The asymptotic behavior of fraudulent algorithms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914649145868288 |
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| author | Benaïm, Michel Miclo, Laurent |
| author_facet | Benaïm, Michel Miclo, Laurent |
| contents | Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \geq 2,$ satisfying $\min U=0$ and let $\mathcal{U} = \{x \in M \: : U(x) = 0\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(β)}:=(X^{(β)}(t))_{t\geq 0}$ defined on $N = M \setminus \mathcal{U},$ whose generator is$U Δ\cdot-β\langle \nabla U,\nabla \cdot\rangle$, where $β\in\RR$ is a real parameter.We show that for $β>\frac{m}{2}-1,$ $X^{(β)}(t)$ converges a.s.\ as $t \rightarrow \infty$, toward a point $p \in \mathcal{U}$ and that each $p \in \mathcal{U}$ has a positive probability to be selected. On the other hand, for $β< \frac{m}{2}-1,$ the law of $(X^{(β)}(t))$ converges in total variation (at an exponential rate) toward the probability measure $π_β$ having density proportional to $U(x)^{-1-β}$ with respect to the Riemannian measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The asymptotic behavior of fraudulent algorithms Benaïm, Michel Miclo, Laurent Probability Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \geq 2,$ satisfying $\min U=0$ and let $\mathcal{U} = \{x \in M \: : U(x) = 0\}$ be the set of global minimizers. Consider the stochastic algorithm $X^{(β)}:=(X^{(β)}(t))_{t\geq 0}$ defined on $N = M \setminus \mathcal{U},$ whose generator is$U Δ\cdot-β\langle \nabla U,\nabla \cdot\rangle$, where $β\in\RR$ is a real parameter.We show that for $β>\frac{m}{2}-1,$ $X^{(β)}(t)$ converges a.s.\ as $t \rightarrow \infty$, toward a point $p \in \mathcal{U}$ and that each $p \in \mathcal{U}$ has a positive probability to be selected. On the other hand, for $β< \frac{m}{2}-1,$ the law of $(X^{(β)}(t))$ converges in total variation (at an exponential rate) toward the probability measure $π_β$ having density proportional to $U(x)^{-1-β}$ with respect to the Riemannian measure. |
| title | The asymptotic behavior of fraudulent algorithms |
| topic | Probability |
| url | https://arxiv.org/abs/2401.12605 |