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SUMMARY:Poincare Inequality for Local Log-Polyak-Łojasiewicz Measures: No
 n-asymptotic Analysis in Low-temperature Regime
DTSTART:20250326T111500
DTEND:20250326T121500
DTSTAMP:20260501T144714Z
UID:4f70bf2ffb46cf4e8cca546a082f2aebd6ff124cc7cc8700ddfffdf0
CATEGORIES:Conferences - Seminars
DESCRIPTION:Zebang Shen (ETHZ)\nPotential functions in highly pertinent ap
 plications\, such as deep learning in over-parameterized regime\, are empi
 rically observed to admit non-isolated minima. To understand the convergen
 ce behavior of stochastic dynamics in such landscapes\, we propose to stud
 y the class of log-PŁ∘ measures μϵ∝exp(−V/ϵ)\, where the poten
 tial V satisfies a local Polyak-Łojasiewicz (PŁ) inequality\, and its 
 set of local minima is provably connected. Notably\, potentials in this cl
 ass can exhibit local maxima and we characterize its optimal set S to be
  a compact C2 embedding submanifold of Rd without boundary. The non-co
 ntractibility of S distinguishes our function class from the classical c
 onvex setting topologically. Moreover\, the embedding structure induces a 
 naturally defined Laplacian-Beltrami operator on S\, and we show that its
  first non-trivial eigenvalue provides an ϵ-independent lower bound for 
 the Poincaré constant in the Poincaré inequality of μϵ. As a direct c
 onsequence\, Langevin dynamics with such non-convex potential V and diff
 usion coefficient ϵ converges to its equilibrium μϵ at a rate of O
 ̃ (1/ϵ)\, provided ϵ is sufficiently small. Here Õ  hides logar
 ithmic terms.\n 
LOCATION:CM 1 517 https://plan.epfl.ch/?room==CM%201%20517
STATUS:CONFIRMED
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