On the optimal relaxation rate for the Metropolis algorithm in one dimension

TL;DR

This work analyzes the relaxation dynamics of the Metropolis algorithm for a particle in a one-dimensional confining potential with symmetric jump proposals. It reveals a novel charge density wave regime arising for two-peaked jumps, and constructs a phase diagram showing diffusion, CDW, and rejection phases, with the optimal convergence rate attained at their triple point. A variational framework is developed to estimate the next-to-largest eigenvalue and the optimal jump distribution, demonstrating that multi-peak designs can surpass traditional diffusion–rejection balances. The findings suggest practical strategies to engineer jump proposals, including multi-peak and sequence-based schemes, to accelerate equilibration in stochastic sampling tasks.

Abstract

We study the relaxation of the Metropolis Monte Carlo algorithm corresponding to a single particle trapped in a one-dimensional confining potential, with even jump distributions that ensure that the dynamics verifies detailed balance. Previous work suggested that, for smooth jump distributions, the fastest relaxation rate is obtained as a result of the competition between diffusive and rejection-dominated dynamics. In this work, we show that a new regime comes into play for two-peaked jump distributions, where the relaxation dynamics is neither dominated by diffusion nor rejection: the eigenmodes adopt an oscillatory form, reminiscent of charge density waves (CDW) -- thus we term this new regime the CDW regime. Using a combination of numerical and analytical techniques, the parameter regions corresponding to diffusion, rejection, and CDW are characterised, as well as the transition lines between them -- i.e. a phase diagram is built. The optimal relaxation rate is located at the triple point of phase coexistence, where the transition lines (diffusive-rejection, diffusive-CDW, and CDW-rejection) intersect. Our theoretical framework is checked versus the numerical diagonalisation of the master equation. We also briefly discuss more sophisticated attempts at optimising the relaxation rate to equilibrium.

Figures (10)

• Figure 1: Leading eigenvalues $\lambda_{\nu}$, $\nu=1,\ldots,9$, of the master equation \ref{['eq:master-equation']} for the Metropolis algorithm, for a particle confined in a harmonic potential $U(x)=\chi x^2/2$. In both panels, we have employed the jump distribution $\omega^{(a,\alpha)}(\eta)$ from Eq. \ref{['eq:alpha-a-distribution']}, and we have plotted the eigenvalues against $a$ for two fixed values of $\alpha$: (left) $\alpha = 1$ and (right) $\alpha = 3$. Purple curves correspond to the maximal rejection probability, with the light purple regions accounting for the continuum of eigenvalues obtained upon the localisation transition. The dashed, vertical lines denote the position of either the optimal value $a_{\text{opt}}$ and/or the point $a^*$ at which the localisation transition takes place.
• Figure 2: Leading eigenfunction for the Metropolis algorithm. It has been obtained via numerical diagonalisation of the master equation, for: (upper left) $\omega^{(a,\sigma)}(\eta)$, with $(a,\sigma) = (1.4,0.1)$, (upper right) $\omega^{(a,\alpha)}(\eta)$, with $(a,\alpha) = (1.75,6)$, and (lower left) $\omega^{(a.b)}(\eta)$ with $(a,b) = (1.85,0.15)$. The insets present the ratio between the leading eigenfunction and the equilibrium distribution $\phi_0(x)$. The dashed curves correspond to the best fit of such ratios to the function $\cos(k_1^* x)$, with $k_1^*=2\pi/a^*$, with (upper left) $a^* = 1.40$, (upper right), $a^* = 1.56$, and (lower left) $a^* = 1.85$, respectively. The corresponding jump distributions---only for $\eta>0$, recall that they are even functions of $\eta$---are shown in the lower right panel: $\omega^{(a,\sigma)}(\eta)$ in red, $\omega^{(a,\alpha)}(\eta)$ in blue, and $\omega^{(a,b)}(\eta)$ in purple.
• Figure 3: Heat maps of the fidelity $\mathcal{F}$ (left) and the IPR (right). Both are plotted in the $(a,\alpha)$ plane of parameters for the algebraic jump distribution in Eq. \ref{['eq:alpha-a-distribution']}. Both order parameters have been evaluated by numerically obtaining the leading eigenfunction of the master equation. The transition lines between the different phases are plotted with dashed lines and intersect at the tricritical point $(a_c,\alpha_c) = (2.08,1.22)$, which is represented by a black dot. These lines correspond to (i) $a_1(\alpha)$ for $\alpha > \alpha_c$ and $a<a_c$, (ii) $a_2(\alpha)$ for $\alpha > \alpha_c$ and $a>a_c$, and (iii) $a_3(\alpha)$ for $\alpha < \alpha_c$ and $a>a_c$, respectively, as defined in the text.
• Figure 4: Heat map of the next-to-largest eigenvalue $\Lambda$ in the $(a,\alpha)$ plane. As in Fig. \ref{['fig9']}, the dashed curves correspond to the transition lines between phases. The black point indicates the coordinates of the global minimum, $(a_c,\alpha_c) = (2.08,1.22)$, which coincides with the tricritical point at which the transition lines intersect.
• Figure 5: Fidelity $\mathcal{F}$ and IPR as functions of $a$, for a fixed value of $\alpha=3>\alpha_c$---similar to the right panel of Fig. \ref{['fig_eigenvalue-crossing']}. Specifically, we have employed the jump distribution $\omega^{(a,\alpha)}(\eta)$ in Eq. \ref{['eq:alpha-a-distribution']}. The dashed, vertical lines delimit the three different phases explored in this work: for $a<a_1(\alpha = 3)=1.37$, the Schrödinger diffusion-like regime, for $a_1(\alpha = 3)=1.37<a<a_2(\alpha = 3)=2.72$, upon eigenvalue crossing, the CDW regime, and for $a>a_2(\alpha = 3)=2.72$, the rejection regime.