For $(Q_t,P_t)$ governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a $\sigma$-strongly log-concave target $\pi(dq) = \frac{1}{Z}e^{-V(q)}dq$ as \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq e^{-\sqrt{\sigma} t}\operatorname{KL}(\operatorname{Law}(Q_{0}, P_{0})\, \|\, \Pi_0), \end{align*} where $\Pi_0$ denotes an appropriately selected reference measure. When $\pi$ is merely log-concave, the estimate \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq \frac{\tau^2}{t^2}\operatorname{KL}(\operatorname{Law}(Q_{\tau}, P_{\tau})\, \|\, \Pi_{\tau}) \end{align*} is derived, where $\Pi_\tau$ denotes another correspondingly chosen reference measure at time $\tau>0$. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in $\mathbb{R}^d$. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.
In sampling, different flavours of annealing, tempering, or other successive approximation approaches are widely used and studied. In this work we investigate annealed underdamped Langevin sampling for logconcave distributions of the form $\pi(x) \propto e^{-\Psi(x)}$ for a potential $\Psi:\mathbb{R}^d \rightarrow \mat...
Laurenz Nagler, Alexander Falk, Andreas Habring· 0 citations
We study the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA) for $\pi(\,\mathrm{d} x)\propto e^{-f(x)-g(x)}\,\mathrm{d} x$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g:\mathbb{R}^d\to\mathbb{R}$ is convex and globally $G$-Lipschitz. For the Moreau-smoothed t...
We investigate the hypercyclicity of Toeplitz operators on the Bergman space $L_{A}^{2}(\mathbb{D})$ with symbols of the form $\Psi(z) = \gamma\bar{z}+\psi(z)$, where $\gamma \in \mathbb{C} \setminus \{0\}$ and $\psi$ is analytic on an open neighborhood of the closed unit disc $\overline{\mathbb{D}}$. Our approach bypa...
Let $\pi(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $\kappa=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds...
We study the maximal existence time $T^{*}(\varrho)$ of the solution of the semilinear wave equation $u_{tt}-\Delta u=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\varrho f,\varrho g)$, where $1<q<p$ and the amplitude $\varrho$ is large. For nontrivial $f$ and...
Consider the stabilizable finite-dimensional linear control system $\dot x=Ax+Bu$. For a prescribed decay rate $\delta>0$, we define the optimal stabilization prefactor $$ \hat{C}(\delta) := \inf_{K\in\mathbb{R}^{m\times n}} \sup_{t\ge0}e^{\delta t}\|e^{(A+BK)t}\|. $$ We determine its asymptotic behavior as $\delta$ ap...
Chang-Qin Quan, Geng-Sheng Wang· 1 citation· ⚡1
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