# A Distributional Optimisation Perspective on Combining Models in Deep Learning

arXiv: https://arxiv.org/abs/2609.24328 · HTML: https://mlsanity.com/paper/2609.24328/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 43 of its 61 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 30 of 43)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Merging Models with Fisher-Weighted Averaging](https://mlsanity.com/paper/2111.09832/) (2021)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [LLM-Adapters: An Adapter Family for Parameter-Efficient Fine-Tuning of Large Language Models](https://mlsanity.com/paper/2304.01933/) (2023)
- [Learning Factored Representations in a Deep Mixture of Experts](https://mlsanity.com/paper/1312.4314/) (2013)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)
- [Mixture of LoRA Experts](https://mlsanity.com/paper/2404.13628/) (2024)
- [Hyperparameter Ensembles for Robustness and Uncertainty Quantification](https://mlsanity.com/paper/2006.13570/) (2020)
- [A Review of Sparse Expert Models in Deep Learning](https://mlsanity.com/paper/2209.01667/) (2022)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Deep Ensembles Work, But Are They Necessary?](https://mlsanity.com/paper/2202.06985/) (2022)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [No One Representation to Rule Them All: Overlapping Features of Training Methods](https://mlsanity.com/paper/2110.12899/) (2021)

## Abstract (arXiv)

Combining predictions from different models can improve performance at machine learning tasks, but the training of the individual models and the rule used to combine them are typically chosen separately, and by ad hoc means. Recent advances in distributional optimisation (i.e. where the optimisation occurs over the set of probability distributions) offer an opportunity for principled joint training, viewing the collection of models as a discrete distribution whose support points are to be optimised, but the potential of these methods is not well-understood. In this paper we (1) cast two standard combination strategies - ensembles and low-rank adapter averaging - as entropy-regularised distributional optimisation, observing that the resulting objective is convex in the ensemble case but not in the adapter-averaging case, so that existing convergence guarantees for mean field Langevin dynamics transfer only to the former; (2) assess existing and novel algorithms for this task, including a functional variant of variational gradient descent; and (3) report an empirical study spanning synthetic classification tasks and fine-tuning of large language models on a commonsense reasoning benchmark.