# Soft Quality-Diversity Optimization

arXiv: https://arxiv.org/abs/2512.00810 · HTML: https://mlsanity.com/paper/2512.00810/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 63 references that are in the map.
- First posted on arXiv 30 Nov 2025.

## Builds on (most cited first, 30 of 46)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [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)
- [Covariance Matrix Adaptation for the Rapid Illumination of Behavior Space](https://mlsanity.com/paper/1912.02400/) (2019)
- [Differentiable Quality Diversity](https://mlsanity.com/paper/2106.03894/) (2021)
- [Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional Archive of Phenotypic Elites Algorithm](https://mlsanity.com/paper/1610.05729/) (2016)
- [Quality-Diversity Optimization: a novel branch of stochastic optimization](https://mlsanity.com/paper/2012.04322/) (2020)
- [Diversity Policy Gradient for Sample Efficient Quality-Diversity Optimization](https://mlsanity.com/paper/2006.08505/) (2020)
- [Illuminating Mario Scenes in the Latent Space of a Generative Adversarial Network](https://mlsanity.com/paper/2007.05674/) (2020)
- [Message Passing Stein Variational Gradient Descent](https://mlsanity.com/paper/1711.04425/) (2017)
- [Stein Variational Policy Gradient](https://mlsanity.com/paper/1704.02399/) (2017)
- [Modern Evolution Strategies for Creativity: Fitting Concrete Images and Abstract Concepts](https://mlsanity.com/paper/2109.08857/) (2021)
- [Deep Surrogate Assisted Generation of Environments](https://mlsanity.com/paper/2206.04199/) (2022)
- [Quality-Diversity through AI Feedback](https://mlsanity.com/paper/2310.13032/) (2023)
- [Data-Efficient Design Exploration through Surrogate-Assisted Illumination](https://mlsanity.com/paper/1806.05865/) (2018)
- [Discovering Policies with DOMiNO: Diversity Optimization Maintaining Near Optimality](https://mlsanity.com/paper/2205.13521/) (2022)
- [Mapping Hearthstone Deck Spaces through MAP-Elites with Sliding Boundaries](https://mlsanity.com/paper/1904.10656/) (2019)
- [Stein Variational Gradient Descent With Matrix-Valued Kernels](https://mlsanity.com/paper/1910.12794/) (2019)
- [pyribs: A Bare-Bones Python Library for Quality Diversity Optimization](https://mlsanity.com/paper/2303.00191/) (2023)
- [Deep Surrogate Assisted MAP-Elites for Automated Hearthstone Deckbuilding](https://mlsanity.com/paper/2112.03534/) (2021)
- [Covariance Matrix Adaptation MAP-Annealing](https://mlsanity.com/paper/2205.10752/) (2022)
- [Unsupervised Learning and Exploration of Reachable Outcome Space](https://mlsanity.com/paper/1909.05508/) (2019)
- [Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions](https://mlsanity.com/paper/1901.08659/) (2019)
- [MAP-Elites with Descriptor-Conditioned Gradients and Archive Distillation into a Single Policy](https://mlsanity.com/paper/2303.03832/) (2023)
- [Proximal Policy Gradient Arborescence for Quality Diversity Reinforcement Learning](https://mlsanity.com/paper/2305.13795/) (2023)
- [Quality-Weighted Vendi Scores And Their Application To Diverse Experimental Design](https://mlsanity.com/paper/2405.02449/) (2024)
- [Training Diverse High-Dimensional Controllers by Scaling Covariance Matrix Adaptation MAP-Annealing](https://mlsanity.com/paper/2210.02622/) (2022)

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

- [Quality-Diversity Optimization as Multi-Objective Optimization](https://mlsanity.com/paper/2602.00478/) (2026)

## Abstract (arXiv)

Quality-Diversity (QD) algorithms constitute a branch of optimization that is concerned with discovering a diverse and high-quality set of solutions to an optimization problem. Current QD methods commonly maintain diversity by dividing the behavior space into discrete regions, ensuring that solutions are distributed across different parts of the space. The QD problem is then solved by searching for the best solution in each region. This approach to QD optimization poses challenges in large solution spaces, where storing many solutions is impractical, and in high-dimensional behavior spaces, where discretization becomes ineffective due to the curse of dimensionality. We present an alternative framing of the QD problem, called Soft QD, that sidesteps the need for discretizations. We validate this formulation by demonstrating its desirable properties, such as monotonicity, and by relating its limiting behavior to the widely used QD Score metric. Furthermore, we leverage it to derive a novel differentiable QD algorithm, Soft QD Using Approximated Diversity (SQUAD), and demonstrate empirically that it is competitive with current state of the art methods on standard benchmarks while offering better scalability to higher dimensional problems. Source code is available at https://github.com/conflictednerd/soft-qd.