space volume?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2tMHog1Vo3", "reviewer": "Reviewer_83tg", "summary": "This work deals with high-dimensional quality-diversity optimisation by proposing a so-called discount model search. It aims to guide exploration\nwith a model that provides a smooth, continuous representation of discount scalar values.", "questions": "I would be grateful and may potentially increase my score if the author(s) could address my comments in the weakness field.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The work is well-motivated - indeed, a major issue of QD is its difficulty in solving high-dimensional problems. \n\nThe paper is well presented, and well-structured. \n\nThe result presents the effectiveness of the proposed method.", "weaknesses": "It seems to me that a major issue is that the method needs additional information (measures) to distinguish between solutions plateaus in QD. It would be more interesting and useful if the work could directly work on algorithm design of QD itself to address the high-dimension issue, rather than resorting to new information (which apparently can tell solutions apart). I am not sure my idea is doable, but it would be very useful if it could.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "S1x8L8F6pb", "reviewer": "Reviewer_TnHM", "summary": "This paper proposes Discount Model Search, a Quality-Diversity algorithm to address stagnation in high-dimensional measure spaces caused by discrete histograms in state-of-the-art methods like CMA-MAE. DMS uses a smooth neural network-based discount model to distinguish solutions with similar measures, guiding exploration effectively. It also introduces the Quality Diversity with Datasets of Measures setting, where users specify desired measures via datasets instead of hand-designing measure functions. DMS is evaluated on 7 benchmark domains and 3 QDDM domains , outperforming baselines like CMA-MAE, DDS, and MAP-Elites in most cases.", "questions": "1. Could you provide a detailed comparison between DMS’s learnable continuous discount model and Kent et al. (2022)’s continuous QD Score, explaining how DMS’s model-guided exploration advances beyond the static metric in Kent et al.’s work, especially in addressing high-dimensional measure space distortion?\n2. For the discount model architecture (e.g., MLPs with [k, 128, 128, 1] layers), could you provide justification for why MLPs are superior to alternative models (e.g., kernel methods for low-dimensional measures, transformers for high-dimensional image measures)?\n3. In the LSI (Hiker) domain, DMS only achieves 3.77% archive coverage. Could you analyze the cause of this low coverage? Additionally, propose and test potential strategies to improve coverage.\n4. The authors attribute DMS’s similar performance to CMA-MAE in TA (MNIST) to “discount model noise.” Could you quantify this noise (e.g., mean squared error between predicted and ideal discount values) and explicitly analyze how it disrupts improvement rankings and emitter updates? Also, explain why this noise has a more pronounced impact in TA (MNIST) than in other domains.\n5. Could you quantify the computational overhead of DMS’s discount model training (e.g., per-iteration runtime, GPU memory usage) relative to CMA-MAE across all domains (e.g., 10D LP, LSI)? For large QDDM datasets (e.g., 10,000 LHQ images), also clarify the measures taken to ensure DMS remains computationally efficient.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. By replacing CMA-MAE’s discrete histogram with a continuous, learnable discount model, DMS addresses the limitation of identical discount values for similar measures in high-dimensional spaces.\n2. The QDDM framework eliminates the need for tedious manual measure function design by leveraging datasets. This expands QD’s applicability to vision and creative domains.\n3. Experiments cover both standard QD benchmarks (testing distortion and scalability) and novel QDDM tasks (testing real-world applicability). Ablation studies rigorously verify key components (e.g., empty points prevent arbitrary discount values, archive learning rate α balances quality/diversity).", "weaknesses": "1. The paper only mentions Kent et al. (2022)’s continuous QD Score as an “evaluation tool” but fails to deeply contrast DMS’s core innovation with Kent et al.’s static metric.\\\n2. DMS uses MLPs for the discount model for non-image measures) but provides no justification for why MLPs outperform other models (e.g., kernel methods, transformers for image measures). For high-dimensional image measures in LSI, the MLP’s ability to capture complex semantic relationships is unproven.\n3. Uses Euclidean distance and LSI uses CLIP scores for centroid matching, but the paper does not compare these choices with alternatives (e.g., cosine similarity for images) or validate their impact on performance. \n4. The LP benchmark only tests up to 16-dimensional measure spaces. \n5. DMS adds discount model training, but the paper does not quantify its computational cost (e.g., runtime, GPU memory) relative to CMA-MAE.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)