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.

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