Poincaré Inequality for Local Log-Polyak-Lojasiewicz Measures : Non-asymptotic Analysis in Low-temperature Regime
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| Format: | Preprint |
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2025
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| _version_ | 1866916616820752384 |
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| author | Gong, Yun Shen, Zebang He, Niao |
| author_facet | Gong, Yun Shen, Zebang He, Niao |
| contents | Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such landscapes, we propose to study the class of \logPLmeasure\ measures $μ_ε\propto \exp(-V/ε)$, where the potential $V$ satisfies a local Polyak-Łojasiewicz (PŁ) inequality, and its set of local minima is provably \emph{connected}. Notably, potentials in this class can exhibit local maxima and we characterize its optimal set S to be a compact $\mathcal{C}^2$ \emph{embedding submanifold} of $\mathbb{R}^d$ without boundary. The \emph{non-contractibility} of S distinguishes our function class from the classical convex 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 \emph{$ε$-independent} lower bound for the \Poincare\ constant in the \Poincare\ inequality of $μ_ε$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $ε$ converges to its equilibrium $μ_ε$ at a rate of $\tilde{\mathcal{O}}(1/ε)$, provided $ε$ is sufficiently small. Here $\tilde{\mathcal{O}}$ hides logarithmic terms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_06862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poincaré Inequality for Local Log-Polyak-Lojasiewicz Measures : Non-asymptotic Analysis in Low-temperature Regime Gong, Yun Shen, Zebang He, Niao Machine Learning Classical Analysis and ODEs Functional Analysis Probability Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such landscapes, we propose to study the class of \logPLmeasure\ measures $μ_ε\propto \exp(-V/ε)$, where the potential $V$ satisfies a local Polyak-Łojasiewicz (PŁ) inequality, and its set of local minima is provably \emph{connected}. Notably, potentials in this class can exhibit local maxima and we characterize its optimal set S to be a compact $\mathcal{C}^2$ \emph{embedding submanifold} of $\mathbb{R}^d$ without boundary. The \emph{non-contractibility} of S distinguishes our function class from the classical convex 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 \emph{$ε$-independent} lower bound for the \Poincare\ constant in the \Poincare\ inequality of $μ_ε$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $ε$ converges to its equilibrium $μ_ε$ at a rate of $\tilde{\mathcal{O}}(1/ε)$, provided $ε$ is sufficiently small. Here $\tilde{\mathcal{O}}$ hides logarithmic terms. |
| title | Poincaré Inequality for Local Log-Polyak-Lojasiewicz Measures : Non-asymptotic Analysis in Low-temperature Regime |
| topic | Machine Learning Classical Analysis and ODEs Functional Analysis Probability |
| url | https://arxiv.org/abs/2502.06862 |