TL;DR
This paper introduces the mirror mean-field Langevin dynamics (MMFLD), an extension of MFLD for constrained domains, providing convergence guarantees and propagation of chaos results.
Contribution
It extends mean-field Langevin dynamics to constrained domains using mirror maps, with theoretical convergence and stability guarantees.
Findings
Linear convergence guarantees via a uniform log-Sobolev inequality
Uniform-in-time propagation of chaos results
Extension of MFLD to constrained probability measures
Abstract
The mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional on the Wasserstein space over , and has gained attention recently as a model for the gradient descent dynamics of interacting particle systems such as infinite-width two-layer neural networks. However, many problems of interest have constrained domains, which are not solved by existing mean-field algorithms due to the global diffusion term. We study the optimization of probability measures constrained to a convex subset of by proposing the \emph{mirror mean-field Langevin dynamics} (MMFLD), an extension of MFLD to the mirror Langevin framework. We obtain linear convergence guarantees for the continuous MMFLD via a uniform log-Sobolev inequality, and uniform-in-time propagation of chaos results for its time- and particle-discretized counterpart.
Peer Reviews
Decision·Submitted to ICLR 2026
This paper provides a clean and novel analysis of mirror mean-field Langevin dynamics as a generalization of mean-field Langevin, and the numerical illustration demonstrates that this can be a better idea to solve distributional optimization problems compared to projected mean-field Langevin. This is interesting in particular since currently there are not many well-studied algorithms for distributional optimization that work well in high dimensions beyond the mean-field Langevin dynamics.
* I think the authors can better motivate the study of mirror mean-field Langevin by showing what new settings can be unlocked by their analysis, e.g. for training weight-constrained two-layer neural networks or for generative modeling. * Since the discretization cost of Step 5 is not analyzed, it could potentially be helpful to have, perhaps an informal, discussion of why simulating this step is easier than simulating a Brownian motion on $\mathcal{X}$ (and consequently performing MFLD on $\ma
* The authors target a fairly important and surprisingly open problem, since mean-field dynamics are used to understand two-layer neural networks and mirror descent is frequently used in constrained optimisation. * The paper is very well-written. * The guarantees are strong and are under relatively standard conditions in this area (e.g., uniform LSI).
* The majority of the proof techniques appear to be borrowed or adapted from other papers (e.g., Nitanda et al., 2022; Jiang, 2021 and Nitanda, 2024), so the work may have limited technical novelty at the proof level. * The analysis of the discretized algorithm assumes that the pure diffusion step (Algorithm 1, step 5) can be **simulated exactly**. The authors note this is for "simplicity of exposition", but this is rarely possible in practice and creates a gap between the theory and the impleme
- The paper is generally well written and easy to understand. The theorem statements are generally clear to me and the proofs are readable and easy to follow. - This paper represents a natural extension of mean-field Langevin dynamics using a mirror map. As was done in the non-mean-field case, this extension has desirable properties of relying on relative Lipschitz types of assumptions, and also naturally maintains the constraints of the problem. - The results utilize the full range of available
- The theorems given seem to be standard extensions of existing results in the literature. The heavy lifting seems to have been done in past works like Nitanda et al '22, Nitanda '24, and Nitanda et al '25. Because of this, I am worried that this work is more of a synthesis work than giving some novel and new ideas that would be sufficient for publication in ICLR. - Coupled with the above limited theoretical novelty, there is a lack of experimental evidence. The authors only give one low-dimensi
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Taxonomy
TopicsQuantum Information and Cryptography · Mechanical and Optical Resonators · Cold Atom Physics and Bose-Einstein Condensates
MethodsSoftmax · Attention Is All You Need · Diffusion
