Sticky coupling as a control variate for sensitivity analysis

Abstract

We present and analyze a control variate strategy based on couplings to reduce the variance of finite difference estimators of sensitivity coefficients, called transport coefficients in the physics literature. We study the bias and variance of a sticky-coupling and a synchronous-coupling based estimator as the finite difference parameter $η$ goes to zero. For diffusions with elliptic additive noise, we show that when the drift is contractive outside a compact the bias of a sticky-coupling based estimator is bounded as $η\to 0$ and its variance behaves like $η^{-1}$, compared to the standard estimator whose bias and variance behave like $η^{-1}$ and $η^{-2}$, respectively. Under the stronger assumption that the drift is contractive everywhere, we additionally show that the bias and variance of the synchronous-coupling based estimator are both bounded as $η\to 0$. Our hypotheses include overdamped Langevin dynamics with many physically relevant non-convex potentials. We illustrate our theoretical results with numerical examples, including overdamped Langevin dynamics with a highly non-convex Lennard-Jones potential to demonstrate both failure of synchronous coupling and the effectiveness of sticky coupling in the not globally contractive setting.

Figures (11)

• Figure 1: Sticky coupling of a one-dimensional particle in a double well potential perturbed by a constant force to the right, i.e. $b(x) = -4x\left(x^2-1\right)$ and $\eta F(x) = 1$. Left: histogram of the occupation density of the coupled process; Right: segment of trajectory of the coupled process with the perturbed marginal $X^\eta$ plotted in blue and the reference marginal $Y^0$ plotted in orange.
• Figure 2: The possible behavior of the coupled trajectories in one-step
• Figure 3: Variance of the coupled estimators for the two-dimensional strongly convex potential \ref{['eqn:quadpot']}. We highlight the different scales of the $y$-axis in the two plots.
• Figure 4: Convergence of coupled estimators to true value of $\alpha_{R_{\mathrm{cov}}}$.
• Figure 5: Variance of the coupled estimators in the two-dimensional potential which is only strongly convex outside of a compact set.

Theorems & Definitions (44)

• Remark 1
• Proposition 1
• Lemma 1
• proof
• Proposition 2
• Lemma 2
• proof
• proof : Proof of Proposition \ref{['prop:std_estimator']}
• Proposition 3
• Theorem 3