Convex Analysis of the Mean Field Langevin Dynamics
- URL: http://arxiv.org/abs/2201.10469v1
- Date: Tue, 25 Jan 2022 17:13:56 GMT
- Title: Convex Analysis of the Mean Field Langevin Dynamics
- Authors: Atsushi Nitanda, Denny Wu, Taiji Suzuki
- Abstract summary: convergence rate analysis of the mean field Langevin dynamics is presented.
$p_q$ associated with the dynamics allows us to develop a convergence theory parallel to classical results in convex optimization.
- Score: 49.66486092259375
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: As an example of the nonlinear Fokker-Planck equation, the mean field
Langevin dynamics attracts attention due to its connection to (noisy) gradient
descent on infinitely wide neural networks in the mean field regime, and hence
the convergence property of the dynamics is of great theoretical interest. In
this work, we give a simple and self-contained convergence rate analysis of the
mean field Langevin dynamics with respect to the (regularized) objective
function in both continuous and discrete time settings. The key ingredient of
our proof is a proximal Gibbs distribution $p_q$ associated with the dynamics,
which, in combination of techniques in [Vempala and Wibisono (2019)], allows us
to develop a convergence theory parallel to classical results in convex
optimization. Furthermore, we reveal that $p_q$ connects to the duality gap in
the empirical risk minimization setting, which enables efficient empirical
evaluation of the algorithm convergence.
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