# How to Fill the Optimum Set? Population Gradient Descent with Harmless Diversity

arXiv: https://arxiv.org/abs/2202.08376 · HTML: https://mlsanity.com/paper/2202.08376/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 42 references that are in the map.
- First posted on arXiv 16 Feb 2022.

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

- [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)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Illuminating search spaces by mapping elites](https://mlsanity.com/paper/1504.04909/) (2015)
- [Text2Mesh: Text-Driven Neural Stylization for Meshes](https://mlsanity.com/paper/2112.03221/) (2021)
- [DRIT++: Diverse Image-to-Image Translation via Disentangled Representations](https://mlsanity.com/paper/1905.01270/) (2019)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [Improving Adversarial Robustness via Promoting Ensemble Diversity](https://mlsanity.com/paper/1901.08846/) (2019)
- [Robots that can adapt like animals](https://mlsanity.com/paper/1407.3501/) (2014)
- [Learning Gradient Fields for Molecular Conformation Generation](https://mlsanity.com/paper/2105.03902/) (2021)
- [Effective Diversity in Population Based Reinforcement Learning](https://mlsanity.com/paper/2002.00632/) (2020)
- [Improving Exploration in Evolution Strategies for Deep Reinforcement Learning via a Population of Novelty-Seeking Agents](https://mlsanity.com/paper/1712.06560/) (2017)
- [Differentiable Quality Diversity](https://mlsanity.com/paper/2106.03894/) (2021)
- [FuseDream: Training-Free Text-to-Image Generation with Improved CLIP+GAN Space Optimization](https://mlsanity.com/paper/2112.01573/) (2021)
- [Bayesian Optimization with Gradients](https://mlsanity.com/paper/1703.04389/) (2017)
- [Fast and stable MAP-Elites in noisy domains using deep grids](https://mlsanity.com/paper/2006.14253/) (2020)
- [Quality Diversity Through Surprise](https://mlsanity.com/paper/1807.02397/) (2018)
- [Multimodal Trajectory Optimization for Motion Planning](https://mlsanity.com/paper/2003.07054/) (2020)
- [Monte Carlo Elites: Quality-Diversity Selection as a Multi-Armed Bandit Problem](https://mlsanity.com/paper/2104.08781/) (2021)
- [Evolution of Swarm Robotics Systems with Novelty Search](https://mlsanity.com/paper/1304.3362/) (2013)
- [Diversity-Promoting Bayesian Learning of Latent Variable Models](https://mlsanity.com/paper/1711.08770/) (2017)

## Built on by (most cited first, 4 of 4)

- [Diffusion-based Molecule Generation with Informative Prior Bridges](https://mlsanity.com/paper/2209.00865/) (2022)
- [Fast Point Cloud Generation with Straight Flows](https://mlsanity.com/paper/2212.01747/) (2022)
- [Pathologies of Predictive Diversity in Deep Ensembles](https://mlsanity.com/paper/2302.00704/) (2023)
- [Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspective](https://mlsanity.com/paper/2602.01182/) (2026)

## Abstract (arXiv)

Although traditional optimization methods focus on finding a single optimal solution, most objective functions in modern machine learning problems, especially those in deep learning, often have multiple or infinite numbers of optima. Therefore, it is useful to consider the problem of finding a set of diverse points in the optimum set of an objective function. In this work, we frame this problem as a bi-level optimization problem of maximizing a diversity score inside the optimum set of the main loss function, and solve it with a simple population gradient descent framework that iteratively updates the points to maximize the diversity score in a fashion that does not hurt the optimization of the main loss. We demonstrate that our method can efficiently generate diverse solutions on a variety of applications, including text-to-image generation, text-to-mesh generation, molecular conformation generation and ensemble neural network training.