Efficient infinite-dimensional optimization over measures
October 2025-October 2030
Optimization over probability measures has become a powerful approach for solving complex problems
that involve probabilistic modeling. It extends finite-dimensional optimization to the infinite-dimensional
space of probability measures. This framework provides a principled way to address in particular the
task of sampling, that refers to the process of drawing samples from a complex probability distribution,
either to approximate it or generate new data. In Bayesian machine learning for instance, we can model
uncertainty of predictions by sampling the model parameters. Similarly, in generative modeling, sampling
is crucial for producing new data such as images or text.
Existing methods struggle with complex measures exhibiting high dimensionality and multimodality, are
difficult to evaluate, and are limited to Euclidean spaces (making them unsuitable to handle infinite-dimensional ones like distributions or operators). Also, their high computational cost limits their use
in sequential sampling tasks. The goal of Optinfinite project is to create a unified framework to design and evaluate efficient
methods for sampling measures over general spaces.
Central to this approach is the use of tools from optimal transport and information geometry, which provide ways to compare measures and design optimization dynamics.
There are key challenges to adress: 1) developing optimization objectives and geometries
suited to the space of measures over general (possibly infinite-dimensional) spaces to design tractable
schemes and metrics, and 2) learning to solve advanced optimization problems, such as sequences of optimization tasks.
Optinfinite will yield novel sampling methods, whose efficiency will be evaluated
using optimization tools. It will also enable to compare measures and reveal where and how they are
different. Ultimately, its goal is to provide a clear methodology and toolset applicable across multiple
domains.