# Discount Model Search for Quality Diversity Optimization in High-Dimensional Measure Spaces

arXiv: https://arxiv.org/abs/2601.01082 · HTML: https://mlsanity.com/paper/2601.01082/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 67 of its 99 references that are in the map.
- First posted on arXiv 3 Jan 2026.

## Builds on (most cited first, 30 of 67)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Interpreting the Latent Space of GANs for Semantic Face Editing](https://mlsanity.com/paper/1907.10786/) (2019)
- [Illuminating search spaces by mapping elites](https://mlsanity.com/paper/1504.04909/) (2015)
- [Closed-Form Factorization of Latent Semantics in GANs](https://mlsanity.com/paper/2007.06600/) (2020)
- [InterFaceGAN: Interpreting the Disentangled Face Representation Learned by GANs](https://mlsanity.com/paper/2005.09635/) (2020)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Unsupervised Discovery of Interpretable Directions in the GAN Latent Space](https://mlsanity.com/paper/2002.03754/) (2020)
- [Glaze: Protecting Artists from Style Mimicry by Text-to-Image Models](https://mlsanity.com/paper/2302.04222/) (2023)
- [Rainbow Teaming: Open-Ended Generation of Diverse Adversarial Prompts](https://mlsanity.com/paper/2402.16822/) (2024)
- [Robots that can adapt like animals](https://mlsanity.com/paper/1407.3501/) (2014)
- [Defining and Characterizing Reward Hacking](https://mlsanity.com/paper/2209.13085/) (2022)
- [Quality and Diversity Optimization: A Unifying Modular Framework](https://mlsanity.com/paper/1708.09251/) (2017)
- [Regularizing Generative Adversarial Networks under Limited Data](https://mlsanity.com/paper/2104.03310/) (2021)
- [A Tutorial on Kernel Density Estimation and Recent Advances](https://mlsanity.com/paper/1704.03924/) (2017)
- [Controlling generative models with continuous factors of variations](https://mlsanity.com/paper/2001.10238/) (2020)
- [Covariance Matrix Adaptation for the Rapid Illumination of Behavior Space](https://mlsanity.com/paper/1912.02400/) (2019)
- [Evolving Mario Levels in the Latent Space of a Deep Convolutional Generative Adversarial Network](https://mlsanity.com/paper/1805.00728/) (2018)
- [Differentiable Quality Diversity](https://mlsanity.com/paper/2106.03894/) (2021)
- [Aligning Latent and Image Spaces to Connect the Unconnectable](https://mlsanity.com/paper/2104.06954/) (2021)
- [Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional Archive of Phenotypic Elites Algorithm](https://mlsanity.com/paper/1610.05729/) (2016)
- [EGAD! an Evolved Grasping Analysis Dataset for diversity and reproducibility in robotic manipulation](https://mlsanity.com/paper/2003.01314/) (2020)
- [Autonomous skill discovery with Quality-Diversity and Unsupervised Descriptors](https://mlsanity.com/paper/1905.11874/) (2019)
- [Diversity Policy Gradient for Sample Efficient Quality-Diversity Optimization](https://mlsanity.com/paper/2006.08505/) (2020)

## Built on by (most cited first, 1 of 1)

- [AutoSOTA: An End-to-End Automated Research System for State-of-the-Art AI Model Discovery](https://mlsanity.com/paper/2604.05550/) (2026)

## Abstract (arXiv)

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.